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*Work done at DeepMind. +Interpretability research aims to build tools for understanding machine learning (ML) models. However, +such tools are inherently hard to evaluate because we do not have ground truth information about +how ML models actually work. In this work, we propose to build transformer models manually as a +testbed for interpretability research. We introduce Tracr, a “compiler” for translating human-readable +programs into weights of a transformer model. Tracr takes code written in RASP, a domain-specific +language (Weiss et al., 2021), and translates it into weights for a standard, decoder-only, GPT-like +transformer architecture. We use Tracr to create a range of ground truth transformers that implement +programs including computing token frequencies, sorting, and Dyck-n parenthesis checking, among +others. We study the resulting models and discuss how this approach can accelerate interpretability +research. To enable the broader research community to explore and use compiled models, we provide +an open-source implementation of Tracr at https://github.com/deepmind/tracr. +Keywords: Interpretability, Transformers, Language Models, RASP, Tracr +1. Introduction +Explanation +Neural +Network +Interpretability +Known +Mechanism +Is the explanation +correct? +Tracr +Figure 1 | Tracr allows us to create models that +implement a known mechanism. We can then +compare this mechanism to explanations an in- +terpretability tool produces. +As deep learning models are becoming more capable and +increasingly deployed in production, improving our ability +to understand how they make decisions is crucial. +Mechanistic interpretability aims to achieve this by +reverse engineering neural networks and producing mech- +anistic explanations of the algorithms a model imple- +ments. This approach has achieved success in convo- +lutional neural networks for image classification. Cam- +marata et al. (2020) explain a range of specific circuits in +InceptionV1 (Szegedy et al., 2015), including curve detec- +tors, high-low frequency detectors, and neurons detecting +more high-level concepts such as dogs or cars. Elhage +et al. (2021) and Wang et al. (2022) achieve early success +in interpreting transformer language models using similar methods. +Despite this success, the toolbox of approaches for generating mechanistic explanations remains +small and poorly understood. Part of the difficulty is that evaluating mechanistic explanations requires +creativity and effort by researchers. It is difficult to evaluate how well an explanation tracks the +actual mechanism used by the model when all our knowledge of the mechanism comes from the +explanation itself. Without access to ground truth about the proposed mechanism, we must verify the +methods used to study it in some other way. +The standard approach for evaluating mechanistic explanations combines evidence from many +ad-hoc experiments (e.g., Olah et al. (2020) and Olsson et al. (2022)). However, since this is expensive +© 2023 DeepMind. All rights reserved +arXiv:2301.05062v1 [cs.LG] 12 Jan 2023 + +DeepMind<>Tracr: Compiled Transformers as a Laboratory for Interpretability +to do, many methods are only evaluated in toy models (e.g., Elhage et al. (2022)) or on a handful +of nontrivial circuits in real models (e.g., Chan et al. (2022)). Systematic evaluation in nontrivial +settings is usually intractable as it requires a lot of researcher time. +The situation is analogous to trying to invent a microscope lens without ever being able to point +it at familiar, well-understood shapes. Through careful reasoning and experimentation, we might +notice regularities in the tiny world seen through the lens, and begin to trust findings made with it; +but if we could look through the lens at something we already understand, we would recognise its +optical properties and correct its flaws. +We propose to directly tackle the absence of ground truth explanations by "compiling" human +readable code to weights of a neural network. In this report, we present Tracr, a proof-of-concept +implementation of such a compiler. Using this approach, we can create models which perform +nontrivial computation with a known implementation. We can then evaluate interpretability tools by +applying them to compiled models and comparing the resulting explanation to the ground truth. +Imagine we want to evaluate a method for locating specific knowledge in transformer models, +such as “causal tracing” (Meng et al., 2022). In real language models, it can be challenging to check +its correctness: the method might point out a location in the model, but we can’t easily independently +verify its claim, since no trusted procedure for establishing such facts about models in the wild exists +yet. With Tracr we can construct models that encode some information in a specific location and +check if our method correctly locates it. We can further explore special cases, such as information +stored redundantly in different places. +In this work, we focus on transformer models (Vaswani et al., 2017) and use RASP, a domain- +specific programming language for describing transformer computations (Weiss et al., 2021). We +develop an approach to compile RASP programs to the weights of a transformer model by combining +hand-coded and fully interpretable model components. We further propose a method that uses +gradient descent to compress the compiled models to make them more efficient and realistic. +More specifically, in this report, we: +• Describe a modified version of the RASP programming language better suited for being compiled +to model weights (Section 3.2) and discuss some limitations of the RASP programming model. +• Introduce Tracr, a “compiler” for translating RASP programs into transformer model weights +(Section 3.4). To describe Tracr, we also introduce craft, its intermediate representation for +expressing linear algebra operations using named basis directions (Section 3.3). +• Showcase several transformer models obtained by using Tracr (Section 4). +• Propose an optimization procedure to “compress” the compiled models and make them more +efficient and realistic (Section 5). We analyse models compressed this way, demonstrating +superposition (Elhage et al., 2022). +• Discuss potential applications and limitations of Tracr and how compiled models can help to +accelerate interpretability research (Section 6). +• Provide an open-source implementation of Tracr (https://github.com/deepmind/tracr). +2. Background +Before describing Tracr, let us recap the transformer architecture and the RASP programming +language. +2 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +is_x = ( tokens +== "x") +prevs = select(indices , indices , <=) +frac_prevs = aggregate (prevs , is_x) +bos x a c x +frac_prevs +indices: 0 +indices: 1 +indices: 2 +indices: 3 +indices: 4 +is_x +one +tokens: a +tokens: b +tokens: bos +tokens: c +tokens: pad +tokens: x +Input +bos x a c x +Attn 1 +bos x a c x +MLP 1 +bos x a c x +Attn 2 +bos x a c x +MLP 2 +Figure 2 | An example RASP program (left) that computes the fraction of previous “x” tokens at each position of the input. +Tracr compiles this program to a transformer model. We show the full residual stream of the compiled model at each layer +for the input sequence “xacx” (right). Attn 1 is a no-op, MLP 1 computes the indicator variable is_x, Attn 2 implements +the select-aggregate operation to compute frac_prevs, and MLP 2 is a no-op again. Section 4 discusses this and other +examples in more detail. +2.1. Transformer Models +A transformer model consists of alternating multi-headed attention (MHA) and multi-layer perceptron +(MLP) layers with residual connections. +Multi-headed attention (Vaswani et al., 2017) computes attention maps on sequences of length 𝑁. +A single attention head 𝑖 first computes an attention pattern +𝐴𝑖 = softmax +� +(𝑥𝑊𝑖 +𝑄)(𝑥𝑊𝑖 +𝐾)𝑇/ +√︁ +𝑑𝑘 +� +∈ ℝ𝑁×𝑁 +for some input 𝑥 ∈ ℝ𝑁×𝑑, where 𝑊𝑖 +𝑄, 𝑊𝑖 +𝐾 ∈ ℝ𝑑×𝑑𝑘 are learnable parameters. Usually, we call the entries +of (𝑥𝑊𝑖 +𝐾) keys, and the entries of (𝑥𝑊𝑖 +𝑄) queries. Multi-headed attention combines 𝐻 attention heads +heads by computing +MHA(𝑥) = Concat +� +𝐴1(𝑥𝑊1 +𝑉 ), . . . , 𝐴𝐻(𝑥𝑊 𝐻 +𝑉 ) +� +𝑊𝑂 +where 𝑊𝑖 +𝑉 ∈ ℝ𝑑×𝑑𝑣 and 𝑊𝑂 ∈ ℝ𝐻𝑑𝑣×𝑑 are another set of learnable parameters. We commonly call the +entries of (𝑥𝑊𝑖 +𝑉) values. +The MLP layers in transformer models compute MLP(𝑥) = 𝜎(𝑥𝑊1)𝑊2 where 𝑊1 ∈ ℝ𝑑×ℎ, 𝑊2 ∈ ℝℎ×𝑑 +are learnable weights, and 𝜎 is a non-linear function, often the Gaussian Error Linear Unit (GeLU; +Hendrycks and Gimpel, 2016). For simplicity we use the Rectified Linear Unit (ReLU; Agarap, 2018). +In this paper, we focus on decoder-only transformers with the popular GPT architecture (Radford +et al., 2018), which consists of alternating blocks of MHA, MLP, and layer normalization (Ba et al., +2016). The input to the model is the sum of a learned embedding of a sequence of input tokens and a +positional embedding. The model is trained to predict the next token using gradient descent. +2.2. Transformer Circuits +We adopt the circuits view of transformers, introduced by Elhage et al. (2021). This view (1) +focuses on the transformer being a residual stream architecture and (2) introduces an alternative +parameterisation for attention operations. Both make it easier to reason about the computation done +by transformers and will help us when assembling transformers manually. +The residual stream view. Transformers have residual connections at each attention and MLP layer. +Elhage et al. (2021) consider the residual connections a core feature of the architecture and describe +3 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +the model in terms of a residual stream that each layer reads from and writes to in sequence. The +residual stream acts as a type of memory that earlier layers can use to pass information to later layers. +Parameterising attention as 𝑊𝑄𝐾 and 𝑊𝑂𝑉. Following Elhage et al. (2021), we parameterise an +attention head by two (low-rank) matrices 𝑊𝑄𝐾𝑖 = 𝑊𝑖 +𝑄(𝑊𝑖 +𝐾)𝑇/√ +𝑑𝑘 ∈ ℝ𝑑×𝑑 and 𝑊𝑂𝑉 𝑖 = 𝑊𝑖 +𝑉𝑊𝑖 +𝑂 ∈ ℝ𝑑×𝑑 +where we split 𝑊𝑂 into different heads, such that 𝑊𝑂 = [𝑊1 +𝑂, . . . 𝑊 𝐻 +𝑂 ], where each 𝑊𝑖 +𝑂 ∈ ℝ𝑑𝑣×𝑑. We can +then write MHA as +𝐴𝑖 = softmax +� +𝑥𝑊𝑄𝐾 +𝑖𝑥𝑇� +MHA(𝑥) = +𝐻 +∑︁ +𝑖=1 +𝐴𝑖𝑥𝑊𝑂𝑉 +𝑖 +Importantly, we can think of MHA as summing over the outputs of 𝐻 independent attention heads, +each parameterised by low-rank matrices 𝑊𝑄𝐾 and 𝑊𝑂𝑉. 𝑊𝑄𝐾 acts as a bilinear operator reading from +the residual stream, and 𝑊𝑂𝑉 is a linear operator both reading from and writing to the residual stream. +The softmax is the only nonlinearity in an attention head. +2.3. The RASP Programming Language +We build on the Restricted Access Sequence Processing Language (RASP), a domain-specific language +for expressing transformer computations. Weiss et al. (2021) propose RASP as a computational model +to describe transformers and provide an interpreter for RASP code. We are primarily interested in +compiling actual transformer models. In this section, we review the main features of RASP; for a +more detailed description, refer to Weiss et al. (2021). +A RASP program can be seen as a computational graph, with each node taking on a particular +value when evaluating the entire graph on a given input token sequence. We usually refer to programs +by the node at the tip of the graph, with the nodes it depends on left implicit. There are two basic node +types, sequence operations and selectors, and two types of RASP operations, elementwise operations +and select-aggregate operations. +Sequence operations. A sequence operation (s-op) represents sequences of values during evaluation. +tokens and indices are built-in primitive s-ops that return a sequence of input tokens or their indices, +respectively. For example: tokens(”hello”) = [h, e, l, l, o], and indices(”hello”) = [0, 1, 2, 3, 4]. S-ops +roughly correspond to the state of the residual stream in transformers. +Elementwise operations. RASP allows arbitrary elementwise operations on s-ops. For example, we +can compute (3*indices)(”hello”) = [0, 3, 6, 9, 12]. Elementwise operations roughly correspond to +MLP layers in transformers. +Select-aggregate operations. To move information between token positions, RASP provides select- +aggregate operations which roughly correspond to attention in transformers. A selector has a graph +dependency on two s-ops and evaluates on inputs of length 𝑁 to a binary matrix of size 𝑁 × 𝑁. To +create a selector, the select operation takes two s-ops and a boolean predicate 𝑝(𝑥, 𝑦). For example: +select(indices, [1, 0, 2], <)(”abc”) = +������ +1 +0 +0 +0 +0 +0 +1 +1 +0 +������ +. +Here, 𝑝(𝑥, 𝑦) = 𝑥 < 𝑦, where 𝑥 comes from indices, and 𝑦 comes from the constant s-op [1, 0, 2]. +The aggregate operation takes as input a selector and an s-op, and produces an s-op that averages +4 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +the value of the s-op weighted by the selection matrix. For example: +aggregate �� +� +������ +1 +0 +0 +0 +0 +0 +1 +1 +0 +������ +, [10, 20, 30]�� +� += [10, 0, 15]. +A selector roughly corresponds to an attention pattern in a transformer. Together a select-aggregate +operation roughly corresponds to an attention head in transformers. +3. Tracr: A Transformer Compiler for RASP +To introduce Tracr, we first describe how RASP maps to the transformer architecture (Section 3.1) +and propose a few modifications to RASP that make this mapping more straightforward (Section 3.2). +Next, we introduce craft, our “assembly language” for transformer models (Section 3.3). Finally, +we describe how Tracr translates RASP programs to transformer weights (Section 3.4). +Appendix A contains some more technical details, and we provide a full open-source implementa- +tion of Tracr at https://github.com/deepmind/tracr. +3.1. Mapping RASP to Tranformers +RASP povides a computational model of transformers. For the most part, we can map RASP operations +directly to the components of a transformer model. +Embeddings. The built-in s-ops tokens and indices correspond to a transformer’s token and +position embeddings. For example, we can embed the tokens and positions as categorical variables in +orthogonal subspaces of the embedding space. +MLP layers. Any elementwise operation in RASP can be approximately computed by an MLP layer +simply because MLPs can approximate any function with accuracy depending on the width and depth +of the MLP (Hornik et al., 1989). +Attention layers. RASP’s select-aggregate operations map to the attention layers in transformer +models. The post-softmax attention pattern needs to match the selection matrix for all inputs to +implement a given selector. So, given a large enough key/query-dimension, an attention head can +implement an arbitrary binary attention pattern using its 𝑊𝑄𝐾 matrix. The 𝑊𝑂𝑉 matrix of the attention +head can then implement the aggregate operation. +3.2. Modifications to RASP +While we can map RASP operations to transformers, we need to make a few modifications to the +RASP language to allow translating it to model weights. +Disallow arbitrary selector combinations. RASP allows to combine selectors using boolean opera- +tions; however, there is no natural analogue for this in real transformers. Combining selectors with +different input variables is particularly problematic. For example, in RASP we can define a selector +select(a, b, ==) and select(c, d, ==) +using four s-ops a,b,c, and d. However, a real attention head only has two inputs. If the model stores +the s-ops in separate subspaces of the residual stream, a single attention head cannot implement this +operation.1 Because of this, we restrict RASP to selectors with only two input variables. In practice, +1We formalise this observation in Appendix C. +5 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +this limitation turns out not to be severe. In particular, we were able to implement programs to solve +all tasks described by Weiss et al. (2021). +Encoding annotations. A compiled model needs to pass information between layers. In a transformer, +it is natural to do this via the residual stream. However, we have to decide how to represent information +in the residual stream. For simplicity, we only use two encodings: categorical and numerical. We +encode categorical variables as one-hot vectors in a dedicated subspace of the residual stream. We +encode numerical variables as the magnitude of a dedicated one-dimensional subspace of the residual +stream. Categorical encoding is generally less efficient when numerical encoding is possible, but some +aggregate operations only work with one type of encoding. For instance, aggregate can compute +a mean across token positions, which is not natural with attention on a one-hot encoded subspace +but straightforward with a numerical one. However, numerically-encoded data is generally harder to +work with, requiring a decoding step. +We require each s-op to be either categorical or numerical and augment RASP with the ability to +annotate s-ops with the desired encoding. By default, we assume s-ops are categorical. +Beginning of sequence token. Transformers often assume any input sequence to start with a +dedicated “beginning of sequence” token (BOS). We make the BOS token mandatory in RASP because +it is crucial when implementing arbitrary attention patterns. In particular, RASP allows selectors that +can produce all-zero rows; this is convenient when programming in RASP, but the softmax makes this +behaviour impossible in a real attention head. In these situations, we use the BOS token as a "default" +position to attend to: it is attended to iff no other token is. This allows the non-BOS part of the sequence +to emulate the intended RASP behaviour. In our case, this choice comes from practical considerations; +but, interestingly, real models sometimes show similar behaviour (e.g., see Elhage et al., 2021). +3.3. craft: An Assembly Language for Transformers +Machine +code +Programming +language +Assembly +RASP +craft +Figure 3 | Tracr translates RASP to craft +and then to model weights, analogous to +how programming languages are first trans- +lated to assembly then to machine code. +If RASP is the high-level language we compile, craft is our +"assembly language", offering slightly more abstraction than +operating on pure weight matrices. +craft represents vector spaces with labelled basis dimen- +sions and operations on them. This allows us to define pro- +jections or other linear operations in terms of basis direction +labels. Importantly, craft abstracts away the need to keep +track of padding in weight matrices. +We implement a transformer in craft that sticks closely to +the transformer circuits view provided by Elhage et al. (2021). +In particular, the residual stream is a vector space 𝑅 with a basis. +An attention head can be defined using a bilinear operator +𝑊𝑄𝐾 : 𝑄 × 𝐾 → ℝ and a linear operator 𝑊𝑂𝑉 : 𝑉 → 𝑂, where +𝑄, 𝐾, 𝑉, 𝑂 ⊂ 𝑅 are the vector spaces that reuse the same basis. +craft then handles the projection of these operators up to +𝑅 × 𝑅 → ℝ and 𝑅 → 𝑅, which corresponds to adding the +requisite padding. +In practice, we first independently translate each RASP computation into a craft component, +then assign components to layers, and finally construct the residual stream space 𝑅, ensuring that all +information needed at a given layer in the model is embedded by previous layers. +Moreover, craft models are independent of concrete transformer implementations. A craft +6 + +JAXTracr: Compiled Transformers as a Laboratory for Interpretability +(a) Steps 1 & 2: Computational graph +with inferred s-op value sets. +(b) Step 3: Nodes translated to MLPs +and attention heads. +(c) Steps 4 & 5: Nodes allocated to +locations in a model. +Figure 4 | Schematic overview of how Tracr compiles the frac_prevs program from Figure 2 with a input vocabulary +{”x”, ”y”} and context size 3. (a) shows the computational graph with value annotations after step 2 of the compilation. (b) +shows how is_x and frac_prevs are translated to model components independently in step 3. (c) shows the assembled +model which has two no-op components because models blocks always need to have one attention and one MLP layer. +model can be translated into weights of any standard GPT-like transformer implementation. +3.4. Compiler Overview +We are now ready to describe Tracr in detail. Tracr comes with an implementation of RASP +embedded in Python. This allows us to write RASP programs in Python and makes it easier to +provide annotations, such as variable encodings. In Tracr, a RASP program is a data structure that +is incrementally constructed by passing in dependencies to each operation. We also do a few basic +simplifications of RASP programs at this stage. For example, we combine consecutive elementwise +operations into a single s-op. +Tracr translates RASP programs to transformer weights in six steps: +1. Construct a computational graph. +2. Infer s-op input and output values. +3. Independently translate s-ops to craft components. +4. Assign components to layers. +5. Construct craft model. +6. Assemble transformer weights. +Let us go through these step by step. Figure 4 gives a schematic overview using an example program. +1. Construct a computational graph. First, we trace the whole program to create a directed graph +representing the computation. The graph has source nodes representing tokens and indices and a +sink node for the output s-op. +2. Infer s-op values. For each s-op, we need to decide how to embed it in the residual stream. To +use categorical encodings, we need to know which values an s-op can take. All nodes have a finite set +of output values because computations are deterministic, and we have a finite input vocabulary and +context size. Therefore, in the second step, we traverse the graph and annotate each node with its +possible outputs. This annotation uses simple heuristics that ensure we find a superset of the values an +s-op will take, though, sometimes, an output set can contain values that the s-op never takes in practice. +3. Independently translate s-ops. Next, we consider each node in the computational graph inde- +pendently and translate it into a craft component. Elementwise operations become MLP blocks, +and select-aggregate operations become attention blocks. We use a library of manually engineered +MLP and attention blocks to approximate arbitrary functions for numerical and categorical inputs +7 + +"x""y"} +[0, 1, 2] +tokens +indices +[0, 1] +is_x +prevs +frac-prevs +[O, 1/3, /2, 1]"x"""y"} +[0, 1, 2] +tokens +indices +[0, 1] +MLP: is_X +prevs +Attn: prevs +[O, 13, /2, 1]Attn: prevs +MLP: is_X +djw do-ou +no-op attr +Input +OutputTracr: Compiled Transformers as a Laboratory for Interpretability +and outputs. MLPs with categorical inputs and outputs function as lookup tables. MLPs with numeri- +cal inputs and outputs use an explicit construction based on the universal function approximation +theorem. For attention layers, we translate a selector into the 𝑊𝑄𝐾 operator and the corresponding +aggregate operation into the 𝑊𝑂𝑉 operator. We only support attention with categorical inputs. For +more details on the MLP and attention blocks, see Appendix A. +4. Assign components to layers. To construct a transformer model, we need to allocate all craft +components in the computational graph to layers. Ideally, we want to find the smallest model to +perform the desired computation. We can generally formulate this as a combinatorial optimization +problem with several constraints: the transformer architecture has alternating attention and MLP +layers, and all computations that depend on each other need to be in the correct order. For scope +reasons, we solve this with a heuristic. First, we compute the longest path from the input to a given +node. This path length is an upper bound for the layer number to which we can allocate the node. +Then we apply additional heuristics to combine layers with blocks that we can compute in parallel. +This approach returns a correct but sometimes suboptimal layer allocation. +5. Construct a craft model. We construct the residual stream space as the direct sum of all model +components’ input and output spaces. In other words, we embed each s-op in its own orthogonal +subspace, which is reserved for its sole use throughout the entire network. Now, we can traverse the +computational graph in the order determined by the layer allocation and stack the components to +obtain a full transformer represented in craft. +6. Assemble transformer weights. Finally, we translate the craft representation of the model +into concrete model weights. First, we combine parallel MLP layers into a single layer and parallel +attention heads into a single layer. In attention layers, we then split up the 𝑊𝑄𝐾 and 𝑊𝑂𝑉 matrices +into 𝑊𝑞, 𝑊𝑘, 𝑊𝑜, 𝑊𝑣 weight matrices. Finally, we adjust the shapes of all weights and connect them to +our transformer architecture. We can then infer the model configuration (depth, layer width, residual +stream size, etc.) to fit the elements we have created. +We base our transformer implementation on the example decoder-only transformer from Haiku +(Hennigan et al., 2020), notably removing the layer norms. Extending Tracr to support any other +transformer implementation is straightforward by reimplementing only step 6. +4. Exploring Compiled Transformers +Having described Tracr, we are now ready to start compiling models. In this section, we walk +through two example programs to illustrate how the compiled models work. Appendix D contains +more examples. Overall, we were able to compile RASP programs for all the tasks described in Weiss +et al. (2021), though we had to modify a few of the programs to only use features supported by +Tracr. +4.1. Example 1: Counting tokens +Figure 2 shows our primary running example, the frac_prevs program, that computes the fraction +of previous "x" tokens. It uses one MLP layer and one attention head. However, because our model +architecture always starts with an attention layer, the compiled model has four layers, with the first +and last layers being no-ops. +The frac_prevs model has a 14 dimensional residual stream, but it uses 12 out of these for the +input embeddings. The computation uses two numerical variables which correspond to the remaining +two dimensions. The input embeddings have a few special dimensions. tokens:bos is the beginning +8 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +smaller = select(tokens , tokens , <=) +target_pos = selector_width (smaller) +sel_sort = select(target_pos , +indices , ==) +sort = aggregate (sel_sort , tokens) +Figure 5 | RASP program that sorts a sequence +of numbers without duplicates. +Attn 1 and MLP +1 +implement +the +selector_width +primitive +(cf. Appendix A) which the program uses to compute +the target position for each token. Attn 2 moves the +tokens to the desired position, and MLP 2 is a no-op. +bos 3 5 4 2 +indices: 0 +indices: 1 +indices: 2 +indices: 3 +indices: 4 +one +sort: 1 +sort: 2 +sort: 3 +sort: 4 +sort: 5 +target_pos: 0 +target_pos: 1 +target_pos: 2 +target_pos: 3 +target_pos: 4 +target_pos: 5 +target_pos_80_selector_width_attn_output +tokens: 1 +tokens: 2 +tokens: 3 +tokens: 4 +tokens: 5 +tokens: bos +tokens: pad +Input +bos 3 5 4 2 +Attn 1 +bos 3 5 4 2 +MLP 1 +bos 3 5 4 2 +Attn 2 +bos 3 5 4 2 +MLP 2 +of sequence token which we need to implement arbitrary attention patterns (cf. Section 3.2), and +one is an input dimension that is fixed to 1. The model uses this dimension as a constant, e.g., to add +a bias in MLP layers. +4.2. Example 2: Sorting +As a second example, let us consider sorting a sequence of numbers. Figure 5 shows a sort_unique +program that sorts a sequence of unique tokens. +The program computes the target position of each token by using the selector_width primitive +in RASP, which computes the number of elements in each row of a selector that with the value 1. +selector_width can be implemented in terms of other RASP operations (Weiss et al., 2021), but +not using our variant of RASP, so we treat it as a primitive that compiles directly to an attention and +MLP layer (here Attn 1 and MLP 1). See Appendix A for more details. +Weiss et al. (2021) propose a sort program that can handle duplicates (cf. their Figure 13). +However, that implementation uses a selector +smaller = select(tokens , tokens , <) +or (select(key , key , ==) and select(indices , indices , <)) +to treat duplicates, which is not supported by Tracr (see Section 3.2). In Appendix D, we provide an +alternative implementation of sort that handles duplicates by adding a small multiple of indices to +the keys and then applying sort_unique. +4.3. More examples +Tracr can compile a wide range of RASP programs. In Appendix D, we discuss a few more examples, +leading up to checking balanced parentheses (Dyck-n). Our open-source Tracr implementation +contains a library of even more example programs to compile. +9 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +Figure 6 | Training setup for compressing a compiled transformer model. At each layer, we use the same matrix 𝑊 ∈ ℝ𝐷×𝑑 +to embed the disentangled 𝐷-dimensional residual stream to 𝑑 ≤ 𝐷 dimensions. We freeze the layer weights and only train +𝑊 to compress the model. +5. Compressing Compiled Transformers +Tracr models can be sparse and inefficient because they reserve an orthogonal subspace of the +residual stream for each s-op. In this section, we propose an experimental approach for “compressing” +the resulting models and making them more efficient. This feature is presented as preliminary work +and is not yet provided in the Tracr library. Here, we present two case studies of compressing +compiled models. +In addition to making Tracr models more efficient, the compressed models allow us to study +how real neural networks might compress 𝐷 features into a representation space with fewer than 𝐷 +dimensions. This phenomenon is called superposition (Elhage et al., 2022); however, to our knowledge, +it has not been studied in models deeper than two layers. +5.1. Gradient Descent Based Compression +We use a single linear projection 𝑊 ∈ ℝ𝐷×𝑑 to compress the disentangled residual stream with size 𝐷 +to a smaller space with dimension 𝑑 < 𝐷. We modify the model to apply 𝑊𝑇 whenever it reads from +and 𝑊 whenever it writes to the residual stream (see Figure 6). We freeze the weights of all layers +and train only 𝑊 using stochastic gradient descent (SGD). +Since vanilla Tracr models are sparse and have orthogonal features, this process can be viewed +as learning the projection from a "hypothetical disentangled model" to the "observed model" described +by Elhage et al. (2022). +We want the compressed model to minimise loss under the constraint that it implements the same +computation as the original model. To achieve this, we train 𝑊 to minimise 𝔼𝑥[L(𝑊, 𝑥)], where +L(𝑊, 𝑥) = Lout(𝑊, 𝑥) + Llayer(𝑊, 𝑥) +Lout = loss( 𝑓 (𝑥), ˆ𝑓𝑊(𝑥)) +Llayer = +∑︁ +layer 𝑖 +(ℎ𝑖(𝑥) − ˆℎ𝑊,𝑖(𝑥))2 +where 𝑓 (𝑥) is the output of the compiled model for input 𝑥, ˆ𝑓𝑊(𝑥) is the output of the compressed +model, and ℎ𝑖(𝑥) and ˆℎ𝑊,𝑖(𝑥) are the output vectors at layer 𝑖 of the respective models. +For categorical outputs, Lout is the softmax cross-entropy loss, whereas, for numerical outputs, it +is the mean-squared error. Llayer is a regularization term that incentives the compressed model to +match the per-layer outputs of the original model. To minimise this loss, the compressed model will +10 + +Attn +MLP +Attn +MLP +h2 +h3 +h1 +M +WT +M +WT +M +WT +W +WT +Input +M +OutputTracr: Compiled Transformers as a Laboratory for Interpretability +0 +1 +2 +3 +training steps +×105 +10−2 +100 +output loss +d = 4 +d = 8 +d = 12 +5 +10 +embedding size d +0.00 +0.02 +0.04 +0.06 +final output loss +Figure 7 | Loss of compressed Tracr models for the frac_prevs program from Figure 2. The left plot shows the loss +during training for different embedding sizes 𝑑; the right plot shows the final loss for different embedding sizes 𝑑. After +about 𝑑 = 6 the compressed model solves the task essentially as well as the original compiled model which uses 𝐷 = 14 +dimensions. Both plots are averaged over 10 random seeds. +have to approximate the computation of the original model but with a smaller residual stream. +We could set up this compression in other ways. For example, we could use a different projection +at each layer, use different matrices for embedding and unembedding, or modify weights other than +𝑊 when compressing the model. These design choices come with a tradeoff between making the +model more expressible and potentially more realistic and enforcing the ground truth computation. +For simplicity, we use a shared 𝑊 for embedding/unembedding at every layer, and we already observe +a rich structure in models compressed with this procedure. +Appendix B contains more details on the training setup, hyperparameters, and resources used. +5.2. What does the compression learn? +As our first case study, Figure 7 shows the example model from Figure 2, that computes the fraction of +token “x”. By learning an embedding matrix 𝑊, we can reduce the residual dimension from 𝐷 = 14 to +𝑑 = 6 without hurting performance. Once we reduce 𝑑 further, the model’s performance starts to suffer. +To understand the compression better, we can study how 𝑊 embeds the original 𝐷 features in +𝑑 < 𝐷 dimensions. We can only do this because we started with a compiled model with known +features. Figure 8 shows 𝑊𝑇𝑊 for compressing the model to 𝑑 = 8. We can compare this to using +principle component analysis (PCA) to compress the model. To interpret the results, we need to use +our knowledge of the algorithm the model implements. The input tokens:x and the variables is_x +and frac_prevs are crucial for computing the fraction of tokens that is “x”, and we find that these +variables mostly get separate dimensions in the compressed residual stream. The other input tokens +stored in tokens:a, tokens:b, tokens:c are not necessary for solving the task, and so they are +discarded in the compressed model. Other variables, such as the indices embeddings, are stored +in non-orthogonal dimensions in the compressed space. This is consistent with existing findings on +superposition as the indices embeddings are sparse and do not occur together (Elhage et al., 2022). +However, some of our results go beyond previous work on superposition. For example, Tracr +models often have multiple variables that depend on each other and encode shared information. In our +running example is_x is an indicator variable that essentially contains the same information as the +input dimension tokens:x.2 In Figure 8, we see that the embeddings of is_x and tokens:x share +part of the embedding space. Intuitively, this occurs because the variables encode similar information. +2They are not exactly the same because is_x is only populated in a later layer. But, if is_x = 1, then tokens:x = 1. +11 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +(a) SGD Compression +(b) PCA +Figure 8 | 𝑊𝑇𝑊 for the compression procedure +described in Section 5 with 𝑑 = 8 (a), compared +to applying PCA and retaining only the first 8 +components (b). In contrast to PCA, our com- +pression procedure produces a compression ma- +trix 𝑊 that retains features necessary for the +task (e.g., is_x and frac_prevs) and discards +features that are unimportant (e.g., tokens:a). +Compiled Compressed +Error +0 +10 +20 +embedding size d +0.0 +0.5 +1.0 +accuracy +0 +10 +20 +embedding size d +0.0 +0.5 +1.0 +cosine similarity +Figure 9 | We compress the sort_unique program (Figure 5). The two plots on the right show that the compressed model +achieves nearly perfect accuracy, but the layer outputs of the compressed model are different from the original compiled +model. The left plot shows the average layer outputs of the compiled model, the compressed model, and the squared error +between both. The source of the error is that the compressed model seems to learn to use a different (numerical) encoding +for the target_pos variable. +In preliminary experiments, we found that shared information between variables seems to influence +how superposition occurs. For example, varying the data distribution to have two variables share +more or less information changes the correlation patterns between embedded features. Prior models +of superposition do not explain this effect, and we leave fully understanding it for future work. +5.3. Do the compressed models still implement the same computation? +Even if the compressed models successfully achieve a low loss, we need to check if they implement +the same computation as the compiled models, or else we would no longer know the ground truth +mechanisms the models implement. To this end, we evaluate the average cosine similarity between +the output at each layer of the two models. +For the compressed frac_prevs model, the cosine similarity is close to 1, which implies that the +compressed model is consistent with the compiled model (up to differences in norm).3 +3In categorical tasks the compressed model is encouraged to output vectors with a large norm due to the output softmax. +We found that this can sometimes lead to the norm of the outputs at intermediate layers also changing even though the +12 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +However, in other cases, the cosine similarity stays below 1 even as the compressed model gets +close to 100% in accuracy. As an example, Figure 9 shows results from compressing the sort_unique +model. Here, the compressed model achieves almost perfect accuracy on the task, but the average +cosine similarity of the outputs at individual layers stays around 0.8. This suggests that the compressed +model solves the tasks differently from the original compiled model. +By inspecting the models’ outputs at each layer, we can attribute the error to the target_pos +variable. In the Tracr model, target_pos is encoded categorically, with a dimension allocated per +position. However, the compiled model only uses one of these dimensions. This suggests that the +compressed model moves the tokens to the target position with a numerical encoding of the target +position rather than a categorical encoding. During training, this reduces the output loss at the cost +of increasing the layer output regulariser. +This case shows that even in this fairly restrictive compression setup, the compressed model can +learn a different computation to be more efficient. This is both encouraging and problematic: it is +evidence that we can achieve meaningful compression with a simple approach; however, even in +this restrictive setting, the compressed model is not guaranteed to be faithful to the original RASP +program, undermining the value provided by the compiler as a source of ground truth. +Overall, using SGD on top of compiled models seems promising to make them more efficient and +naturalistic. We hope that future work can make this training setup more robust and that we can +ultimately fully integrate it in a future version of Tracr. +6. Discussion +We provide an open-source implementation of Tracr because we think it has many potential appli- +cations in interpretability research. In this section, we discuss applications we see for Tracr and +compiled transformers more generally and reflect on the current limitations of Tracr and how they +can be addressed. +6.1. Applications of compiled models in interpretability research +Compilers like Tracr allow researchers to set up controlled experiments that test specific hypotheses +about the computational structure of transformers. In this way, it acts as a laboratory for research in +interpretability, enabling research that might otherwise be intractable. +Test cases for interpretability tools. Compiled models serve as a natural foundation for testing the +faithfulness (Jacovi and Goldberg, 2020) of an explanation, and provide a way to falsify (Leavitt +and Morcos, 2020) the explanations given by interpretability techniques. Ultimately, they could be +used to build libraries of test cases for interpretability tools, which could in turn enable quantitative +evaluation metrics. For example, Meng et al. (2022) propose a method to locate factual knowledge +in transformers. Tracr could allow us to test what this or similar methods can locate in a range of +models implementing different algorithms, contextualising its result in real models. +Replacing model components. Another way to evaluate our understanding of how a model works +is to replace parts of the model with hand-coded components. For example, Nanda and Lieberum +(2022) test their understanding of how a transformer implements modular addition by replacing +components of the model with their own idealised implementation and find that this can increase +downstream performance, which is strong evidence that the proposed explanation is correct. While +Tracr compiles an algorithm into a full transformer model, it could be adapted to only compile part +cosine similarity is 1. +13 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +of a model to replace part of a trained model. This could make it easier to evaluate our understanding +of a large model. +Understanding model phenomena and developing new techniques. Beyond evaluation, compiled +models can be used as a testbed for studying circuits-level phenomena and developing new approaches +for interpreting transformer models. For example, in Section 5 we successfully induced superposition +in compressed Tracr models. Future work could analyse superposition in Tracr models, extending +previous work in toy models (Elhage et al., 2022; Scherlis et al., 2022). In particular, Tracr allows +studying how the structure of computation implemented by a model affects which features will be +stored in superposition. One goal for this line of research could be to predict how a specific Tracr +model will be compressed, which features will be stored in superposition and how. A complementary +approach is to try reversing the superposition induced by a compression procedure, e.g., using ideas +from compressed sensing and dictionary learning (Aharon et al., 2006; Donoho, 2006). +6.2. Limitations of RASP and Tracr +RASP and Tracr are limited in terms of expressivity, efficiency and realism compared to real trans- +former models. Many of these limitations could be overcome in future versions of Tracr. +Expressivity. RASP is designed for algorithmic tasks that map an input sequence to a discrete output +sequence. However, current language models usually map a sequence of input tokens to a probability +distribution over the next token. Circuits in real models often consist of components that increase or +decrease the probability of some tokens based on previous tokens (Wang et al., 2022). RASP, and +hence Tracr, cannot model such "probabilistic" computation, but could potentially be extended to +support it. RASP only uses binary attention patterns, which inherently limits the range of algorithms +it can implement (Merrill et al., 2022). A way to extend RASP to support numeric attention patterns +is discussed in Weiss et al. (2021). +Efficiency. Tracr models store all variables in orthogonal subspaces of the residual stream. Even +if a variable is only used in part of the computation, Tracr reserves a subspace of the residual +stream for it in all layers of the model. Real models use a more compressed representation and likely +reuse dimensions for multiple features. Improved versions of the compression procedure discussed in +Section 5 could address this limitation, as would using a constraint optimisation solver instead of a +heuristic for layer allocation. +Realism. Tracr constructs layers from hand-coded parameter matrices. This is both unrealistic and +inefficient, but could be addressed by learning the layers in isolation, then assembling them into +a full model manually. Similarly, instead of manually splitting the 𝑊𝑄𝐾 and 𝑊𝑂𝑉 matrices, matrix +factorisation could be used to get more efficient solutions. Also, Tracr models align their features +with the computational basis. This is unrealistic, and makes the resulting models easy to interpret +just by inspecting the residual stream activations. Rotating the basis of the compiled model is a +straightforward way to address this if obfuscation is needed; compression would be an even more +comprehensive approach. +While all of these issues could be overcome in a more sophisticated compiler, there are fundamental +limitations on the role compiled models can play. Compiled models are an intermediate step between +very simple toy models and real learned models. They help us understand ideas and methods, but +results in compiled models do not necessarily generalise to real models. Compared with real models, +compiled models will always be simpler. For example, we will likely never compile full-fledged +language models. Compiled models will be more likely to be intepretable (e.g., the axis-aligned +orthogonal residual stream bases in Tracr), and more likely to fit into existing paradigms for thinking +about transformers. When using them to evaluate interpretability tools, we should be careful to make +14 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +sure that the tools do not exploit this, treating such evaluations as a minimum bar rather than a full +validation of a technique. Conversely, some methods might conceivably rely on features present in +real models but not in compiled models. +7. Conclusion +In this work, we proposed manually constructing neural network weights and using them to develop +and evaluate new interpretability tools. To this end, we developed Tracr, a tool for compiling +human-readable code to the weights of a transformer model. +We outlined our vision for the use of compiled models in interpretability, and there may other +potential applications of Tracr within and beyond interpretability research. We are looking forward +to seeing other researchers use it, and we hope studying compiled models will help to increase our +understanding of neural networks. +Acknowledgements +We thank Avraham Ruderman, Jackie Kay, Michela Paganini, Tom Lieberum, and Geoffrey Irving for +valuable discussions, Victoria Krakovna and Marlene Staib for collaborating on early experiments +with compiling RASP, and Chris Olah and Tristan Hume for feedback on an early draft of this paper. +Author Contributions +VM proposed the initial idea for Tracr and wrote our RASP implementation. DL, VM, JK and MR +designed and developed Tracr. DL designed, implemented, and ran the compression experiments in +Section 5. MR wrote documentation and led the open-sourcing process. JK derived the theoretical +results in Appendix C. TM and VM advised on research direction. DL and VM wrote the manuscript. +DL led the project. +References +A. F. Agarap. Deep learning using rectified linear units (RELU). arXiv preprint arXiv:1803.08375, +2018. +M. Aharon, M. Elad, and A. Bruckstein. K-SVD: An algorithm for designing overcomplete dictionaries +for sparse representation. IEEE Transactions on signal processing, 54(11):4311–4322, 2006. +J. L. Ba, J. R. Kiros, and G. E. Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016. +N. Cammarata, S. Carter, G. Goh, C. Olah, M. Petrov, L. Schubert, C. Voss, B. Egan, and S. K. Lim. +Thread: Circuits. Distill, 2020. doi: 10.23915/distill.00024. https://distill.pub/2020/ +circuits. +L. Chan, A. Garriga-Alonso, N. Goldowsky-Dill, R. Greenblatt, J. Nitishinskaya, A. Radhakrishnan, +B. Shlegeris, and N. Thomas. 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Shlegeris, and J. Steinhardt. Interpretability in the wild: a +circuit for indirect object identification in GPT-2 small. arXiv preprint arXiv:2211.00593, 2022. +G. Weiss, Y. Goldberg, and E. Yahav. Thinking like transformers. In International Conference on +Machine Learning (ICML), 2021. +17 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +A. Tracr Implementation Details +This section highlights a few more implementation details of Tracr. We describe how we construct +MLP and attention blocks, how we implement the selector width primitive, and how we extend RASP +and Tracr to use causal attention. For the full implementation and documentation, refer to the code +repository at https://github.com/deepmind/tracr. +A.1. MLP and Attention Blocks +For MLP layers, we distinguish between Map operations with a single input and output and SequenceMap +operations with two inputs and one output. We can recursively represent functions with more than +two inputs using SequenceMaps. +We translate Maps with categorical inputs and outputs to MLPs that act as a lookup table. +SequenceMaps with categorical inputs and outputs become MLPs where the first layer maps to +an encoding of all pairs of inputs and the second layer acts as a lookup table. +For numerical inputs and outputs, we explicitly construct MLP layers as universal function approx- +imators. In these MLPs, the first layer discretises the input, and the second layer maps each discrete +bucket to a corresponding output value. We know which input/output values can occur, so we can +choose the discretisation around these known input values to minimise the approximation error. +We construct the 𝑊𝑄𝐾 matrix to implement the desired attention pattern. Here we ensure that if a +token does not attend to any other token in RASP, it will attend to the BOS token in the Tracr model. +The 𝑊𝑂𝑉 matrix maps the value input to the corresponding output dimensions. Attention layers only +support categorical key and query inputs. The value inputs can be numerical or categorical. We can +only use categorical values if the head never attends to more than one token. +A.2. Selector Width Primitive +RASP provides the selector width primitive, which counts the number of 1s in each row of a selector. +It provides an alternative to aggregate for processing selectors. +Weiss et al. (2021) provide a selector width implementation in pure RASP, making it not necessarily +a language primitive. However, the most efficient implementation uses the BOS token, which exists +Tracr but is not exposed to the RASP program. +Therefore, Tracr translates selector width directly into an efficient implementation in craft +consisting of an attention layer and an MLP layer. The attention layer implements an attention pattern +that matches the selector to compute the width of. It uses the BOS token as value input, resulting in +the attention head computing 𝑥 = 1/(1 + 𝑤) where 𝑤 is the desired selector width output. The next +MLP layer then computes 𝑤 = 1/𝑥 − 1 and cleans the BOS token position. +A.3. Casual Attention +Most transformer models used in practice use causal attention, i.e., they apply a mask to the attention +patterns that allows the model to attend only to previous tokens. This allows training the models +autoregressively. However, RASP assumes non-causal (i.e. bidirectional) attention by default. While +all models discussed in the main paper use non-causal attention, Tracr also supports causal attention. +To enable this, we extend RASP to support causal attention via a flag set during evaluation. To +evaluate a RASP program in the causal evaluation mode, we apply a causal mask to the output of +18 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +each selector. Causal evaluation changes the semantics of some RASP operations, and, in general, it is +necessary to adapt RASP programs to function with causal attention. For example, the frac_prevs +program no longer needs to compute a causal mask manually. However, for example, the length +implementation by Weiss et al. (2021) no longer correctly computes the length of a sequence because +it requires attending to future tokens. +Similarly, Tracr has a flag to enable causal compilation. Most of the compilation process does +not change, and we only need to ensure to compile selectors to causal attention heads. +B. Compression Training Details +We implemented the compression described in Section 5 in Jax on top of the Haiku transformer +implementation that comes with Tracr. We train 𝑊 using the AdamW optimizer (implemented in +Optax) with a weight decay factor of 0.1, and parameters 𝛽1 = 0.9, 𝛽2 = 0.99. We train for 3 × 105 +steps with a batch size of 256. We decay the learning rate linearly from 10−3 to 10−6 over the first +half of training. Each compression run requires between 1 and 4 hours of run time on two CPU cores +(depending on the size of the model to compress). +C. Theoretical Results on Combining Attention Heads +In this section, we study how we could implement combinations of selectors with more than two +inputs, which are allowed in RASP. We focus on combining selectors with an and operation, but the +results generalize to other boolean operations. +Consider the following selectors: +simple_selector = select(tokens , indices , <=) +simplifiable_selector = select(tokens , indices , <=) and +select(tokens , "a", ==) +simplified_selector = select(tokens , indices , q <= k and q == "a") +compound_selector = select(a, b, <=) and +select(c, d, <=) +where a, b, c and d are different s-ops. The simple selector depends on only two s-ops and is +straightforward to implement. The simplifiable selector is syntactically defined using the and operator +but can be converted into the simplified selector, which still only depends on two s-ops. This section +concerns selectors like the compound_selector above, which irreducibly depend on more than two +different s-ops. +An attention head can be parameterized by a 𝑊𝑄𝐾 matrix and an 𝑊𝑂𝑉 matrix. In this section, we +focus on 𝑊𝑄𝐾 only, i.e., on the circuit responsible for the attention patterns. The standard view of +𝑊𝑄𝐾 is as a matrix that computes the keys and queries from the residual stream space 𝑅 ⊆ ℝ𝑑 and +computes their dot product. We instead interpret it as a bilinear operator 𝑊𝑄𝐾 : 𝑄 × 𝐾− > ℝ acting +on two subspaces of the residual stream, 𝑄, 𝐾 ⊂ 𝑅, which are spanned by orthogonal bases {𝑞𝑗}, {𝑘𝑖}. +The elements of these bases correspond to elements of the value sets of s-ops in a one-hot encoding. +We call those value sets 𝑄, 𝐾 as well to ease notation. +A selector is a function 𝑆 : 𝑄 × 𝐾 → {0, 1}. In Tracr, an attention head implements a selector +if 𝑆(𝑞, 𝑘) = 𝑊𝑄𝐾(𝑞, 𝑘) := 𝑞𝑇𝑊𝑄𝐾𝑘 for all 𝑞 ∈ 𝑄, 𝑘 ∈ 𝐾. (We can ignore the softmax without loss of +generality as we can rescale the norm of 𝑊𝑄𝐾 to recover boolean outputs.) +Suppose we have two selectors 𝐴 and 𝐵 implemented by attention heads with query-key matrices +𝑊 𝐴 +𝑄𝐾 and 𝑊 𝐵 +𝑄𝐾. They each read from residual subspaces 𝑄𝐴 × 𝐾𝐴 and 𝑄𝐵 × 𝐾𝐵. The straightforward +way to implement a combined selector 𝐴 ∧ 𝐵 would be to define an attention head with a 𝑊𝑄𝐾 acting +19 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +on (𝑄𝐴 ⊕ 𝑄𝐵) × (𝐾𝐴 ⊕ 𝐾𝐵) with attention logits that are the boolean and of the ones from 𝐴 and 𝐵. +Unfortunately, this is only possible in trivial cases because the operator needs to be bilinear. +Lemma 1. There is no 𝑊 𝐴∧𝐵 +𝑄𝐾 +operating over (𝑄𝐴 ⊕ 𝑄𝐵) × (𝐾𝐴 ⊕ 𝐾𝐵) such that +(q𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) = (q⊺ +𝑎 𝑊 𝐴 +𝑄𝐾k𝑎)(q⊺ +𝑏 𝑊 𝐵 +𝑄𝐾k𝑏) +for all q𝑎 ∈ 𝑄𝐴, q𝑏 ∈ 𝑄𝐵, k𝑎 ∈ 𝐾𝐴, and k𝑏 ∈ 𝐾𝐵. +Proof. Assume such a 𝑊 𝐴∧𝐵 +𝑄𝐾 +exists. Then consider evaluating the combined attention head on a more +complex query, i.e. to change (q𝑎 + q𝑏) to (q𝑎 + q𝑏) + (q′ +𝑎 + q′ +𝑏) in the LHS above. Then, we have +((q𝑎 + q𝑏) + (q′ +𝑎 + q′ +𝑏))⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) += (q𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) + (q′ +𝑎 + q′ +𝑏)⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) += (q⊺ +𝑎 𝑊 𝐴 +𝑄𝐾k𝑎)(q⊺ +𝑏 𝑊 𝐵 +𝑄𝐾k𝑏) + (q′⊺ +𝑎 𝑊 𝐴 +𝑄𝐾k𝑎)(q′⊺ +𝑏 𝑊 𝐵 +𝑄𝐾k𝑏) +But if we distribute the first line differently, we also find that +((q𝑎 + q𝑏) + (q′ +𝑎 + q′ +𝑏))⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) += (q𝑎 + q′ +𝑏)⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) + (q′ +𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) += (q⊺ +𝑎 𝑊 𝐴 +𝑄𝐾k𝑎)(q′⊺ +𝑏 𝑊 𝐵 +𝑄𝐾k𝑏) + (q′⊺ +𝑎 𝑊 𝐴 +𝑄𝐾k𝑎)(q⊺ +𝑏 𝑊 𝐵 +𝑄𝐾k𝑏). +By subtracting both results from each other, we can follow that +(q𝑎 − q′ +𝑎)⊺𝑊 𝐴 +𝑄𝐾k𝑎(q𝑏 − q′ +𝑏)⊺𝑊 𝐵 +𝑄𝐾k𝑏 = 0 +Thus, one of the original attention heads 𝐴 or 𝐵 must have query-invariant attention logits. By an +analogous argument, at least one of the heads must have key-invariant attention logits. +Hence, either one of the heads’ attention logits are constant, or one of them only depends on the +key and the other only on the value. Importantly, we cannot find 𝑊 𝐴∧𝐵 +𝑄𝐾 +for arbitrary 𝑊 𝐴 +𝑄𝐾 and 𝑊 𝐵 +𝑄𝐾. +□ +We could work around this, for example, by extending the combined attention head to act +(𝑄𝐴 ⊗ 𝑄𝐵) × (𝐾𝐴 ⊗ 𝐾𝐵). Unfortunately, this would result in an explosion of dimensions, requiring +|𝑄𝐴||𝐾𝐴| + |𝑄𝐵||𝐾𝐵| dimensions. +Lemma 2. We can construct 𝑊 𝐴∧𝐵 +𝑄𝐾 +operating over (𝑄𝐴 ⊗ 𝑄𝐵) × (𝐾𝐴 ⊗ 𝐾𝐵), such that +(q𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 + k𝑏) = (q⊺ +𝑎 𝑊 𝐴 +𝑄𝐾k𝑎)(q⊺ +𝑏 𝑊 𝐵 +𝑄𝐾k𝑏) +for all q𝑎 ∈ 𝑄𝐴, q𝑏 ∈ 𝑄𝐵, k𝑎 ∈ 𝐾𝐴, and k𝑏 ∈ 𝐾𝐵. +Proof. Let 𝑊 𝐴∧𝐵 +𝑄𝐾 += 𝑊 𝐴 +𝑄𝐾 ⊗ 𝑊 𝐵 +𝑄𝐾 be the tensor product of the bilinear maps defined by 𝑊 𝐴 +𝑄𝐾 and 𝑊 𝐵 +𝑄𝐾. +Then for q𝑎, q𝑏, k𝑎, k𝑏, we get (q𝑎 ⊗ q𝑏)⊺𝑊 𝐴∧𝐵 +𝑄𝐾 (k𝑎 ⊗ k𝑏) = (q⊺ +𝑎 𝑊 𝐴 +𝑄𝐾k𝑎)(q⊺ +𝑏 𝑊 𝐵 +𝑄𝐾k𝑏). +□ +20 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +D. More Compiled Models +Here, we present a few additional RASP programs and the compiled Tracr models. +Figure 10 shows and extended sort program. It works similarly to the sort_unique program in +Figure 5, but sorts any sequence of values by a sequence of keys and can handle duplicates occurring +in the keys. +Figure 11 shows the pair_balance program, which computes the difference in the fraction of +open and closed parenthesis tokens. We can now use it as a subroutine for the dyck-n program, +which checks if a sequence of 𝑛 different types of parentheses is balanced: +Input: pairs +1 +# Compute +running +balance of each type of parenthesis +2 +balances = [pair_balance(pair) for pair in pairs] +3 +4 +# If balances +were +negative +anywhere -> parentheses +not +balanced +5 +any_negative = balances [0] < 0 +6 +for balance in balances [1:]: +7 +any_negative = any_negative or (balance < 0) +8 +9 +select_all = select (1, 1, ==) +10 +has_neg = aggregate(select_all , any_negative) +11 +12 +# If all +balances +are 0 at the end -> closed all +parentheses +13 +all_zero = balances [0] == 0 +14 +for balance in balances [1:]: +15 +all_zero = all_zero +and (balance == 0) +16 +17 +select_last = select(indices , length - 1, ==) +18 +last_zero = aggregate(select_last , all_zero) +19 +20 +dyck_n = (last_zero +and not +has_neg) +Figure 12 shows the compiled dyck-2 model for pairs = (“()”, “{}”). +21 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +Input: keys, vals, min_key, context_length +1 +keys = (keys + indices + min_key) / context_length +2 +smaller = select(keys , keys , <=) +3 +target_pos = selector_width (smaller) +4 +sel_sort = select(target_pos , indices , ==) +5 +sort = aggregate(sel_sort , vals) +bos 4 3 3 4 +indices: 0 +indices: 1 +indices: 2 +indices: 3 +indices: 4 +one +sequence_map: 1.0 +sequence_map: 1.2 +sequence_map: 1.4 +sequence_map: 1.6 +sequence_map: 1.8 +sequence_map: 2.0 +sequence_map: 2.2 +sequence_map: 2.4 +sequence_map: 2.6 +sequence_map: 2.8 +sequence_map: 3.0 +sequence_map: 3.2 +sequence_map: 3.4 +sequence_map: 3.6 +sequence_map: 3.8 +sequence_map: 4.0 +sequence_map: 4.2 +sequence_map: 4.4 +sequence_map: 4.6 +sequence_map: 4.8 +sequence_map: 5.0 +sequence_map: 5.2 +sequence_map: 5.4 +sequence_map: 5.6 +sequence_map: 5.8 +sort: 1 +sort: 2 +sort: 3 +sort: 4 +sort: 5 +target_pos: 0 +target_pos: 1 +target_pos: 2 +target_pos: 3 +target_pos: 4 +target_pos: 5 +target_pos_75_selector_width_attn_output +tokens: 1 +tokens: 2 +tokens: 3 +tokens: 4 +tokens: 5 +tokens: bos +tokens: pad +Input +bos 4 3 3 4 +Attn 1 +bos 4 3 3 4 +MLP 1 +bos 4 3 3 4 +Attn 2 +bos 4 3 3 4 +MLP 2 +bos 4 3 3 4 +Attn 3 +bos 4 3 3 4 +MLP 3 +Figure 10 | Compiled sort program. Attn 1 is a no-op, MLP 1 adds a small multiple of indices to the keys, and the rest of +the model essentially implements sort_unique. +22 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +Input: open_token, close_token +1 +bools_open = (tokens == open_token) +2 +opens = frac_prevs(bools_open) +3 +bools_close = (tokens == close_token) +4 +closes = frac_prevs(bools_close) +5 +pair_balance = opens - closes +bos ( ( ) ( +bools_close +bools_open +closes +indices: 0 +indices: 1 +indices: 2 +indices: 3 +indices: 4 +one +opens +pair_balance +tokens: ( +tokens: ) +tokens: bos +tokens: pad +Input +bos ( ( ) ( +Attn 1 +bos ( ( ) ( +MLP 1 +bos ( ( ) ( +Attn 2 +bos ( ( ) ( +MLP 2 +Figure 11 | RASP program that uses frac_prevs as a subroutine to compute the fraction of open and closed parenthesis +tokens and computes the difference. The compiled model uses open_token = “(” and close_token = “)”. Note that the +compiled model has the same number of layers as the single frac_prevs model in Figure 2. Attn 1 is still a no-op, MLP 1 +and Attn 2 compute both calls to frac_prevs in parallel, and MLP 2 computes the final result. +23 + +Tracr: Compiled Transformers as a Laboratory for Interpretability +bos { } { } +any_negative_14 +balance_()_16 +balance_{}_17 +bools_close_29 +bools_close_33 +bools_open_27 +bools_open_31 +closes_21 +closes_23 +has_neg_9 +indices: 0 +indices: 1 +indices: 2 +indices: 3 +indices: 4 +last_zero_5: False +last_zero_5: True +length_15: 0 +length_15: 1 +length_15: 2 +length_15: 3 +length_15: 4 +length_15: 5 +length_15_selector_width_attn_output +map_10: -1 +map_10: 0 +map_10: 1 +map_10: 2 +map_10: 3 +map_10: 4 +map_11: False +map_11: True +map_12: False +map_12: True +map_24: False +map_24: True +map_25: False +map_25: True +not_has_neg_6: False +not_has_neg_6: True +one +opens_20 +opens_22 +sequence_map_18: False +sequence_map_18: True +sequence_map_8: False +sequence_map_8: True +shuffle_dyck_4: False +shuffle_dyck_4: True +tokens: ( +tokens: ) +tokens: bos +tokens: pad +tokens: { +tokens: } +Input +bos { } { } +Attn 1 +bos { } { } +MLP 1 +bos { } { } +Attn 2 +bos { } { } +MLP 2 +bos { } { } +Attn 3 +bos { } { } +MLP 3 +bos { } { } +Attn 4 +bos { } { } +MLP 4 +bos { } { } +Attn 5 +bos { } { } +MLP 5 +bos { } { } +Attn 6 +bos { } { } +MLP 6 +bos { } { } +Attn 7 +bos { } { } +MLP 7 +bos { } { } +Attn 8 +bos { } { } +MLP 8 +Figure 12 | Compiled dyck-2 program for pairs = (“()”, “{}”). +24 + diff --git a/1tE4T4oBgHgl3EQfaQwc/content/tmp_files/load_file.txt b/1tE4T4oBgHgl3EQfaQwc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..28d8cbada464c736b4622237853329d297a442e3 --- /dev/null +++ b/1tE4T4oBgHgl3EQfaQwc/content/tmp_files/load_file.txt @@ -0,0 +1,1187 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf,len=1186 +page_content='2023-1-13 Tracr: Compiled Transformers as a Laboratory for Interpretability David Lindner1*, János Kramár2, Matthew Rahtz2, Thomas McGrath2 and Vladimir Mikulik2 1ETH Zurich, 2DeepMind, *Work done at DeepMind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Interpretability research aims to build tools for understanding machine learning (ML) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, such tools are inherently hard to evaluate because we do not have ground truth information about how ML models actually work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this work, we propose to build transformer models manually as a testbed for interpretability research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We introduce Tracr, a “compiler” for translating human-readable programs into weights of a transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr takes code written in RASP, a domain-specific language (Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2021), and translates it into weights for a standard, decoder-only, GPT-like transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We use Tracr to create a range of ground truth transformers that implement programs including computing token frequencies, sorting, and Dyck-n parenthesis checking, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We study the resulting models and discuss how this approach can accelerate interpretability research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To enable the broader research community to explore and use compiled models, we provide an open-source implementation of Tracr at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='com/deepmind/tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Keywords: Interpretability, Transformers, Language Models, RASP, Tracr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Introduction Explanation Neural Network Interpretability Known Mechanism Is the explanation correct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr Figure 1 | Tracr allows us to create models that implement a known mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can then compare this mechanism to explanations an in- terpretability tool produces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' As deep learning models are becoming more capable and increasingly deployed in production, improving our ability to understand how they make decisions is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Mechanistic interpretability aims to achieve this by reverse engineering neural networks and producing mech- anistic explanations of the algorithms a model imple- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This approach has achieved success in convo- lutional neural networks for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Cam- marata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2020) explain a range of specific circuits in InceptionV1 (Szegedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2015), including curve detec- tors, high-low frequency detectors, and neurons detecting more high-level concepts such as dogs or cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021) and Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2022) achieve early success in interpreting transformer language models using similar methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Despite this success, the toolbox of approaches for generating mechanistic explanations remains small and poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Part of the difficulty is that evaluating mechanistic explanations requires creativity and effort by researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' It is difficult to evaluate how well an explanation tracks the actual mechanism used by the model when all our knowledge of the mechanism comes from the explanation itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Without access to ground truth about the proposed mechanism, we must verify the methods used to study it in some other way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The standard approach for evaluating mechanistic explanations combines evidence from many ad-hoc experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', Olah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2020) and Olsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, since this is expensive © 2023 DeepMind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' All rights reserved arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='05062v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='LG] 12 Jan 2023 DeepMind<>Tracr: Compiled Transformers as a Laboratory for Interpretability to do, many methods are only evaluated in toy models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2022)) or on a handful of nontrivial circuits in real models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Systematic evaluation in nontrivial settings is usually intractable as it requires a lot of researcher time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The situation is analogous to trying to invent a microscope lens without ever being able to point it at familiar, well-understood shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Through careful reasoning and experimentation, we might notice regularities in the tiny world seen through the lens, and begin to trust findings made with it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' but if we could look through the lens at something we already understand, we would recognise its optical properties and correct its flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We propose to directly tackle the absence of ground truth explanations by "compiling" human readable code to weights of a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this report, we present Tracr, a proof-of-concept implementation of such a compiler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Using this approach, we can create models which perform nontrivial computation with a known implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can then evaluate interpretability tools by applying them to compiled models and comparing the resulting explanation to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Imagine we want to evaluate a method for locating specific knowledge in transformer models, such as “causal tracing” (Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In real language models, it can be challenging to check its correctness: the method might point out a location in the model, but we can’t easily independently verify its claim, since no trusted procedure for establishing such facts about models in the wild exists yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' With Tracr we can construct models that encode some information in a specific location and check if our method correctly locates it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can further explore special cases, such as information stored redundantly in different places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this work, we focus on transformer models (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2017) and use RASP, a domain- specific programming language for describing transformer computations (Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We develop an approach to compile RASP programs to the weights of a transformer model by combining hand-coded and fully interpretable model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We further propose a method that uses gradient descent to compress the compiled models to make them more efficient and realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' More specifically, in this report, we: Describe a modified version of the RASP programming language better suited for being compiled to model weights (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2) and discuss some limitations of the RASP programming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Introduce Tracr, a “compiler” for translating RASP programs into transformer model weights (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To describe Tracr, we also introduce craft, its intermediate representation for expressing linear algebra operations using named basis directions (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Showcase several transformer models obtained by using Tracr (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Propose an optimization procedure to “compress” the compiled models and make them more efficient and realistic (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We analyse models compressed this way, demonstrating superposition (Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Discuss potential applications and limitations of Tracr and how compiled models can help to accelerate interpretability research (Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Provide an open-source implementation of Tracr (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='com/deepmind/tracr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Background Before describing Tracr, let us recap the transformer architecture and the RASP programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2 Tracr: Compiled Transformers as a Laboratory for Interpretability is_x = ( tokens == "x") prevs = select(indices ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' indices ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' <=) frac_prevs = aggregate (prevs ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' is_x) bos x a c x frac_prevs indices: 0 indices: 1 indices: 2 indices: 3 indices: 4 is_x one tokens: a tokens: b tokens: bos tokens: c tokens: pad tokens: x Input bos x a c x Attn 1 bos x a c x MLP 1 bos x a c x Attn 2 bos x a c x MLP 2 Figure 2 | An example RASP program (left) that computes the fraction of previous “x” tokens at each position of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr compiles this program to a transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We show the full residual stream of the compiled model at each layer for the input sequence “xacx” (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Attn 1 is a no-op, MLP 1 computes the indicator variable is_x, Attn 2 implements the select-aggregate operation to compute frac_prevs, and MLP 2 is a no-op again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Section 4 discusses this and other examples in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Transformer Models A transformer model consists of alternating multi-headed attention (MHA) and multi-layer perceptron (MLP) layers with residual connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Multi-headed attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2017) computes attention maps on sequences of length 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A single attention head 𝑖 first computes an attention pattern 𝐴𝑖 = softmax � (𝑥𝑊𝑖 𝑄)(𝑥𝑊𝑖 𝐾)𝑇/ √︁ 𝑑𝑘 � ∈ ℝ𝑁×𝑁 for some input 𝑥 ∈ ℝ𝑁×𝑑, where 𝑊𝑖 𝑄, 𝑊𝑖 𝐾 ∈ ℝ𝑑×𝑑𝑘 are learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Usually, we call the entries of (𝑥𝑊𝑖 𝐾) keys, and the entries of (𝑥𝑊𝑖 𝑄) queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Multi-headed attention combines 𝐻 attention heads heads by computing MHA(𝑥) = Concat � 𝐴1(𝑥𝑊1 𝑉 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' , 𝐴𝐻(𝑥𝑊 𝐻 𝑉 ) � 𝑊𝑂 where 𝑊𝑖 𝑉 ∈ ℝ𝑑×𝑑𝑣 and 𝑊𝑂 ∈ ℝ𝐻𝑑𝑣×𝑑 are another set of learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We commonly call the entries of (𝑥𝑊𝑖 𝑉) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The MLP layers in transformer models compute MLP(𝑥) = 𝜎(𝑥𝑊1)𝑊2 where 𝑊1 ∈ ℝ𝑑×ℎ, 𝑊2 ∈ ℝℎ×𝑑 are learnable weights, and 𝜎 is a non-linear function, often the Gaussian Error Linear Unit (GeLU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Hendrycks and Gimpel, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For simplicity we use the Rectified Linear Unit (ReLU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Agarap, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this paper, we focus on decoder-only transformers with the popular GPT architecture (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2018), which consists of alternating blocks of MHA, MLP, and layer normalization (Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The input to the model is the sum of a learned embedding of a sequence of input tokens and a positional embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The model is trained to predict the next token using gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Transformer Circuits We adopt the circuits view of transformers, introduced by Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This view (1) focuses on the transformer being a residual stream architecture and (2) introduces an alternative parameterisation for attention operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Both make it easier to reason about the computation done by transformers and will help us when assembling transformers manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The residual stream view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Transformers have residual connections at each attention and MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021) consider the residual connections a core feature of the architecture and describe 3 Tracr: Compiled Transformers as a Laboratory for Interpretability the model in terms of a residual stream that each layer reads from and writes to in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The residual stream acts as a type of memory that earlier layers can use to pass information to later layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Parameterising attention as 𝑊𝑄𝐾 and 𝑊𝑂𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Following Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021), we parameterise an attention head by two (low-rank) matrices 𝑊𝑄𝐾𝑖 = 𝑊𝑖 𝑄(𝑊𝑖 𝐾)𝑇/√ 𝑑𝑘 ∈ ℝ𝑑×𝑑 and 𝑊𝑂𝑉 𝑖 = 𝑊𝑖 𝑉𝑊𝑖 𝑂 ∈ ℝ𝑑×𝑑 where we split 𝑊𝑂 into different heads, such that 𝑊𝑂 = [𝑊1 𝑂, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 𝑊 𝐻 𝑂 ], where each 𝑊𝑖 𝑂 ∈ ℝ𝑑𝑣×𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can then write MHA as 𝐴𝑖 = softmax � 𝑥𝑊𝑄𝐾 𝑖𝑥𝑇� MHA(𝑥) = 𝐻 ∑︁ 𝑖=1 𝐴𝑖𝑥𝑊𝑂𝑉 𝑖 Importantly, we can think of MHA as summing over the outputs of 𝐻 independent attention heads, each parameterised by low-rank matrices 𝑊𝑄𝐾 and 𝑊𝑂𝑉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 𝑊𝑄𝐾 acts as a bilinear operator reading from the residual stream, and 𝑊𝑂𝑉 is a linear operator both reading from and writing to the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The softmax is the only nonlinearity in an attention head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The RASP Programming Language We build on the Restricted Access Sequence Processing Language (RASP), a domain-specific language for expressing transformer computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021) propose RASP as a computational model to describe transformers and provide an interpreter for RASP code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We are primarily interested in compiling actual transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this section, we review the main features of RASP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' for a more detailed description, refer to Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A RASP program can be seen as a computational graph, with each node taking on a particular value when evaluating the entire graph on a given input token sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We usually refer to programs by the node at the tip of the graph, with the nodes it depends on left implicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' There are two basic node types, sequence operations and selectors, and two types of RASP operations, elementwise operations and select-aggregate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Sequence operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A sequence operation (s-op) represents sequences of values during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' tokens and indices are built-in primitive s-ops that return a sequence of input tokens or their indices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example: tokens(”hello”) = [h, e, l, l, o], and indices(”hello”) = [0, 1, 2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' S-ops roughly correspond to the state of the residual stream in transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Elementwise operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' RASP allows arbitrary elementwise operations on s-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, we can compute (3*indices)(”hello”) = [0, 3, 6, 9, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Elementwise operations roughly correspond to MLP layers in transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Select-aggregate operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To move information between token positions, RASP provides select- aggregate operations which roughly correspond to attention in transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A selector has a graph dependency on two s-ops and evaluates on inputs of length 𝑁 to a binary matrix of size 𝑁 × 𝑁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To create a selector, the select operation takes two s-ops and a boolean predicate 𝑝(𝑥, 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example: select(indices, [1, 0, 2], <)(”abc”) = ������ 1 0 0 0 0 0 1 1 0 ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Here, 𝑝(𝑥, 𝑦) = 𝑥 < 𝑦, where 𝑥 comes from indices, and 𝑦 comes from the constant s-op [1, 0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The aggregate operation takes as input a selector and an s-op, and produces an s-op that averages 4 Tracr: Compiled Transformers as a Laboratory for Interpretability the value of the s-op weighted by the selection matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example: aggregate �� � ������ 1 0 0 0 0 0 1 1 0 ������ , [10, 20, 30]�� � = [10, 0, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A selector roughly corresponds to an attention pattern in a transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Together a select-aggregate operation roughly corresponds to an attention head in transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr: A Transformer Compiler for RASP To introduce Tracr, we first describe how RASP maps to the transformer architecture (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1) and propose a few modifications to RASP that make this mapping more straightforward (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Next, we introduce craft, our “assembly language” for transformer models (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Finally, we describe how Tracr translates RASP programs to transformer weights (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Appendix A contains some more technical details, and we provide a full open-source implementa- tion of Tracr at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='com/deepmind/tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Mapping RASP to Tranformers RASP povides a computational model of transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For the most part, we can map RASP operations directly to the components of a transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The built-in s-ops tokens and indices correspond to a transformer’s token and position embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, we can embed the tokens and positions as categorical variables in orthogonal subspaces of the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Any elementwise operation in RASP can be approximately computed by an MLP layer simply because MLPs can approximate any function with accuracy depending on the width and depth of the MLP (Hornik et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Attention layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' RASP’s select-aggregate operations map to the attention layers in transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The post-softmax attention pattern needs to match the selection matrix for all inputs to implement a given selector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' So, given a large enough key/query-dimension, an attention head can implement an arbitrary binary attention pattern using its 𝑊𝑄𝐾 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The 𝑊𝑂𝑉 matrix of the attention head can then implement the aggregate operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Modifications to RASP While we can map RASP operations to transformers, we need to make a few modifications to the RASP language to allow translating it to model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Disallow arbitrary selector combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' RASP allows to combine selectors using boolean opera- tions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' however, there is no natural analogue for this in real transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Combining selectors with different input variables is particularly problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, in RASP we can define a selector select(a, b, ==) and select(c, d, ==) using four s-ops a,b,c, and d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, a real attention head only has two inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' If the model stores the s-ops in separate subspaces of the residual stream, a single attention head cannot implement this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1 Because of this, we restrict RASP to selectors with only two input variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In practice, 1We formalise this observation in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 5 Tracr: Compiled Transformers as a Laboratory for Interpretability this limitation turns out not to be severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In particular, we were able to implement programs to solve all tasks described by Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Encoding annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A compiled model needs to pass information between layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In a transformer, it is natural to do this via the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, we have to decide how to represent information in the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For simplicity, we only use two encodings: categorical and numerical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We encode categorical variables as one-hot vectors in a dedicated subspace of the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We encode numerical variables as the magnitude of a dedicated one-dimensional subspace of the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Categorical encoding is generally less efficient when numerical encoding is possible, but some aggregate operations only work with one type of encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For instance, aggregate can compute a mean across token positions, which is not natural with attention on a one-hot encoded subspace but straightforward with a numerical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, numerically-encoded data is generally harder to work with, requiring a decoding step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We require each s-op to be either categorical or numerical and augment RASP with the ability to annotate s-ops with the desired encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' By default, we assume s-ops are categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Beginning of sequence token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Transformers often assume any input sequence to start with a dedicated “beginning of sequence” token (BOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We make the BOS token mandatory in RASP because it is crucial when implementing arbitrary attention patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In particular, RASP allows selectors that can produce all-zero rows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' this is convenient when programming in RASP, but the softmax makes this behaviour impossible in a real attention head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In these situations, we use the BOS token as a "default" position to attend to: it is attended to iff no other token is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This allows the non-BOS part of the sequence to emulate the intended RASP behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In our case, this choice comes from practical considerations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' but, interestingly, real models sometimes show similar behaviour (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', see Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' craft: An Assembly Language for Transformers Machine code Programming language Assembly RASP craft Figure 3 | Tracr translates RASP to craft and then to model weights, analogous to how programming languages are first trans- lated to assembly then to machine code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' If RASP is the high-level language we compile, craft is our "assembly language", offering slightly more abstraction than operating on pure weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' craft represents vector spaces with labelled basis dimen- sions and operations on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This allows us to define pro- jections or other linear operations in terms of basis direction labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Importantly, craft abstracts away the need to keep track of padding in weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We implement a transformer in craft that sticks closely to the transformer circuits view provided by Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In particular, the residual stream is a vector space 𝑅 with a basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' An attention head can be defined using a bilinear operator 𝑊𝑄𝐾 : 𝑄 × 𝐾 → ℝ and a linear operator 𝑊𝑂𝑉 : 𝑉 → 𝑂, where 𝑄, 𝐾, 𝑉, 𝑂 ⊂ 𝑅 are the vector spaces that reuse the same basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' craft then handles the projection of these operators up to 𝑅 × 𝑅 → ℝ and 𝑅 → 𝑅, which corresponds to adding the requisite padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In practice, we first independently translate each RASP computation into a craft component, then assign components to layers, and finally construct the residual stream space 𝑅, ensuring that all information needed at a given layer in the model is embedded by previous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Moreover, craft models are independent of concrete transformer implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A craft 6 JAXTracr: Compiled Transformers as a Laboratory for Interpretability (a) Steps 1 & 2: Computational graph with inferred s-op value sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (b) Step 3: Nodes translated to MLPs and attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (c) Steps 4 & 5: Nodes allocated to locations in a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Figure 4 | Schematic overview of how Tracr compiles the frac_prevs program from Figure 2 with a input vocabulary {”x”, ”y”} and context size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (a) shows the computational graph with value annotations after step 2 of the compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (b) shows how is_x and frac_prevs are translated to model components independently in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (c) shows the assembled model which has two no-op components because models blocks always need to have one attention and one MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' model can be translated into weights of any standard GPT-like transformer implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compiler Overview We are now ready to describe Tracr in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr comes with an implementation of RASP embedded in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This allows us to write RASP programs in Python and makes it easier to provide annotations, such as variable encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In Tracr, a RASP program is a data structure that is incrementally constructed by passing in dependencies to each operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We also do a few basic simplifications of RASP programs at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, we combine consecutive elementwise operations into a single s-op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr translates RASP programs to transformer weights in six steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Construct a computational graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Infer s-op input and output values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Independently translate s-ops to craft components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Assign components to layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Construct craft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Assemble transformer weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Let us go through these step by step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Figure 4 gives a schematic overview using an example program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Construct a computational graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' First, we trace the whole program to create a directed graph representing the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The graph has source nodes representing tokens and indices and a sink node for the output s-op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Infer s-op values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For each s-op, we need to decide how to embed it in the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To use categorical encodings, we need to know which values an s-op can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' All nodes have a finite set of output values because computations are deterministic, and we have a finite input vocabulary and context size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Therefore, in the second step, we traverse the graph and annotate each node with its possible outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This annotation uses simple heuristics that ensure we find a superset of the values an s-op will take, though, sometimes, an output set can contain values that the s-op never takes in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Independently translate s-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Next, we consider each node in the computational graph inde- pendently and translate it into a craft component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Elementwise operations become MLP blocks, and select-aggregate operations become attention blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We use a library of manually engineered MLP and attention blocks to approximate arbitrary functions for numerical and categorical inputs 7 "x""y"} [0, 1, 2] tokens indices [0, 1] is_x prevs frac-prevs [O, 1/3, /2, 1]"x"""y"} [0, 1, 2] tokens indices [0, 1] MLP: is_X prevs Attn: prevs [O, 13, /2, 1]Attn: prevs MLP: is_X djw do-ou no-op attr Input OutputTracr: Compiled Transformers as a Laboratory for Interpretability and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' MLPs with categorical inputs and outputs function as lookup tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' MLPs with numeri- cal inputs and outputs use an explicit construction based on the universal function approximation theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For attention layers, we translate a selector into the 𝑊𝑄𝐾 operator and the corresponding aggregate operation into the 𝑊𝑂𝑉 operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We only support attention with categorical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For more details on the MLP and attention blocks, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Assign components to layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To construct a transformer model, we need to allocate all craft components in the computational graph to layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Ideally, we want to find the smallest model to perform the desired computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can generally formulate this as a combinatorial optimization problem with several constraints: the transformer architecture has alternating attention and MLP layers, and all computations that depend on each other need to be in the correct order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For scope reasons, we solve this with a heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' First, we compute the longest path from the input to a given node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This path length is an upper bound for the layer number to which we can allocate the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Then we apply additional heuristics to combine layers with blocks that we can compute in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This approach returns a correct but sometimes suboptimal layer allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Construct a craft model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We construct the residual stream space as the direct sum of all model components’ input and output spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In other words, we embed each s-op in its own orthogonal subspace, which is reserved for its sole use throughout the entire network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Now, we can traverse the computational graph in the order determined by the layer allocation and stack the components to obtain a full transformer represented in craft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Assemble transformer weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Finally, we translate the craft representation of the model into concrete model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' First, we combine parallel MLP layers into a single layer and parallel attention heads into a single layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In attention layers, we then split up the 𝑊𝑄𝐾 and 𝑊𝑂𝑉 matrices into 𝑊𝑞, 𝑊𝑘, 𝑊𝑜, 𝑊𝑣 weight matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Finally, we adjust the shapes of all weights and connect them to our transformer architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can then infer the model configuration (depth, layer width, residual stream size, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=') to fit the elements we have created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We base our transformer implementation on the example decoder-only transformer from Haiku (Hennigan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2020), notably removing the layer norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Extending Tracr to support any other transformer implementation is straightforward by reimplementing only step 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Exploring Compiled Transformers Having described Tracr, we are now ready to start compiling models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this section, we walk through two example programs to illustrate how the compiled models work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Appendix D contains more examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Overall, we were able to compile RASP programs for all the tasks described in Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021), though we had to modify a few of the programs to only use features supported by Tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Example 1: Counting tokens Figure 2 shows our primary running example, the frac_prevs program, that computes the fraction of previous "x" tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' It uses one MLP layer and one attention head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, because our model architecture always starts with an attention layer, the compiled model has four layers, with the first and last layers being no-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The frac_prevs model has a 14 dimensional residual stream, but it uses 12 out of these for the input embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The computation uses two numerical variables which correspond to the remaining two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The input embeddings have a few special dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' tokens:bos is the beginning 8 Tracr: Compiled Transformers as a Laboratory for Interpretability smaller = select(tokens , tokens , <=) target_pos = selector_width (smaller) sel_sort = select(target_pos , indices , ==) sort = aggregate (sel_sort , tokens) Figure 5 | RASP program that sorts a sequence of numbers without duplicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Attn 1 and MLP 1 implement the selector_width primitive (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Appendix A) which the program uses to compute the target position for each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Attn 2 moves the tokens to the desired position, and MLP 2 is a no-op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bos 3 5 4 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='indices: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='indices: ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='tokens: pad ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bos 3 5 4 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='Attn 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bos 3 5 4 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='MLP 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bos 3 5 4 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='Attn 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bos 3 5 4 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='MLP 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='of sequence token which we need to implement arbitrary attention patterns (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2), and one is an input dimension that is fixed to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The model uses this dimension as a constant, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', to add a bias in MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Example 2: Sorting As a second example, let us consider sorting a sequence of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Figure 5 shows a sort_unique program that sorts a sequence of unique tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The program computes the target position of each token by using the selector_width primitive in RASP, which computes the number of elements in each row of a selector that with the value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' selector_width can be implemented in terms of other RASP operations (Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2021), but not using our variant of RASP, so we treat it as a primitive that compiles directly to an attention and MLP layer (here Attn 1 and MLP 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' See Appendix A for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021) propose a sort program that can handle duplicates (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' their Figure 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, that implementation uses a selector smaller = select(tokens , tokens , <) or (select(key , key , ==) and select(indices , indices , <)) to treat duplicates, which is not supported by Tracr (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In Appendix D, we provide an alternative implementation of sort that handles duplicates by adding a small multiple of indices to the keys and then applying sort_unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' More examples Tracr can compile a wide range of RASP programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In Appendix D, we discuss a few more examples, leading up to checking balanced parentheses (Dyck-n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Our open-source Tracr implementation contains a library of even more example programs to compile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 9 Tracr: Compiled Transformers as a Laboratory for Interpretability Figure 6 | Training setup for compressing a compiled transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' At each layer, we use the same matrix 𝑊 ∈ ℝ𝐷×𝑑 to embed the disentangled 𝐷-dimensional residual stream to 𝑑 ≤ 𝐷 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We freeze the layer weights and only train 𝑊 to compress the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compressing Compiled Transformers Tracr models can be sparse and inefficient because they reserve an orthogonal subspace of the residual stream for each s-op.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this section, we propose an experimental approach for “compressing” the resulting models and making them more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This feature is presented as preliminary work and is not yet provided in the Tracr library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Here, we present two case studies of compressing compiled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In addition to making Tracr models more efficient, the compressed models allow us to study how real neural networks might compress 𝐷 features into a representation space with fewer than 𝐷 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This phenomenon is called superposition (Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' however, to our knowledge, it has not been studied in models deeper than two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Gradient Descent Based Compression We use a single linear projection 𝑊 ∈ ℝ𝐷×𝑑 to compress the disentangled residual stream with size 𝐷 to a smaller space with dimension 𝑑 < 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We modify the model to apply 𝑊𝑇 whenever it reads from and 𝑊 whenever it writes to the residual stream (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We freeze the weights of all layers and train only 𝑊 using stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Since vanilla Tracr models are sparse and have orthogonal features, this process can be viewed as learning the projection from a "hypothetical disentangled model" to the "observed model" described by Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We want the compressed model to minimise loss under the constraint that it implements the same computation as the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To achieve this, we train 𝑊 to minimise 𝔼𝑥[L(𝑊, 𝑥)], where L(𝑊, 𝑥) = Lout(𝑊, 𝑥) + Llayer(𝑊, 𝑥) Lout = loss( 𝑓 (𝑥), ˆ𝑓𝑊(𝑥)) Llayer = ∑︁ layer 𝑖 (ℎ𝑖(𝑥) − ˆℎ𝑊,𝑖(𝑥))2 where 𝑓 (𝑥) is the output of the compiled model for input 𝑥, ˆ𝑓𝑊(𝑥) is the output of the compressed model, and ℎ𝑖(𝑥) and ˆℎ𝑊,𝑖(𝑥) are the output vectors at layer 𝑖 of the respective models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For categorical outputs, Lout is the softmax cross-entropy loss, whereas, for numerical outputs, it is the mean-squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Llayer is a regularization term that incentives the compressed model to match the per-layer outputs of the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To minimise this loss, the compressed model will 10 Attn MLP Attn MLP h2 h3 h1 M WT M WT M WT W WT Input M OutputTracr: Compiled Transformers as a Laboratory for Interpretability 0 1 2 3 training steps ×105 10−2 100 output loss d = 4 d = 8 d = 12 5 10 embedding size d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='06 final output loss Figure 7 | Loss of compressed Tracr models for the frac_prevs program from Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The left plot shows the loss during training for different embedding sizes 𝑑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' the right plot shows the final loss for different embedding sizes 𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' After about 𝑑 = 6 the compressed model solves the task essentially as well as the original compiled model which uses 𝐷 = 14 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Both plots are averaged over 10 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' have to approximate the computation of the original model but with a smaller residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We could set up this compression in other ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, we could use a different projection at each layer, use different matrices for embedding and unembedding, or modify weights other than 𝑊 when compressing the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' These design choices come with a tradeoff between making the model more expressible and potentially more realistic and enforcing the ground truth computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For simplicity, we use a shared 𝑊 for embedding/unembedding at every layer, and we already observe a rich structure in models compressed with this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Appendix B contains more details on the training setup, hyperparameters, and resources used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' What does the compression learn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' As our first case study, Figure 7 shows the example model from Figure 2, that computes the fraction of token “x”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' By learning an embedding matrix 𝑊, we can reduce the residual dimension from 𝐷 = 14 to 𝑑 = 6 without hurting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Once we reduce 𝑑 further, the model’s performance starts to suffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To understand the compression better, we can study how 𝑊 embeds the original 𝐷 features in 𝑑 < 𝐷 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can only do this because we started with a compiled model with known features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Figure 8 shows 𝑊𝑇𝑊 for compressing the model to 𝑑 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can compare this to using principle component analysis (PCA) to compress the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To interpret the results, we need to use our knowledge of the algorithm the model implements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The input tokens:x and the variables is_x and frac_prevs are crucial for computing the fraction of tokens that is “x”, and we find that these variables mostly get separate dimensions in the compressed residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The other input tokens stored in tokens:a, tokens:b, tokens:c are not necessary for solving the task, and so they are discarded in the compressed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Other variables, such as the indices embeddings, are stored in non-orthogonal dimensions in the compressed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This is consistent with existing findings on superposition as the indices embeddings are sparse and do not occur together (Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, some of our results go beyond previous work on superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, Tracr models often have multiple variables that depend on each other and encode shared information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In our running example is_x is an indicator variable that essentially contains the same information as the input dimension tokens:x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 In Figure 8, we see that the embeddings of is_x and tokens:x share part of the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Intuitively, this occurs because the variables encode similar information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 2They are not exactly the same because is_x is only populated in a later layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' But, if is_x = 1, then tokens:x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 11 Tracr: Compiled Transformers as a Laboratory for Interpretability (a) SGD Compression (b) PCA Figure 8 | 𝑊𝑇𝑊 for the compression procedure described in Section 5 with 𝑑 = 8 (a), compared to applying PCA and retaining only the first 8 components (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In contrast to PCA, our com- pression procedure produces a compression ma- trix 𝑊 that retains features necessary for the task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', is_x and frac_prevs) and discards features that are unimportant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', tokens:a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compiled Compressed Error 0 10 20 embedding size d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 accuracy 0 10 20 embedding size d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 cosine similarity Figure 9 | We compress the sort_unique program (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The two plots on the right show that the compressed model achieves nearly perfect accuracy, but the layer outputs of the compressed model are different from the original compiled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The left plot shows the average layer outputs of the compiled model, the compressed model, and the squared error between both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The source of the error is that the compressed model seems to learn to use a different (numerical) encoding for the target_pos variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In preliminary experiments, we found that shared information between variables seems to influence how superposition occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, varying the data distribution to have two variables share more or less information changes the correlation patterns between embedded features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Prior models of superposition do not explain this effect, and we leave fully understanding it for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Do the compressed models still implement the same computation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Even if the compressed models successfully achieve a low loss, we need to check if they implement the same computation as the compiled models, or else we would no longer know the ground truth mechanisms the models implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To this end, we evaluate the average cosine similarity between the output at each layer of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For the compressed frac_prevs model, the cosine similarity is close to 1, which implies that the compressed model is consistent with the compiled model (up to differences in norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3 3In categorical tasks the compressed model is encouraged to output vectors with a large norm due to the output softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We found that this can sometimes lead to the norm of the outputs at intermediate layers also changing even though the 12 Tracr: Compiled Transformers as a Laboratory for Interpretability However, in other cases, the cosine similarity stays below 1 even as the compressed model gets close to 100% in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' As an example, Figure 9 shows results from compressing the sort_unique model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Here, the compressed model achieves almost perfect accuracy on the task, but the average cosine similarity of the outputs at individual layers stays around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This suggests that the compressed model solves the tasks differently from the original compiled model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' By inspecting the models’ outputs at each layer, we can attribute the error to the target_pos variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In the Tracr model, target_pos is encoded categorically, with a dimension allocated per position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, the compiled model only uses one of these dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This suggests that the compressed model moves the tokens to the target position with a numerical encoding of the target position rather than a categorical encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' During training, this reduces the output loss at the cost of increasing the layer output regulariser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This case shows that even in this fairly restrictive compression setup, the compressed model can learn a different computation to be more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This is both encouraging and problematic: it is evidence that we can achieve meaningful compression with a simple approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' however, even in this restrictive setting, the compressed model is not guaranteed to be faithful to the original RASP program, undermining the value provided by the compiler as a source of ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Overall, using SGD on top of compiled models seems promising to make them more efficient and naturalistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We hope that future work can make this training setup more robust and that we can ultimately fully integrate it in a future version of Tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Discussion We provide an open-source implementation of Tracr because we think it has many potential appli- cations in interpretability research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this section, we discuss applications we see for Tracr and compiled transformers more generally and reflect on the current limitations of Tracr and how they can be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Applications of compiled models in interpretability research Compilers like Tracr allow researchers to set up controlled experiments that test specific hypotheses about the computational structure of transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this way, it acts as a laboratory for research in interpretability, enabling research that might otherwise be intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Test cases for interpretability tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compiled models serve as a natural foundation for testing the faithfulness (Jacovi and Goldberg, 2020) of an explanation, and provide a way to falsify (Leavitt and Morcos, 2020) the explanations given by interpretability techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Ultimately, they could be used to build libraries of test cases for interpretability tools, which could in turn enable quantitative evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, Meng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2022) propose a method to locate factual knowledge in transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr could allow us to test what this or similar methods can locate in a range of models implementing different algorithms, contextualising its result in real models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Replacing model components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Another way to evaluate our understanding of how a model works is to replace parts of the model with hand-coded components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, Nanda and Lieberum (2022) test their understanding of how a transformer implements modular addition by replacing components of the model with their own idealised implementation and find that this can increase downstream performance, which is strong evidence that the proposed explanation is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' While Tracr compiles an algorithm into a full transformer model, it could be adapted to only compile part cosine similarity is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 13 Tracr: Compiled Transformers as a Laboratory for Interpretability of a model to replace part of a trained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This could make it easier to evaluate our understanding of a large model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Understanding model phenomena and developing new techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Beyond evaluation, compiled models can be used as a testbed for studying circuits-level phenomena and developing new approaches for interpreting transformer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, in Section 5 we successfully induced superposition in compressed Tracr models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Future work could analyse superposition in Tracr models, extending previous work in toy models (Elhage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Scherlis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In particular, Tracr allows studying how the structure of computation implemented by a model affects which features will be stored in superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' One goal for this line of research could be to predict how a specific Tracr model will be compressed, which features will be stored in superposition and how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A complementary approach is to try reversing the superposition induced by a compression procedure, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', using ideas from compressed sensing and dictionary learning (Aharon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Donoho, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Limitations of RASP and Tracr RASP and Tracr are limited in terms of expressivity, efficiency and realism compared to real trans- former models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Many of these limitations could be overcome in future versions of Tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Expressivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' RASP is designed for algorithmic tasks that map an input sequence to a discrete output sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, current language models usually map a sequence of input tokens to a probability distribution over the next token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Circuits in real models often consist of components that increase or decrease the probability of some tokens based on previous tokens (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' RASP, and hence Tracr, cannot model such "probabilistic" computation, but could potentially be extended to support it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' RASP only uses binary attention patterns, which inherently limits the range of algorithms it can implement (Merrill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A way to extend RASP to support numeric attention patterns is discussed in Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr models store all variables in orthogonal subspaces of the residual stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Even if a variable is only used in part of the computation, Tracr reserves a subspace of the residual stream for it in all layers of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Real models use a more compressed representation and likely reuse dimensions for multiple features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Improved versions of the compression procedure discussed in Section 5 could address this limitation, as would using a constraint optimisation solver instead of a heuristic for layer allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Realism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr constructs layers from hand-coded parameter matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This is both unrealistic and inefficient, but could be addressed by learning the layers in isolation, then assembling them into a full model manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Similarly, instead of manually splitting the 𝑊𝑄𝐾 and 𝑊𝑂𝑉 matrices, matrix factorisation could be used to get more efficient solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Also, Tracr models align their features with the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This is unrealistic, and makes the resulting models easy to interpret just by inspecting the residual stream activations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Rotating the basis of the compiled model is a straightforward way to address this if obfuscation is needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' compression would be an even more comprehensive approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' While all of these issues could be overcome in a more sophisticated compiler, there are fundamental limitations on the role compiled models can play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compiled models are an intermediate step between very simple toy models and real learned models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' They help us understand ideas and methods, but results in compiled models do not necessarily generalise to real models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compared with real models, compiled models will always be simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, we will likely never compile full-fledged language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compiled models will be more likely to be intepretable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', the axis-aligned orthogonal residual stream bases in Tracr), and more likely to fit into existing paradigms for thinking about transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' When using them to evaluate interpretability tools, we should be careful to make 14 Tracr: Compiled Transformers as a Laboratory for Interpretability sure that the tools do not exploit this, treating such evaluations as a minimum bar rather than a full validation of a technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Conversely, some methods might conceivably rely on features present in real models but not in compiled models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Conclusion In this work, we proposed manually constructing neural network weights and using them to develop and evaluate new interpretability tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To this end, we developed Tracr, a tool for compiling human-readable code to the weights of a transformer model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We outlined our vision for the use of compiled models in interpretability, and there may other potential applications of Tracr within and beyond interpretability research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We are looking forward to seeing other researchers use it, and we hope studying compiled models will help to increase our understanding of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Acknowledgements We thank Avraham Ruderman, Jackie Kay, Michela Paganini, Tom Lieberum, and Geoffrey Irving for valuable discussions, Victoria Krakovna and Marlene Staib for collaborating on early experiments with compiling RASP, and Chris Olah and Tristan Hume for feedback on an early draft of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Author Contributions VM proposed the initial idea for Tracr and wrote our RASP implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' DL, VM, JK and MR designed and developed Tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' DL designed, implemented, and ran 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Conmy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Shlegeris, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Steinhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Interpretability in the wild: a circuit for indirect object identification in GPT-2 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='00593, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Weiss, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Goldberg, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Yahav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Thinking like transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In International Conference on Machine Learning (ICML), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 17 Tracr: Compiled Transformers as a Laboratory for Interpretability A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Tracr Implementation Details This section highlights a few more implementation details of Tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We describe how we construct MLP and attention blocks, how we implement the selector width primitive, and how we extend RASP and Tracr to use causal attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For the full implementation and documentation, refer to the code repository at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='com/deepmind/tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' MLP and Attention Blocks For MLP layers, we distinguish between Map operations with a single input and output and SequenceMap operations with two inputs and one output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can recursively represent functions with more than two inputs using SequenceMaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We translate Maps with categorical inputs and outputs to MLPs that act as a lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' SequenceMaps with categorical inputs and outputs become MLPs where the first layer maps to an encoding of all pairs of inputs and the second layer acts as a lookup table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For numerical inputs and outputs, we explicitly construct MLP layers as universal function approx- imators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In these MLPs, the first layer discretises the input, and the second layer maps each discrete bucket to a corresponding output value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We know which input/output values can occur, so we can choose the discretisation around these known input values to minimise the approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We construct the 𝑊𝑄𝐾 matrix to implement the desired attention pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Here we ensure that if a token does not attend to any other token in RASP, it will attend to the BOS token in the Tracr model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The 𝑊𝑂𝑉 matrix maps the value input to the corresponding output dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Attention layers only support categorical key and query inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The value inputs can be numerical or categorical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can only use categorical values if the head never attends to more than one token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Selector Width Primitive RASP provides the selector width primitive, which counts the number of 1s in each row of a selector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' It provides an alternative to aggregate for processing selectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021) provide a selector width implementation in pure RASP, making it not necessarily a language primitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, the most efficient implementation uses the BOS token, which exists Tracr but is not exposed to the RASP program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Therefore, Tracr translates selector width directly into an efficient implementation in craft consisting of an attention layer and an MLP layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The attention layer implements an attention pattern that matches the selector to compute the width of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' It uses the BOS token as value input, resulting in the attention head computing 𝑥 = 1/(1 + 𝑤) where 𝑤 is the desired selector width output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The next MLP layer then computes 𝑤 = 1/𝑥 − 1 and cleans the BOS token position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Casual Attention Most transformer models used in practice use causal attention, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', they apply a mask to the attention patterns that allows the model to attend only to previous tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This allows training the models autoregressively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, RASP assumes non-causal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' bidirectional) attention by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' While all models discussed in the main paper use non-causal attention, Tracr also supports causal attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To enable this, we extend RASP to support causal attention via a flag set during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' To evaluate a RASP program in the causal evaluation mode, we apply a causal mask to the output of 18 Tracr: Compiled Transformers as a Laboratory for Interpretability each selector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Causal evaluation changes the semantics of some RASP operations, and, in general, it is necessary to adapt RASP programs to function with causal attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' For example, the frac_prevs program no longer needs to compute a causal mask manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' However, for example, the length implementation by Weiss et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (2021) no longer correctly computes the length of a sequence because it requires attending to future tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Similarly, Tracr has a flag to enable causal compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Most of the compilation process does not change, and we only need to ensure to compile selectors to causal attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Compression Training Details We implemented the compression described in Section 5 in Jax on top of the Haiku transformer implementation that comes with Tracr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We train 𝑊 using the AdamW optimizer (implemented in Optax) with a weight decay factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1, and parameters 𝛽1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='9, 𝛽2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We train for 3 × 105 steps with a batch size of 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We decay the learning rate linearly from 10−3 to 10−6 over the first half of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Each compression run requires between 1 and 4 hours of run time on two CPU cores (depending on the size of the model to compress).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Theoretical Results on Combining Attention Heads In this section, we study how we could implement combinations of selectors with more than two inputs, which are allowed in RASP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We focus on combining selectors with an and operation, but the results generalize to other boolean operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Consider the following selectors: simple_selector = select(tokens , indices , <=) simplifiable_selector = select(tokens , indices , <=) and select(tokens , "a", ==) simplified_selector = select(tokens , indices , q <= k and q == "a") compound_selector = select(a, b, <=) and select(c, d, <=) where a, b, c and d are different s-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The simple selector depends on only two s-ops and is straightforward to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The simplifiable selector is syntactically defined using the and operator but can be converted into the simplified selector, which still only depends on two s-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' This section concerns selectors like the compound_selector above, which irreducibly depend on more than two different s-ops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' An attention head can be parameterized by a 𝑊𝑄𝐾 matrix and an 𝑊𝑂𝑉 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In this section, we focus on 𝑊𝑄𝐾 only, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=', on the circuit responsible for the attention patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The standard view of 𝑊𝑄𝐾 is as a matrix that computes the keys and queries from the residual stream space 𝑅 ⊆ ℝ𝑑 and computes their dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We instead interpret it as a bilinear operator 𝑊𝑄𝐾 : 𝑄 × 𝐾− > ℝ acting on two subspaces of the residual stream, 𝑄, 𝐾 ⊂ 𝑅, which are spanned by orthogonal bases {𝑞𝑗}, {𝑘𝑖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The elements of these bases correspond to elements of the value sets of s-ops in a one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We call those value sets 𝑄, 𝐾 as well to ease notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' A selector is a function 𝑆 : 𝑄 × 𝐾 → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' In Tracr, an attention head implements a selector if 𝑆(𝑞, 𝑘) = 𝑊𝑄𝐾(𝑞, 𝑘) := 𝑞𝑇𝑊𝑄𝐾𝑘 for all 𝑞 ∈ 𝑄, 𝑘 ∈ 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' (We can ignore the softmax without loss of generality as we can rescale the norm of 𝑊𝑄𝐾 to recover boolean outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=') Suppose we have two selectors 𝐴 and 𝐵 implemented by attention heads with query-key matrices 𝑊 𝐴 𝑄𝐾 and 𝑊 𝐵 𝑄𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' They each read from residual subspaces 𝑄𝐴 × 𝐾𝐴 and 𝑄𝐵 × 𝐾𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' The straightforward way to implement a combined selector 𝐴 ∧ 𝐵 would be to define an attention head with a 𝑊𝑄𝐾 acting 19 Tracr: Compiled Transformers as a Laboratory for Interpretability on (𝑄𝐴 ⊕ 𝑄𝐵) × (𝐾𝐴 ⊕ 𝐾𝐵) with attention logits that are the boolean and of the ones from 𝐴 and 𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Unfortunately, this is only possible in trivial cases because the operator needs to be bilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' There is no 𝑊 𝐴∧𝐵 𝑄𝐾 operating over (𝑄𝐴 ⊕ 𝑄𝐵) × (𝐾𝐴 ⊕ 𝐾𝐵) such that (q𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) = (q⊺ 𝑎 𝑊 𝐴 𝑄𝐾k𝑎)(q⊺ 𝑏 𝑊 𝐵 𝑄𝐾k𝑏) for all q𝑎 ∈ 𝑄𝐴, q𝑏 ∈ 𝑄𝐵, k𝑎 ∈ 𝐾𝐴, and k𝑏 ∈ 𝐾𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Assume such a 𝑊 𝐴∧𝐵 𝑄𝐾 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Then consider evaluating the combined attention head on a more complex query, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' to change (q𝑎 + q𝑏) to (q𝑎 + q𝑏) + (q′ 𝑎 + q′ 𝑏) in the LHS above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Then, we have ((q𝑎 + q𝑏) + (q′ 𝑎 + q′ 𝑏))⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) = (q𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) + (q′ 𝑎 + q′ 𝑏)⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) = (q⊺ 𝑎 𝑊 𝐴 𝑄𝐾k𝑎)(q⊺ 𝑏 𝑊 𝐵 𝑄𝐾k𝑏) + (q′⊺ 𝑎 𝑊 𝐴 𝑄𝐾k𝑎)(q′⊺ 𝑏 𝑊 𝐵 𝑄𝐾k𝑏) But if we distribute the first line differently, we also find that ((q𝑎 + q𝑏) + (q′ 𝑎 + q′ 𝑏))⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) = (q𝑎 + q′ 𝑏)⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) + (q′ 𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) = (q⊺ 𝑎 𝑊 𝐴 𝑄𝐾k𝑎)(q′⊺ 𝑏 𝑊 𝐵 𝑄𝐾k𝑏) + (q′⊺ 𝑎 𝑊 𝐴 𝑄𝐾k𝑎)(q⊺ 𝑏 𝑊 𝐵 𝑄𝐾k𝑏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' By subtracting both results from each other, we can follow that (q𝑎 − q′ 𝑎)⊺𝑊 𝐴 𝑄𝐾k𝑎(q𝑏 − q′ 𝑏)⊺𝑊 𝐵 𝑄𝐾k𝑏 = 0 Thus, one of the original attention heads 𝐴 or 𝐵 must have query-invariant attention logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' By an analogous argument, at least one of the heads must have key-invariant attention logits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Hence, either one of the heads’ attention logits are constant, or one of them only depends on the key and the other only on the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Importantly, we cannot find 𝑊 𝐴∧𝐵 𝑄𝐾 for arbitrary 𝑊 𝐴 𝑄𝐾 and 𝑊 𝐵 𝑄𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' □ We could work around this, for example, by extending the combined attention head to act (𝑄𝐴 ⊗ 𝑄𝐵) × (𝐾𝐴 ⊗ 𝐾𝐵).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Unfortunately, this would result in an explosion of dimensions, requiring |𝑄𝐴||𝐾𝐴| + |𝑄𝐵||𝐾𝐵| dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can construct 𝑊 𝐴∧𝐵 𝑄𝐾 operating over (𝑄𝐴 ⊗ 𝑄𝐵) × (𝐾𝐴 ⊗ 𝐾𝐵), such that (q𝑎 + q𝑏)⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 + k𝑏) = (q⊺ 𝑎 𝑊 𝐴 𝑄𝐾k𝑎)(q⊺ 𝑏 𝑊 𝐵 𝑄𝐾k𝑏) for all q𝑎 ∈ 𝑄𝐴, q𝑏 ∈ 𝑄𝐵, k𝑎 ∈ 𝐾𝐴, and k𝑏 ∈ 𝐾𝐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Let 𝑊 𝐴∧𝐵 𝑄𝐾 = 𝑊 𝐴 𝑄𝐾 ⊗ 𝑊 𝐵 𝑄𝐾 be the tensor product of the bilinear maps defined by 𝑊 𝐴 𝑄𝐾 and 𝑊 𝐵 𝑄𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Then for q𝑎, q𝑏, k𝑎, k𝑏, we get (q𝑎 ⊗ q𝑏)⊺𝑊 𝐴∧𝐵 𝑄𝐾 (k𝑎 ⊗ k𝑏) = (q⊺ 𝑎 𝑊 𝐴 𝑄𝐾k𝑎)(q⊺ 𝑏 𝑊 𝐵 𝑄𝐾k𝑏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' □ 20 Tracr: Compiled Transformers as a Laboratory for Interpretability D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' More Compiled Models Here, we present a few additional RASP programs and the compiled Tracr models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Figure 10 shows and extended sort program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' It works similarly to the sort_unique program in Figure 5, but sorts any sequence of values by a sequence of keys and can handle duplicates occurring in the keys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Figure 11 shows the pair_balance program, which computes the difference in the fraction of open and closed parenthesis tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' We can now use it as a subroutine for the dyck-n program,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' which checks if a sequence of 𝑛 different types of parentheses is balanced: Input: pairs 1 # Compute running balance of each type of parenthesis 2 balances = [pair_balance(pair) for pair in pairs] 3 4 # If balances were negative anywhere -> parentheses not balanced 5 any_negative = balances [0] < 0 6 for balance in balances [1:]: 7 any_negative = any_negative or (balance < 0) 8 9 select_all = select (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' ==) 10 has_neg = aggregate(select_all ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' any_negative) 11 12 # If all balances are 0 at the end -> closed all parentheses 13 all_zero = balances [0] == 0 14 for balance in balances [1:]: 15 all_zero = all_zero and (balance == 0) 16 17 select_last = select(indices ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' length - 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' ==) 18 last_zero = aggregate(select_last ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' all_zero) 19 20 dyck_n = (last_zero and not has_neg) Figure 12 shows the compiled dyck-2 model for pairs = (“()”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' “{}”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 21 Tracr: Compiled Transformers as a Laboratory for Interpretability Input: keys, vals, min_key, context_length 1 keys = (keys + indices + min_key) / context_length 2 smaller = select(keys , keys , <=) 3 target_pos = selector_width (smaller) 4 sel_sort = select(target_pos , indices , ==) 5 sort = aggregate(sel_sort , vals) bos 4 3 3 4 indices: 0 indices: 1 indices: 2 indices: 3 indices: 4 one sequence_map: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 sequence_map: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 sequence_map: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4 sequence_map: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='6 sequence_map: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='8 sequence_map: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 sequence_map: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 sequence_map: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4 sequence_map: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='6 sequence_map: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='8 sequence_map: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 sequence_map: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 sequence_map: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4 sequence_map: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='6 sequence_map: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='8 sequence_map: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 sequence_map: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 sequence_map: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4 sequence_map: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='6 sequence_map: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='8 sequence_map: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 sequence_map: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 sequence_map: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4 sequence_map: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='6 sequence_map: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='8 sort: 1 sort: 2 sort: 3 sort: 4 sort: 5 target_pos: 0 target_pos: 1 target_pos: 2 target_pos: 3 target_pos: 4 target_pos: 5 target_pos_75_selector_width_attn_output tokens: 1 tokens: 2 tokens: 3 tokens: 4 tokens: 5 tokens: bos tokens: pad Input bos 4 3 3 4 Attn 1 bos 4 3 3 4 MLP 1 bos 4 3 3 4 Attn 2 bos 4 3 3 4 MLP 2 bos 4 3 3 4 Attn 3 bos 4 3 3 4 MLP 3 Figure 10 | Compiled sort program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' Attn 1 is a no-op, MLP 1 adds a small multiple of indices to the keys, and the rest of the model essentially implements sort_unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' 22 Tracr: Compiled Transformers as a Laboratory for Interpretability Input: open_token,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content=' close_token ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bools_open = (tokens == open_token) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='opens = frac_prevs(bools_open) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bools_close = (tokens == close_token) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='closes = frac_prevs(bools_close) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='pair_balance = opens - closes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bos ( ( ) ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bools_close ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='bools_open ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='closes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='indices: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='indices: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='indices: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfaQwc/content/2301.05062v1.pdf'} 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claims in meta-analysis studies of COVID +quarantine (stay-at-home) orders + +S. Stanley Young1 and Warren B. Kindzierski2 + +1 CGStat, Raleigh, NC, USA +2 Independent consultant, St Albert, Alberta, Canada + +Correspondence: Warren B. Kindzierski, 12 Hart Place, St Albert, Alberta, T8N 5R1, Canada. +Email: wbk@shaw.ca or warrenk@ualberta.ca. + + + + + + +Abstract + +The coronavirus pandemic (COVID) has been an extraordinary test of modern government +scientific procedures that inform and shape policy. Many governments implemented COVID +quarantine (stay-at-home) orders on the notion that this nonpharmaceutical intervention would +delay and flatten the epidemic peak and largely benefit public health outcomes. The overall +research capacity response to COVID since late 2019 has been massive. Given lack of research +transparency, only a small fraction of published research has been judged by others to be +reproducible before COVID. Independent evaluation of published meta-analysis on a common +research question can be used to assess the reproducibility of a claim coming from that field of +research. We used a p-value plotting statistical method to independently evaluate reproducibility +of specific research claims made in four meta-analysis studies related to benefits/risks of COVID +quarantine orders. Outcomes we investigated included: mortality, mental health symptoms, +incidence of domestic violence, and suicidal ideation (thoughts of killing yourself). Three of the +four meta-analyses that we evaluated (mortality, mental health symptoms, incidence of domestic +violence) raise further questions about benefits/risks of this form of intervention. The fourth +meta-analysis study (suicidal ideation) is unreliable. Given lack of research transparency and +irreproducibility of published research, independent evaluation of meta-analysis studies using p- +value plotting is offered as a way to strengthen or refute (falsify) claims made in COVID +research. + +Keywords: COVID, stay-at-home orders, health outcomes, meta-analysis, reproducibility + +1. Introduction +1.1 Background +Since late 2019, the coronavirus pandemic (COVID) has been an extraordinary test of modern +government scientific procedures that inform and shape policy. Governments worldwide were +faced with a disease whose severity was uncertain and was infecting millions. Governments were +forced to act quickly given further uncertainties in the capacity of their health care systems to +deal with the virus. In many cases, governments relied on public health experts for their policy, +and more broadly to the established mechanisms by which scientific and medical expertise +inform government policy. + +On 11 March of 2020, the World Health Organization (WHO) officially declared COVID a +pandemic (Lavezzo et al., 2020; Members, 2020). Many governments subsequently adopted +aggressive pandemic policies. Examples of these policies, imposed as large-scale restrictions on +people, included (Gostin et al. 2020; Jenson 2020, Magness 2021): quarantine (stay-at-home) +orders, masking orders in community settings, nighttime curfews, closures of schools, +universities and many businesses, and bans on large gatherings. + +Mathematical modelling studies using simulated pandemic scenarios were used to justify +durations of restrictions imposed on people, ranging from 2 weeks to months (CDC 2017, +Jenson, 2020). These restrictions were intended to “flatten the epidemic curve” (Matrajt & +Leung, 2020). The term – flatten the epidemic curve – was originally utilized by the US Centers +of Disease Control for pandemic planning (CDC, 2007) to warrant use of targeted antiviral +medications and nonpharmaceutical interventions (NPIs) to delay and flatten the epidemic peak. + + +A key aspect of flattening the epidemic curve in a pandemic was being able to spread health care +demands resulting from a high incidence peak that could potentially overwhelm health care +utilization capacity (Jenson, 2020). The restrictions implemented by governments, however, +were lengthy as public health official policy targets shifted (Magness 2021). In United States, +political influence dominated both the initiation and ultimate duration of these restrictions +(Kosnik & Bellas, 2020). + +1.2 Research reproducibility +The overall research capacity response to COVID since late 2019 has been massive (Kinsella et +al., 2020; Chu et al., 2021; Ioannidis et al., 2022). To present an estimate of the magnitude of this +response, we used the Advanced Search Builder capabilities of freely available PubMed search +engine (pubmed.ncbi.nlm.nih.gov/advanced/). We used the terms covid[Title] OR sars-cov- +2[Title] for the period 2020-2023 (search performed November 23, 2022). Our search returned +247,597 listings in the National Library of Medicine data base. + +As reported in literature, only a small fraction of published research has been judged by others to +be reproducible before COVID (Ioannidis, 2005, 2022; Ioannidis et al., 2011; Keown, 2012; +Iqbal et al., 2016; Randall & Welser, 2018; Stodden et al., 2018). Landis et al. (2012) suggest +that the inability to reproduce findings is due to a lack of research transparency. + +Research transparency permits openness of study design, verification of results, synthesis of new +findings with previous knowledge, and effective inquiry of research (Munafo et al., 2017). +Causes of poor reproducibility of published research are related to aspects of lack of research +transparency such as (Ware & Munafo, 2015): biased study designs, flexibility in research +practices, low statistical power, and chasing statistical significance. + +As indicated above, many research studies have been published in response to COVID. +However, there remains concerns about reproducibility of COVID research, particularly where +observational data are used to generate results (Bramstedt, 2020; Peng & Hicks, 2021). The +current situation of irreproducible research may be that not much has changed during COVID +(e.g., Gustot, 2020; Sumner et al., 2020; Paez, 2021). + +1.3 Meta-analysis +Meta-analysis is a systematic procedure for statistically combining data (test statistics) from +multiple studies that address a common research question (Egger et al., 2001), for example, +whether an intervention (or risk factor) is causal of a health outcome. A meta-analysis examines +a claim by taking a summary statistic along with a measure of its reliability from multiple +individual intervention/risk factor—health outcome studies (called base papers) found in the +literature. These statistics are combined to give what is supposed to be a more reliable estimate +of an effect (Young & Kindzierski, 2019). + +One aspect of replication—the performance of another study statistically confirming the same +hypothesis or claim—is a cornerstone of science and replication of research claims is important +before causal inference can be made (Moonesinghe et al., 2007). If a replication study result does +not conform to a prevailing paradigm, it might not be submitted for publication. Also, if a similar + +flawed methodology is used in a replication study as in an original study, or if studies with +negative findings are not submitted for publication whereas studies with positive findings are, +then a false claim can be canonized (Nissen et al., 2016). + +Meta-analysis has been placed at the top of the medical evidence-based pyramid – above case– +control and cohort studies, and randomized trials (Murad et al., 2016). A key assumption of a +meta-analysis is that estimates drawn from the base papers for the analysis are unbiased +estimates of the effect of interest (Boos & Stefanski, 2013). Given these attributes, independent +evaluation of published meta-analysis on a common research question can be used to assess the +reproducibility of a claim coming from that field of research (Young & Kindzierski, 2019; +Kindzierski et al., 2021; Young & Kindzierski, 2022a). + +The objective of this study was to use a p-value plotting statistical method (after Schweder & +Spjøtvoll, 1982) to independently evaluate specific research claims related to COVID quarantine +(stay-at-home) orders in published meta-analysis studies. This was done in an attempt to +illustrate the importance of reproducibility of research claims arising from this +nonpharmaceutical intervention in the context of the surge of COVID papers in literature over +the past few years. + +2. Methods +We first wanted to gauge the number of reports of meta-analysis studies in literature related to +some aspect of COVID. To do this we again used the Advanced Search Builder capabilities of +the PubMed search engine. On November 20, 2022 we used the terms ((covid[Title]) OR (sars- +cov-2[Title]) AND (2020:2023[pdat])) AND (meta-analysis[Title] AND (2020:2023[pdat])). Our +search returned 3,204 listings in the National Library of Medicine data base. This included 633 +listings for 2020, 1,301 listings for 2021, and 1,270 listings thus far for 2022. We find these +counts astonishing in that a meta-analysis is a summary of available papers. + +Given our understanding of pre-COVID research reproducibility of published literature discussed +above, we speculated that there may be numerous meta-analysis studies relating to COVID that +are irreproducible. We prepared and posted a research plan – Young & Kindzierski (2022b) – on +the Researchers.One platform. This plan can be accessed and downloaded without restrictions +from the platform. Our plan was to use p-value plotting to independently evaluate four selected +published meta-analysis studies specifically relating to possible health outcomes of COVID +quarantine (stay-at-home) orders – also referred to as ‘lockdowns’ or ‘shelter-in-place’ in +literature. + +2.1 Data Sets +As stated in our research plan (Young & Kindzierski, 2022b), we considered four meta-analysis +studies in our evaluation: +• Herby et al. (2022) – mortality +• Prati & Mancini (2021) – psychological impacts (specifically, mental health symptoms) +• Piquero et al. (2021) – reported incidents of domestic violence +• Zhu et al. (2022) – suicidal ideation (thoughts of killing yourself) +Electronic copies of each meta-analysis study (and any corresponding electronic supplementary +information files) were downloaded from the internet and read. + + +The Herby et al. (2022) meta-analysis examined the effect of COVID quarantine (stay-at-home) +orders implemented in 2020 on mortality based on available empirical evidence. These orders +were defined as the imposition of at least one compulsory, non-pharmaceutical intervention. +Herby et al. initially identified 19,646 records that could potentially address their purpose. + +After three levels of screening by Herby et al., 32 studies qualified. Of these, estimates from 22 +studies could be converted to standardized measures for inclusion in their meta-analysis. For our +evaluation, we could only consider results for 20 of the 22 studies (data they provided for two +studies could not be converted to p-values). Their research claim was that “lockdowns in the +spring of 2020 had little to no effect on COVID-19 mortality”. + +The Prati & Mancini (2021) meta-analysis examined the psychological impact of COVID +quarantine (stay-at-home) orders on the general population. This included: mental health +symptoms (such as anxiety and depression), positive psychological functioning (such as well- +being and life-satisfaction), and feelings of loneliness and social support as ancillary outcomes. + +Prati & Mancini initially identified 1,248 separate records that could potentially address their +purpose. After screening, they identified and assessed 63 studies for eligibility and ultimately +considered 25 studies for their meta-analysis. For our evaluation, we used all 20 results they +reported on for mental health symptoms. Their research claim was that “lockdowns do not have +uniformly detrimental effects on mental health and most people are psychologically resilient to +their effects”. + +The Piquero et al. (2021) meta-analysis examined the effect of COVID quarantine (stay-at- +home) orders on reported incidents of domestic violence. They used the following search terms +to identify suitable papers with quantitative data to include in their meta-analysis… “domestic +violence”, “intimate partner violence”, or “violence against women”. + +Piquero et al. initially identified 22,557 records that could potentially address their purpose. +After screening, they assessed 132 studies for eligibility and ultimately considered 18 studies in +their meta-analysis. For our evaluation, we used all 17 results (effect sizes) they reported on from +the 18 studies. Their research claim was that “incidents of domestic violence increased in +response to stay-at-home/lockdown orders”. + +The Zhu et al. (2021) meta-analysis examined the effect of COVID quarantine (stay-at-home) +orders on suicidal ideation and suicide attempts among psychiatric patients in any setting (e.g., +home, institution, etc.). They used the following search terms to identify suitable papers with +quantitative data to include in their meta-analysis… “suicide” or “suicide attempt” or “suicidal +ideation” or “self-harm”, “psychiatric patients” or “psychiatric illness” or “mental disorders” or +“psychiatric hospitalization” or “psychiatric department” or “depressive symptoms” or +“obsessive-compulsive disorder”. + +Zhu et al. initially identified 728 records that could potentially address their purpose. After +screening, they assessed 83 studies for eligibility and ultimately considered 21 studies in their +meta-analysis. For our evaluation, we used all 12 results they reported on for suicidal ideation + +among psychiatric patients. Their research claim was that “estimated prevalence of suicidal +ideation within 12 months [during COVID] was… significantly higher than a world Mental +Health Survey conducted by the World Health Organization (WHO) in 21 countries [conducted +2001−2007]”. + +2.2 P-value Plots +In epidemiology it is traditional to use risk ratios and confidence intervals instead of p-values +from a hypothesis test to demonstrate or interpret statistical significance. Altman & Bland +(2011a,b) show that both confidence intervals and p-values are constructed from the same data +and they are inter-changeable, and one can be calculated from the other. + +Using JMP statistical software (SAS Institute, Cary, NC), we estimated p-values from risk ratios +and confidence intervals for all data in each of the meta-analysis studies. In the case of the Herby +et al. (2022) meta-analysis, standard error (SE) was presented instead of confidence intervals. +Where SE values were not reports, we used the median SE of the other base studies used in the +meta-analysis (6.8). The p-values for each meta-analysis are summarized in an Excel file (.xlsx +format) that can be downloaded at our posted Researchers.One research plan (Young & +Kindzierski, 2022b). + +We then developed p-value plots after Schweder & Spjøtvoll (1982) to inspect the distribution of +the set of p-values for each meta-analysis study. The p-value is a random variable derived from a +distribution of the test statistic used to analyze data and to test a null hypothesis (Young & +Kindzierski, 2022a). + +In a well-designed and conducted study, the p-value is distributed uniformly over the interval 0 +to 1 regardless of sample size under the null hypothesis (Schweder & Spjøtvoll, 1982). A +distribution of true null hypothesis points plotted against their ranks in a p-value plot should +form a 45-degree line when there are no effects (Schweder & Spjøtvoll, 1982; Hung et al., 1997; +Bordewijk et al., 2020). Researchers can use a p-value plot to assess the heterogeneity of the test +statistics combined in meta-analyses. + +The p-value plots we constructed were interpreted as follows (Young & Kindzierski, 2022a): +• Computed p-values were ordered from smallest to largest and plotted against the integers, 1, +2, 3,… +• If p-value points on the plot followed an approximate 45-degree line, we concluded that test +statistics resulted from a random (chance) process and the data supported the null hypothesis +of no significant association or effect. +• If p-value points on the plot followed approximately a line with a flat/shallow slope, where +most (the majority) of p-values were small (< 0.05), then test statistic data set provided +evidence for a real, statistically significant, association or effect. +• If p-value points on the plot exhibited a bilinear shape (divided into two lines), the data set of +test statistics used for meta-analysis is consistent with a two-component mixture and a +general (overall) claim is not supported. In addition, a small p-value reported for the overall +claim in the meta-analysis may not be valid (Schweder & Spjøtvoll, 1982). + + +Examples of p-value plots are provided in Appendix A after Young et al. (2022) to assist in +interpretation of the p-value plots we constructed here. Specifically, the p-value plots in +Appendix A represent ‘plausible true null’ and ‘plausible true alternative’ hypothesis outcomes +based on published meta-analysis studies of observational data sets in the field of environmental +epidemiology. As shown in the p-value plots in Appendix A: +• A plausible true null hypothesis plots as an approximate 45-degree line. +• A plausible true alternative hypothesis plots as a line with a flat/shallow slope, where most +(the majority) of p-values are small (< 0.05). + +The distribution of the p-value under the alternative hypothesis – where p-values are a measure +of evidence against the null hypothesis – is a function of both sample size and the true value or +range of true values of the tested parameter (Hung et al., 1997). The p-value plots presented in +Young et al. (2022) represent examples of distinct (single) sample distributions for each +condition – i.e., for true null associations and true effects between two variables. Evidence for p- +value plots exhibiting behaviors outside of that shown in Young et al. (2022) should initially be +treated as ambiguous (uncertain). + +3. Results + +Mortality +Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on +mortality – the Herby et al. (2022) meta-analysis – is shown in Figure 1. There are 20 studies that +we included in the figure. Six of the 20 studies had p-values below 0.05 while four of the studies +had p-values close to 1.00. Ten studies fell roughly on a 45-degree line implying random results. + +This data set comprises mostly null associations (14) and with five or six possible associations +with effects (1-in-20 could be chance, false, positive association). While not ideal, this data set is +a closer fit to a sample distribution for a true null association between two variables. Our +interpretation of the p-value plot is that COVID quarantine (stay-at-home) orders are not +supported for reducing mortality, consistent with Herby et al. (2022). + + +[Fig 1 to be inserted here] +Figure 1. P-value plot (p-value versus rank) for Herby et al. (2022) meta-analysis of the effect of +COVID quarantine (stay-at-home) orders implemented in 2020 on mortality. Symbols (circles) +are p-values ordered from smallest to largest (n=20). + +Psychological impact (mental health symptoms) +Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on mental +health symptoms – the Prati & Mancini (2021) meta-analysis – is shown in Figure 2. Figure 2 +presents as a bilinear shape showing a two-component mixture. This data set clearly does not +represent a distinct sample distribution for either true null associations or true effects between +two variables. Our interpretation of the p-value plot is that COVID quarantine (stay-at-home) +orders have an ambiguous (uncertain) effect on mental health symptoms. However as discussed +below, there are valid questions their research claim. + + + +[Fig 2 to be inserted here] +Figure 2. P-value plot (p-value versus rank) for Prati & Mancini (2021) meta-analysis of the +effect of COVID quarantine (stay-at-home) orders on mental health symptoms. Symbols (circles) +are p-values ordered from smallest to largest (n=20). + +Incidents of domestic violence +Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on reported +incidents of domestic violence – the Piquero et al. (2021) meta-analysis – is shown in Figure 3. +Thirteen of the 17 studies had p-values less than 0.05. While not shown in the figure, eight of the +p-values were small (<0.001). + +This data set comprises mostly non-null associations (13) and with four possible null +associations. While not perfect, this data set is a closer fit to a sample distribution for a true +alternative association between two variables. Our interpretation of the p-value plot is that +COVID quarantine (stay-at-home) have a negative effect (increase) for reported incidents of +domestic violence. + + +[Fig 3 to be inserted here] +Figure 3. P-value plot (p-value versus rank) for Piquero et al. (2021) meta-analysis of the effect +of COVID quarantine (stay-at-home) orders on reported incidents of domestic violence. Symbols +(circles) are p-values ordered from smallest to largest (n=17). + +Suicidal ideation +Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on suicidal +ideation – the Zhu et al. (2021) meta-analysis – is shown in Figure 4. The p-values for all 12 +studies were less than 0.05. Ten of the 12 studies had p-values less than 0.05. While not shown in +the figure, eight of the p-values were small (<0.001). + +This data set presents as a distinct sample distribution for true effects between two variables. Our +interpretation of the p-value plot is that COVID quarantine (stay-at-home) orders have an effect +on suicidal ideation (thoughts of killing yourself). However as discussed below, there are valid +questions about how the meta-analysis was formulated. + +[Fig 4 to be inserted here] +Figure 4. P-value plot (p-value versus rank) for Zhu et al. (2021) meta-analysis of the effect of +COVID quarantine (stay-at-home) orders on suicidal ideation (thoughts of killing yourself). +Symbols (circles) are p-values ordered from smallest to largest (n=12). + +4. Discussion + +As stated previously, independent evaluation of published meta-analysis on a common research +question can be used to assess the reproducibility of a claim coming from that field of research. +We evaluated four meta-analysis studies of COVID quarantine (stay-at-home) orders +implemented in 2020 and corresponding health benefits and/or harms. Our intent was to illustrate + +the importance of reproducibility of research claims arising from this nonpharmaceutical +intervention in the context of the surge of COVID papers in literature over the past few years. + +Mortality +The Herby et al. (2022) meta-analysis examined the effect of COVID quarantine orders on +mortality. Their research claim was that “lockdowns in the spring of 2020 had little to no effect +on COVID-19 mortality”. Here, they imply that the intervention (COVID quarantine orders) had +little or no effect on reduction of mortality. + +The quantitative data Herby et al. present to put their findings into perspective is that they +estimated the average lockdown in United States (Europe) in the spring of 2020 avoided 16,000 +(23,000) deaths. In contrast, they report that there are about 38,000 (72,000) flu deaths occurring +each year in the United States (Europe). + +Our evidence agrees with their claim. Our p-value plot (Figure 1) is not consistent with expected +behaviour of a distinct sample distribution for a true effect between the intervention (quarantine) +and the outcome (reduction in mortality). More importantly, our plot shows considerable +randomness (many null associations, p-values > 0.05) supporting no consistent effect. Herby et +al. further stated that “costs to society must be compared to the benefits of lockdowns, which our +meta-analysis has shown are little to none”. + +Psychological impact (mental health symptoms) +The Prati & Mancini (2021) meta-analysis examined the psychological impact of COVID +quarantine orders on the general population. Their research claim was that “lockdowns do not +have uniformly detrimental effects on mental health and most people are psychologically +resilient to their effects”. We evaluated a component of psychological impact – i.e., whether +COVID quarantine orders affect mental health symptoms (Figure 2). Figure 2 clearly exhibits a +two-component mixture implying an ambiguous (uncertain) effect on mental health symptoms. +However, our evidence does not necessarily support their claim. + +Digging deep into their study reveals an interesting finding. Their study looked at a variety of +psychological symptoms that differed from study to study. Although not shown here, when they +examined these symptoms separately – a meta-analysis of each symptom – there was a strong +signal for anxiety (p-value less than 0.0001). This is less than a Boos & Stefanski (2011) +proposed p-value action level of 0.001 for expected replicability. Here, the term ‘action level’ +means that if a study is replicated, the replication will give a p-value less than 0.05. + +We also note that Prati & Mancini appear to take absence of evidence of a negative mental health +effect of COVID quarantine orders in their meta-analysis as implying it does not affect mental +health. But absence of evidence does not imply evidence of absence (Altman & Bland, 1995, +Alderson, 2004; Sedgwick, 2014). Just because meta-analysis failed to find an effect, it does not +imply that “…most people are psychologically resilient to their [lockdown] effects”. A more +plausible and valid inference is that this statement of claim is insufficiently researched at this +point. + + + +Incidents of domestic violence +The Piquero et al. (2021) meta-analysis examined COVID quarantine orders on reported +incidents of domestic violence. Their research claim was that “incidents of domestic violence +increased in response to stay-at-home/lockdown orders”. Our evidence suggests agreement with +this claim. Our p-value plot (Figure 3) is more consistent with expected behaviour of a distinct +sample distribution for a true effect between the intervention (quarantine) and the outcome +(increase in incidents of domestic violence). + +Several null association studies exist within their data set. We note that Figure 3 has 13 of 17 p- +values less than 0.05, with eight of these less than 0.001. Our evidence supports that COVID +quarantine orders likely increased incidents of domestic violence. + +Suicidal ideation +The Zhu et al. (2021) meta-analysis examined COVID quarantine orders on suicidal ideation +(thoughts of killing yourself). Their research claim was that “estimated prevalence of suicidal +ideation within 12 months [during COVID] was… significantly higher than a world Mental +Health Survey conducted by the World Health Organization (WHO) in 21 countries [conducted +2001−2007]”. + +The p-value plot (Figure 4) strongly supports their claim. The plot is very consistent with +expected behaviour of a distinct sample distribution for a true effect between the intervention +(quarantine) and the outcome (increased prevalence of suicidal ideation). However, digging deep +into their study reveals a problem in the formulation of their meta-analysis. + +In strong science, a research question being investigated is judged against a control. Zhu et al. +effectively ignores controls in their meta-analysis. They compared incidence of suicidal ideation +against a zero standard and not to control groups. Specifically, the pre-COVID (i.e., background) +suicidal ideation signal is ignored in their meta-analysis. + +Indeed, in their Table 1 they present results from the base papers where data for control groups is +available. For example, the Seifert et al. (2021) base paper notes suicidal ideation presented in +123 of 374 patients in the psychiatric emergency department of Hannover Medical School during +the pandemic, and 141 of 476 in the same department before the pandemic – 32.9%versus +29.6%. The difference is not significant. + +Comparing their Table 1 data set with their Figure 1 forest plot, Zhu et al. only carried 32.9% +into their meta-analysis, in effect ignoring the control data. It is the same situation with all data +set entries in their Figure 1. Zhu et al. only considered pandemic incidence in their meta-analysis, +and they ignored any control data. How they formulated their work calls their claims into serious +question. We conclude that the Zhu et al. results are unreliable. + +Implications +COVID quarantine orders were implemented on the notion that this nonpharmaceutical +intervention would delay and flatten the epidemic peak and benefit public health outcomes +overall. Three of the four meta-analyses that we evaluated raise questions about public health + +benefits/risks of this form of nonpharmaceutical intervention. The fourth meta-analysis study is +unreliable. + +One meta-analysis that we evaluated – Herby et al. (2022) – questions the benefits of this form of +intervention for preventing mortality. Our p-value plot supports their finding that COVID +quarantine orders had little or no effect on reduction of mortality. + +A second meta-analysis – Prati & Mancini (2021) assessment of mental health symptoms – +offers confounding evidence. Our p-value plot clearly exhibits a two-component mixture +implying an ambiguous (uncertain) effect between COVID quarantine orders and mental health +symptoms. However, data for a component of mental health symptoms (anxiety) suggests a +negative effect from COVID quarantine orders. Further, Prati & Mancini (2021) lack evidence to +claim that “…most people are psychologically resilient to their [lockdown] effects”. + +Our evaluation of the Piquero et al. (2021) meta-analysis – assessment of domestic violence +incidents – supports a true effect between the intervention (quarantine) and the outcome +(increase in incidents of domestic violence) with additional confirmatory research needed. +Finally, the meta-analysis of Zhu et al. (2021) on suicidal ideation (thoughts of killing yourself) +is wrongly formulated and should be disregarded until or unless controls are included in the +analysis. + +Standing back and looking at the overall findings of these studies, benefits of COVID quarantine +orders remain uncertain and risks (negative public health consequences) of this intervention +cannot be ruled out. Given that the base studies and the meta-analyses themselves were, for the +most part, rapidly conducted and published, we acknowledge that confirmatory research for +some of the outcomes investigated is warranted. + +Our interpretation of COVID quarantine benefits/risks is consistent, for example, with earlier +research of James (2020) and conventional wisdom, Inglesby et al. 2006. James takes a position +that is it unclear whether there were benefits from this intervention relative to less restrictive +measures aimed at controlling “risky” personal interactions (e.g., mass gatherings and large +clusters of individuals in enclosed spaces). + +James (2020) also notes numerous economic and public health harms in the United States as +May 1, 2020: +• Over 20 million newly unemployed. +• State-wide school closures across the country. +• Increased spouse and child abuse reports. +• Increased divorces. +• Increased backlog of patient needs for mental health services, cancer treatments, dialysis +treatments and everyday visits for routine care. +• Increased acute emergency services. +This is consistent with interim quantitative data as of September 2020 presented by the American +Institute of Economic Research (2020) on the cost and negative public health implications of +pandemic restrictions in United States and around the world. + + +Acknowledgments +No external funding was provided for this study. 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Suicidal ideation and suicide attempts in psychiatric patients +during the COVID-19: A systematic review and meta-analysis. Psychiatry Research, 317, +114837. https://doi.org/10.1016/j.psychres.2022.114837 + + + + + +Figures + + +Figure 1. P-value plot (p-value versus rank) for Herby et al. (2022) meta-analysis of the effect of +COVID quarantine (stay-at-home) orders implemented in 2020 on mortality. Symbols (circles) +are p-values ordered from smallest to largest (n=20). + + +Figure 2. P-value plot (p-value versus rank) for Prati & Mancini (2021) meta-analysis of the +effect of COVID quarantine (stay-at-home) orders on mental health symptoms. Symbols (circles) +are p-values ordered from smallest to largest (n=20). + + +0.9 +0.8 +0.7 +0.6 +p-value +0.5 +0.4 +0.3 +0.2 +0.1 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Rank order0.9 +0.8 +0.7 +0.6 +p-value +0.5 +0.4 +0.3 +0.2 +0.1 +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +Rank order +Figure 3. P-value plot (p-value versus rank) for Piquero et al. (2021) meta-analysis of the effect +of COVID quarantine (stay-at-home) orders on reported incidents of domestic violence. Symbols +(circles) are p-values ordered from smallest to largest (n=17). + +Figure 4. P-value plot (p-value versus rank) for Zhu et al. (2021) meta-analysis of the effect of +COVID quarantine (stay-at-home) orders on suicidal ideation (thoughts of killing yourself). +Symbols (circles) are p-values ordered from smallest to largest (n=12). + + + +1 +0.9 +0.8 +0.7 +0.6 +p-value +0.5 +0.4 +0.3 +0.2 +0.1 +- +0 +3 +5 +6 +7 +8 +10 +11121314151617 +Rankorder1 +0.9 +0.8 +0.7 +0.6 +p-value +0.5 +0.4 +0.3 +0.2 +0.1 +0 +0 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +Rankorder \ No newline at end of file diff --git a/59FKT4oBgHgl3EQfTC3B/content/tmp_files/load_file.txt b/59FKT4oBgHgl3EQfTC3B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..33b44ca927f8889a2d2acdd1665f2b4a426ebcaa --- /dev/null +++ b/59FKT4oBgHgl3EQfTC3B/content/tmp_files/load_file.txt @@ -0,0 +1,952 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf,len=951 +page_content='Reproducibility of health claims in meta-analysis studies of COVID quarantine (stay-at-home) orders S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Stanley Young1 and Warren B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Kindzierski2 1 CGStat, Raleigh, NC, USA 2 Independent consultant, St Albert, Alberta, Canada Correspondence: Warren B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Kindzierski, 12 Hart Place, St Albert, Alberta, T8N 5R1, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Email: wbk@shaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='ca or warrenk@ualberta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Abstract The coronavirus pandemic (COVID) has been an extraordinary test of modern government scientific procedures that inform and shape policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Many governments implemented COVID quarantine (stay-at-home) orders on the notion that this nonpharmaceutical intervention would delay and flatten the epidemic peak and largely benefit public health outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The overall research capacity response to COVID since late 2019 has been massive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Given lack of research transparency, only a small fraction of published research has been judged by others to be reproducible before COVID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Independent evaluation of published meta-analysis on a common research question can be used to assess the reproducibility of a claim coming from that field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We used a p-value plotting statistical method to independently evaluate reproducibility of specific research claims made in four meta-analysis studies related to benefits/risks of COVID quarantine orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Outcomes we investigated included: mortality, mental health symptoms, incidence of domestic violence, and suicidal ideation (thoughts of killing yourself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Three of the four meta-analyses that we evaluated (mortality, mental health symptoms, incidence of domestic violence) raise further questions about benefits/risks of this form of intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The fourth meta-analysis study (suicidal ideation) is unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Given lack of research transparency and irreproducibility of published research, independent evaluation of meta-analysis studies using p- value plotting is offered as a way to strengthen or refute (falsify) claims made in COVID research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Keywords: COVID, stay-at-home orders, health outcomes, meta-analysis, reproducibility 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1 Background Since late 2019, the coronavirus pandemic (COVID) has been an extraordinary test of modern government scientific procedures that inform and shape policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Governments worldwide were faced with a disease whose severity was uncertain and was infecting millions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Governments were forced to act quickly given further uncertainties in the capacity of their health care systems to deal with the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' In many cases, governments relied on public health experts for their policy, and more broadly to the established mechanisms by which scientific and medical expertise inform government policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' On 11 March of 2020, the World Health Organization (WHO) officially declared COVID a pandemic (Lavezzo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Members, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Many governments subsequently adopted aggressive pandemic policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Examples of these policies, imposed as large-scale restrictions on people, included (Gostin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Jenson 2020, Magness 2021): quarantine (stay-at-home) orders, masking orders in community settings, nighttime curfews, closures of schools, universities and many businesses, and bans on large gatherings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Mathematical modelling studies using simulated pandemic scenarios were used to justify durations of restrictions imposed on people, ranging from 2 weeks to months (CDC 2017, Jenson, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' These restrictions were intended to “flatten the epidemic curve” (Matrajt & Leung, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The term – flatten the epidemic curve – was originally utilized by the US Centers of Disease Control for pandemic planning (CDC, 2007) to warrant use of targeted antiviral medications and nonpharmaceutical interventions (NPIs) to delay and flatten the epidemic peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' A key aspect of flattening the epidemic curve in a pandemic was being able to spread health care demands resulting from a high incidence peak that could potentially overwhelm health care utilization capacity (Jenson, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The restrictions implemented by governments, however, were lengthy as public health official policy targets shifted (Magness 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' In United States, political influence dominated both the initiation and ultimate duration of these restrictions (Kosnik & Bellas, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='2 Research reproducibility The overall research capacity response to COVID since late 2019 has been massive (Kinsella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Ioannidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' To present an estimate of the magnitude of this response, we used the Advanced Search Builder capabilities of freely available PubMed search engine (pubmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='ncbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='nlm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='nih.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='gov/advanced/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We used the terms covid[Title] OR sars-cov- 2[Title] for the period 2020-2023 (search performed November 23, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our search returned 247,597 listings in the National Library of Medicine data base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' As reported in literature, only a small fraction of published research has been judged by others to be reproducible before COVID (Ioannidis, 2005, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Ioannidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Keown, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Iqbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Randall & Welser, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Stodden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Landis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2012) suggest that the inability to reproduce findings is due to a lack of research transparency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Research transparency permits openness of study design, verification of results, synthesis of new findings with previous knowledge, and effective inquiry of research (Munafo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Causes of poor reproducibility of published research are related to aspects of lack of research transparency such as (Ware & Munafo, 2015): biased study designs, flexibility in research practices, low statistical power, and chasing statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' As indicated above, many research studies have been published in response to COVID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' However, there remains concerns about reproducibility of COVID research, particularly where observational data are used to generate results (Bramstedt, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Peng & Hicks, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The current situation of irreproducible research may be that not much has changed during COVID (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', Gustot, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Sumner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Paez, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='3 Meta-analysis Meta-analysis is a systematic procedure for statistically combining data (test statistics) from multiple studies that address a common research question (Egger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2001), for example, whether an intervention (or risk factor) is causal of a health outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' A meta-analysis examines a claim by taking a summary statistic along with a measure of its reliability from multiple individual intervention/risk factor—health outcome studies (called base papers) found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' These statistics are combined to give what is supposed to be a more reliable estimate of an effect (Young & Kindzierski, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' One aspect of replication—the performance of another study statistically confirming the same hypothesis or claim—is a cornerstone of science and replication of research claims is important before causal inference can be made (Moonesinghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' If a replication study result does not conform to a prevailing paradigm, it might not be submitted for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Also, if a similar flawed methodology is used in a replication study as in an original study, or if studies with negative findings are not submitted for publication whereas studies with positive findings are, then a false claim can be canonized (Nissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Meta-analysis has been placed at the top of the medical evidence-based pyramid – above case– control and cohort studies, and randomized trials (Murad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' A key assumption of a meta-analysis is that estimates drawn from the base papers for the analysis are unbiased estimates of the effect of interest (Boos & Stefanski, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Given these attributes, independent evaluation of published meta-analysis on a common research question can be used to assess the reproducibility of a claim coming from that field of research (Young & Kindzierski, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Kindzierski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Young & Kindzierski, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The objective of this study was to use a p-value plotting statistical method (after Schweder & Spjøtvoll, 1982) to independently evaluate specific research claims related to COVID quarantine (stay-at-home) orders in published meta-analysis studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This was done in an attempt to illustrate the importance of reproducibility of research claims arising from this nonpharmaceutical intervention in the context of the surge of COVID papers in literature over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Methods We first wanted to gauge the number of reports of meta-analysis studies in literature related to some aspect of COVID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' To do this we again used the Advanced Search Builder capabilities of the PubMed search engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' On November 20, 2022 we used the terms ((covid[Title]) OR (sars- cov-2[Title]) AND (2020:2023[pdat])) AND (meta-analysis[Title] AND (2020:2023[pdat])).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our search returned 3,204 listings in the National Library of Medicine data base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This included 633 listings for 2020, 1,301 listings for 2021, and 1,270 listings thus far for 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We find these counts astonishing in that a meta-analysis is a summary of available papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Given our understanding of pre-COVID research reproducibility of published literature discussed above, we speculated that there may be numerous meta-analysis studies relating to COVID that are irreproducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We prepared and posted a research plan – Young & Kindzierski (2022b) – on the Researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='One platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This plan can be accessed and downloaded without restrictions from the platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our plan was to use p-value plotting to independently evaluate four selected published meta-analysis studies specifically relating to possible health outcomes of COVID quarantine (stay-at-home) orders – also referred to as ‘lockdowns’ or ‘shelter-in-place’ in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1 Data Sets As stated in our research plan (Young & Kindzierski, 2022b), we considered four meta-analysis studies in our evaluation: • Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) – mortality • Prati & Mancini (2021) – psychological impacts (specifically, mental health symptoms) • Piquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) – reported incidents of domestic violence • Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) – suicidal ideation (thoughts of killing yourself) Electronic copies of each meta-analysis study (and any corresponding electronic supplementary information files) were downloaded from the internet and read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) meta-analysis examined the effect of COVID quarantine (stay-at-home) orders implemented in 2020 on mortality based on available empirical evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' These orders were defined as the imposition of at least one compulsory, non-pharmaceutical intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' initially identified 19,646 records that could potentially address their purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' After three levels of screening by Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 32 studies qualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Of these, estimates from 22 studies could be converted to standardized measures for inclusion in their meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' For our evaluation, we could only consider results for 20 of the 22 studies (data they provided for two studies could not be converted to p-values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “lockdowns in the spring of 2020 had little to no effect on COVID-19 mortality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The Prati & Mancini (2021) meta-analysis examined the psychological impact of COVID quarantine (stay-at-home) orders on the general population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This included: mental health symptoms (such as anxiety and depression), positive psychological functioning (such as well- being and life-satisfaction), and feelings of loneliness and social support as ancillary outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Prati & Mancini initially identified 1,248 separate records that could potentially address their purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' After screening, they identified and assessed 63 studies for eligibility and ultimately considered 25 studies for their meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' For our evaluation, we used all 20 results they reported on for mental health symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “lockdowns do not have uniformly detrimental effects on mental health and most people are psychologically resilient to their effects”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The Piquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis examined the effect of COVID quarantine (stay-at- home) orders on reported incidents of domestic violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' They used the following search terms to identify suitable papers with quantitative data to include in their meta-analysis… “domestic violence”, “intimate partner violence”, or “violence against women”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Piquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' initially identified 22,557 records that could potentially address their purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' After screening, they assessed 132 studies for eligibility and ultimately considered 18 studies in their meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' For our evaluation, we used all 17 results (effect sizes) they reported on from the 18 studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “incidents of domestic violence increased in response to stay-at-home/lockdown orders”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis examined the effect of COVID quarantine (stay-at-home) orders on suicidal ideation and suicide attempts among psychiatric patients in any setting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', home, institution, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' They used the following search terms to identify suitable papers with quantitative data to include in their meta-analysis… “suicide” or “suicide attempt” or “suicidal ideation” or “self-harm”, “psychiatric patients” or “psychiatric illness” or “mental disorders” or “psychiatric hospitalization” or “psychiatric department” or “depressive symptoms” or “obsessive-compulsive disorder”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' initially identified 728 records that could potentially address their purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' After screening, they assessed 83 studies for eligibility and ultimately considered 21 studies in their meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' For our evaluation, we used all 12 results they reported on for suicidal ideation among psychiatric patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “estimated prevalence of suicidal ideation within 12 months [during COVID] was… significantly higher than a world Mental Health Survey conducted by the World Health Organization (WHO) in 21 countries [conducted 2001−2007]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='2 P-value Plots In epidemiology it is traditional to use risk ratios and confidence intervals instead of p-values from a hypothesis test to demonstrate or interpret statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Altman & Bland (2011a,b) show that both confidence intervals and p-values are constructed from the same data and they are inter-changeable, and one can be calculated from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Using JMP statistical software (SAS Institute, Cary, NC), we estimated p-values from risk ratios and confidence intervals for all data in each of the meta-analysis studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' In the case of the Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) meta-analysis, standard error (SE) was presented instead of confidence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Where SE values were not reports, we used the median SE of the other base studies used in the meta-analysis (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The p-values for each meta-analysis are summarized in an Excel file (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='xlsx format) that can be downloaded at our posted Researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='One research plan (Young & Kindzierski, 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We then developed p-value plots after Schweder & Spjøtvoll (1982) to inspect the distribution of the set of p-values for each meta-analysis study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The p-value is a random variable derived from a distribution of the test statistic used to analyze data and to test a null hypothesis (Young & Kindzierski, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' In a well-designed and conducted study, the p-value is distributed uniformly over the interval 0 to 1 regardless of sample size under the null hypothesis (Schweder & Spjøtvoll, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' A distribution of true null hypothesis points plotted against their ranks in a p-value plot should form a 45-degree line when there are no effects (Schweder & Spjøtvoll, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Bordewijk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Researchers can use a p-value plot to assess the heterogeneity of the test statistics combined in meta-analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The p-value plots we constructed were interpreted as follows (Young & Kindzierski, 2022a): • Computed p-values were ordered from smallest to largest and plotted against the integers, 1, 2, 3,… • If p-value points on the plot followed an approximate 45-degree line, we concluded that test statistics resulted from a random (chance) process and the data supported the null hypothesis of no significant association or effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • If p-value points on the plot followed approximately a line with a flat/shallow slope, where most (the majority) of p-values were small (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05), then test statistic data set provided evidence for a real, statistically significant, association or effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • If p-value points on the plot exhibited a bilinear shape (divided into two lines), the data set of test statistics used for meta-analysis is consistent with a two-component mixture and a general (overall) claim is not supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' In addition, a small p-value reported for the overall claim in the meta-analysis may not be valid (Schweder & Spjøtvoll, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Examples of p-value plots are provided in Appendix A after Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) to assist in interpretation of the p-value plots we constructed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Specifically, the p-value plots in Appendix A represent ‘plausible true null’ and ‘plausible true alternative’ hypothesis outcomes based on published meta-analysis studies of observational data sets in the field of environmental epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' As shown in the p-value plots in Appendix A: • A plausible true null hypothesis plots as an approximate 45-degree line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • A plausible true alternative hypothesis plots as a line with a flat/shallow slope, where most (the majority) of p-values are small (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The distribution of the p-value under the alternative hypothesis – where p-values are a measure of evidence against the null hypothesis – is a function of both sample size and the true value or range of true values of the tested parameter (Hung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The p-value plots presented in Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) represent examples of distinct (single) sample distributions for each condition – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', for true null associations and true effects between two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Evidence for p- value plots exhibiting behaviors outside of that shown in Young et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) should initially be treated as ambiguous (uncertain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Results Mortality Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on mortality – the Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) meta-analysis – is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' There are 20 studies that we included in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Six of the 20 studies had p-values below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05 while four of the studies had p-values close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Ten studies fell roughly on a 45-degree line implying random results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This data set comprises mostly null associations (14) and with five or six possible associations with effects (1-in-20 could be chance, false, positive association).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' While not ideal, this data set is a closer fit to a sample distribution for a true null association between two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our interpretation of the p-value plot is that COVID quarantine (stay-at-home) orders are not supported for reducing mortality, consistent with Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' [Fig 1 to be inserted here] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' P-value plot (p-value versus rank) for Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) meta-analysis of the effect of COVID quarantine (stay-at-home) orders implemented in 2020 on mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Symbols (circles) are p-values ordered from smallest to largest (n=20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Psychological impact (mental health symptoms) Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on mental health symptoms – the Prati & Mancini (2021) meta-analysis – is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Figure 2 presents as a bilinear shape showing a two-component mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This data set clearly does not represent a distinct sample distribution for either true null associations or true effects between two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our interpretation of the p-value plot is that COVID quarantine (stay-at-home) orders have an ambiguous (uncertain) effect on mental health symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' However as discussed below, there are valid questions their research claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' [Fig 2 to be inserted here] Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' P-value plot (p-value versus rank) for Prati & Mancini (2021) meta-analysis of the effect of COVID quarantine (stay-at-home) orders on mental health symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Symbols (circles) are p-values ordered from smallest to largest (n=20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Incidents of domestic violence Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on reported incidents of domestic violence – the Piquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis – is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Thirteen of the 17 studies had p-values less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' While not shown in the figure, eight of the p-values were small (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This data set comprises mostly non-null associations (13) and with four possible null associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' While not perfect, this data set is a closer fit to a sample distribution for a true alternative association between two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our interpretation of the p-value plot is that COVID quarantine (stay-at-home) have a negative effect (increase) for reported incidents of domestic violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' [Fig 3 to be inserted here] Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' P-value plot (p-value versus rank) for Piquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis of the effect of COVID quarantine (stay-at-home) orders on reported incidents of domestic violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Symbols (circles) are p-values ordered from smallest to largest (n=17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Suicidal ideation Our independent evaluation of the effect of COVID quarantine (stay-at-home) orders on suicidal ideation – the Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis – is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The p-values for all 12 studies were less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Ten of the 12 studies had p-values less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' While not shown in the figure, eight of the p-values were small (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This data set presents as a distinct sample distribution for true effects between two variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our interpretation of the p-value plot is that COVID quarantine (stay-at-home) orders have an effect on suicidal ideation (thoughts of killing yourself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' However as discussed below, there are valid questions about how the meta-analysis was formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' [Fig 4 to be inserted here] Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' P-value plot (p-value versus rank) for Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis of the effect of COVID quarantine (stay-at-home) orders on suicidal ideation (thoughts of killing yourself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Symbols (circles) are p-values ordered from smallest to largest (n=12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Discussion As stated previously, independent evaluation of published meta-analysis on a common research question can be used to assess the reproducibility of a claim coming from that field of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We evaluated four meta-analysis studies of COVID quarantine (stay-at-home) orders implemented in 2020 and corresponding health benefits and/or harms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our intent was to illustrate the importance of reproducibility of research claims arising from this nonpharmaceutical intervention in the context of the surge of COVID papers in literature over the past few years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Mortality The Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) meta-analysis examined the effect of COVID quarantine orders on mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “lockdowns in the spring of 2020 had little to no effect on COVID-19 mortality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Here, they imply that the intervention (COVID quarantine orders) had little or no effect on reduction of mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The quantitative data Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' present to put their findings into perspective is that they estimated the average lockdown in United States (Europe) in the spring of 2020 avoided 16,000 (23,000) deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' In contrast, they report that there are about 38,000 (72,000) flu deaths occurring each year in the United States (Europe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our evidence agrees with their claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our p-value plot (Figure 1) is not consistent with expected behaviour of a distinct sample distribution for a true effect between the intervention (quarantine) and the outcome (reduction in mortality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' More importantly, our plot shows considerable randomness (many null associations, p-values > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05) supporting no consistent effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' further stated that “costs to society must be compared to the benefits of lockdowns, which our meta-analysis has shown are little to none”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Psychological impact (mental health symptoms) The Prati & Mancini (2021) meta-analysis examined the psychological impact of COVID quarantine orders on the general population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “lockdowns do not have uniformly detrimental effects on mental health and most people are psychologically resilient to their effects”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We evaluated a component of psychological impact – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', whether COVID quarantine orders affect mental health symptoms (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Figure 2 clearly exhibits a two-component mixture implying an ambiguous (uncertain) effect on mental health symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' However, our evidence does not necessarily support their claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Digging deep into their study reveals an interesting finding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their study looked at a variety of psychological symptoms that differed from study to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Although not shown here, when they examined these symptoms separately – a meta-analysis of each symptom – there was a strong signal for anxiety (p-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='0001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This is less than a Boos & Stefanski (2011) proposed p-value action level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='001 for expected replicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Here, the term ‘action level’ means that if a study is replicated, the replication will give a p-value less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We also note that Prati & Mancini appear to take absence of evidence of a negative mental health effect of COVID quarantine orders in their meta-analysis as implying it does not affect mental health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' But absence of evidence does not imply evidence of absence (Altman & Bland, 1995, Alderson, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Sedgwick, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Just because meta-analysis failed to find an effect, it does not imply that “…most people are psychologically resilient to their [lockdown] effects”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' A more plausible and valid inference is that this statement of claim is insufficiently researched at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Incidents of domestic violence The Piquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis examined COVID quarantine orders on reported incidents of domestic violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “incidents of domestic violence increased in response to stay-at-home/lockdown orders”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our evidence suggests agreement with this claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our p-value plot (Figure 3) is more consistent with expected behaviour of a distinct sample distribution for a true effect between the intervention (quarantine) and the outcome (increase in incidents of domestic violence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Several null association studies exist within their data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We note that Figure 3 has 13 of 17 p- values less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='05, with eight of these less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our evidence supports that COVID quarantine orders likely increased incidents of domestic violence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Suicidal ideation The Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis examined COVID quarantine orders on suicidal ideation (thoughts of killing yourself).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Their research claim was that “estimated prevalence of suicidal ideation within 12 months [during COVID] was… significantly higher than a world Mental Health Survey conducted by the World Health Organization (WHO) in 21 countries [conducted 2001−2007]”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The p-value plot (Figure 4) strongly supports their claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The plot is very consistent with expected behaviour of a distinct sample distribution for a true effect between the intervention (quarantine) and the outcome (increased prevalence of suicidal ideation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' However, digging deep into their study reveals a problem in the formulation of their meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' In strong science, a research question being investigated is judged against a control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' effectively ignores controls in their meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' They compared incidence of suicidal ideation against a zero standard and not to control groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Specifically, the pre-COVID (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', background) suicidal ideation signal is ignored in their meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Indeed, in their Table 1 they present results from the base papers where data for control groups is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' For example, the Seifert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) base paper notes suicidal ideation presented in 123 of 374 patients in the psychiatric emergency department of Hannover Medical School during the pandemic, and 141 of 476 in the same department before the pandemic – 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='9%versus 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The difference is not significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Comparing their Table 1 data set with their Figure 1 forest plot, Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' only carried 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='9% into their meta-analysis, in effect ignoring the control data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' It is the same situation with all data set entries in their Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' only considered pandemic incidence in their meta-analysis, and they ignored any control data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' How they formulated their work calls their claims into serious question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' We conclude that the Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' results are unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Implications COVID quarantine orders were implemented on the notion that this nonpharmaceutical intervention would delay and flatten the epidemic peak and benefit public health outcomes overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Three of the four meta-analyses that we evaluated raise questions about public health benefits/risks of this form of nonpharmaceutical intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The fourth meta-analysis study is unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' One meta-analysis that we evaluated – Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) – questions the benefits of this form of intervention for preventing mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our p-value plot supports their finding that COVID quarantine orders had little or no effect on reduction of mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' A second meta-analysis – Prati & Mancini (2021) assessment of mental health symptoms – offers confounding evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our p-value plot clearly exhibits a two-component mixture implying an ambiguous (uncertain) effect between COVID quarantine orders and mental health symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' However, data for a component of mental health symptoms (anxiety) suggests a negative effect from COVID quarantine orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Further, Prati & Mancini (2021) lack evidence to claim that “…most people are psychologically resilient to their [lockdown] effects”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our evaluation of the Piquero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) meta-analysis – assessment of domestic violence incidents – supports a true effect between the intervention (quarantine) and the outcome (increase in incidents of domestic violence) with additional confirmatory research needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Finally, the meta-analysis of Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2021) on suicidal ideation (thoughts of killing yourself) is wrongly formulated and should be disregarded until or unless controls are included in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Standing back and looking at the overall findings of these studies, benefits of COVID quarantine orders remain uncertain and risks (negative public health consequences) of this intervention cannot be ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Given that the base studies and the meta-analyses themselves were, for the most part, rapidly conducted and published, we acknowledge that confirmatory research for some of the outcomes investigated is warranted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Our interpretation of COVID quarantine benefits/risks is consistent, for example, with earlier research of James (2020) and conventional wisdom, Inglesby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' James takes a position that is it unclear whether there were benefits from this intervention relative to less restrictive measures aimed at controlling “risky” personal interactions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', mass gatherings and large clusters of individuals in enclosed spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' James (2020) also notes numerous economic and public health harms in the United States as May 1, 2020: • Over 20 million newly unemployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • State-wide school closures across the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • Increased spouse and child abuse reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • Increased divorces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • Increased backlog of patient needs for mental health services, cancer treatments, dialysis treatments and everyday visits for routine care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' • Increased acute emergency services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' This is consistent with interim quantitative data as of September 2020 presented by the American Institute of Economic Research (2020) on the cost and negative public health implications of pandemic restrictions in United States and around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Acknowledgments No external funding was provided for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' The study was conceived based on previous work undertaken by CG Stat for the National Association of Scholars (nas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org), New York, NY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' References Alderson, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Absence of evidence is not evidence of absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' British Medical Journal, 328(7438), 476.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Absence of evidence is not evidence of absence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' British Medical Journal, 311(7003), 485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1136/bmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='7003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='485 Altman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', & Bland, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2011a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' How to obtain a confidence interval from a P value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' British Medical Journal, 343, d2090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1136/bmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='d2090 Altman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', & Bland, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2011b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' How to obtain the P value from a confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' British Medical Journal, 343, d2304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1136/bmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='d2304 American Institute of Economic Research (AIER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Cost of Lockdowns: A Preliminary Report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' AIER, Great Barrington, MA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='aier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/article/cost-of-us-lockdowns-a- preliminary-report/ Boos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', & Stefanski, L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1111/add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='12673 Young, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', Cheng, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Reliability of a meta-analysis of air quality−asthma cohort studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' International Journal of Statistics and Probability, 11(2), 61−76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='5539/ijspv11n2p61 Young, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', & Kindzierski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Evaluation of a meta-analysis of air quality and heart attacks, a case study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Critical Reviews in Toxicology, 49(1), 85–94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1080/10408444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1576587 Young, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', & Kindzierski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Statistical reliability of a diet-disease association meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' International Journal of Statistics and Probability, 11(3), 40–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='5539/ijsp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='v11n3p40 Young, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', & Kindzierski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Research Plan Lockdowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='One.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='one/articles/22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='00005v1 Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=', & Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Suicidal ideation and suicide attempts in psychiatric patients during the COVID-19: A systematic review and meta-analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Psychiatry Research, 317, 114837.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='psychres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='114837 Figures Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' P-value plot (p-value versus rank) for Herby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' (2022) meta-analysis of the effect of COVID quarantine (stay-at-home) orders implemented in 2020 on mortality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Symbols (circles) are p-values ordered from smallest to largest (n=20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' P-value plot (p-value versus rank) for Prati & Mancini (2021) meta-analysis of the effect of COVID quarantine (stay-at-home) orders on mental health symptoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' Symbols (circles) are p-values ordered from smallest to largest (n=20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='8 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='6 p-value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FKT4oBgHgl3EQfTC3B/content/2301.11778v1.pdf'} +page_content='3 0.' metadata={'source': 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@@ +version https://git-lfs.github.com/spec/v1 +oid sha256:39703860509f73df2e123c7ab4d6270c9d923c6a95b222321daf374376de510b +size 2031661 diff --git a/5dE4T4oBgHgl3EQfBQvo/content/tmp_files/2301.04851v1.pdf.txt b/5dE4T4oBgHgl3EQfBQvo/content/tmp_files/2301.04851v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..15b6606555c142b3abe036f928baef0046d35582 --- /dev/null +++ b/5dE4T4oBgHgl3EQfBQvo/content/tmp_files/2301.04851v1.pdf.txt @@ -0,0 +1,884 @@ +How different of shadows of compact objects with and without +horizons? +Xiangyu Wang1, Yehui Hou2, Minyong Guo1∗ +1 Department of Physics, Beijing Normal University, Beijing 100875, P. R. China +2Department of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871, P.R. China +Abstract +In this work, we theoretically assume that a compact object (CO) can have a dark surface so +that the CO is simplified to have no emissions and reflections. Considering that the radius of the +surface can be located inside or outside the photon region, which is closely related to the shadow +curve, we investigate if a CO without an event horizon could produce shadow structures similar +to black holes and figure out how different of shadows of COs with and without horizons. In +particular, by introducing the (possible) observational photon region, we analytically construct +an exact correspondence between the shadow curves with the impact parameters of photons and +find that there are indeed several differences for shadows of COs without horizons and black +holes. +More precisely, We found the shadow curve is still determined by the photon region +when the radius of the surface is small enough to retain a whole photon region outside the shell. +When only part of the photon region remains, the shadow curve is partially determined by the +photon region, and the remaining portion of the shadow curve is partly controlled by the impact +parameters of photons which has a turning point on the surface. When there’s no photon region +outside the surface, the shadow curve is totally controlled by the impact parameters of photons +which has a turning point on the surface. +∗ Corresponding author: minyongguo@bnu.edu.cn +1 +arXiv:2301.04851v1 [gr-qc] 12 Jan 2023 + +1 +Introduction +It is known that due to the strong gravitational field around a black hole, lights have to bend +and form a central dark area in the view of distant observers, dubbed as the black hole shadow. +When it comes to black hole shadows, one of the most apparent features might be the so-called +shadow curve (also referred to as the critical curve in literature [1, 2]). And in most cases, we know +that the shadow curve is closely related to the photon region, which is composed of the spherical +photon orbits 1, even though the essence of a black hole shadow is the existence of an event horizon +that can capture photons with specific impact parameters. +In recent years, the central depression of the emission has been found in the black hole images +photographed by the Event Horizon Telescope (EHT) [6–12]. +There have been many exciting +works on shadows in terms of the EHT [13–51], among which some papers investigated whether +some specific compact objects (COs) without horizons could mimic the black hole shadows [45–51], +that is if the shadow is a sufficient condition for the existence of an event horizon. Along this +line, previous studies mainly focused on the boson stars, which have no hard emitting surface. +Considering that boson stars are illuminated by the around accretion flows which have a cut-off +in the luminance at the inner edge of the accretion disk, the authors have numerically found that +some boson stars, especially Proca stars, could produce images including shadow structures similar +to black holes. +In our work, we would like to consider a CO with a surface and theoretically investigate how +different of shadows of COs with and without horizons are. For simplicity, we focus on a model with +two ideal assumptions. Compared with the luminous accretion flows or other light sources in the +background, we first assume the CO is a non-luminous body; that is, the surface of the CO has no +emissions. Secondly, we take the CO somehow as a dark star so that few lights can reflect from the +surface of the CO. Thus; the reflections can be omitted. In short, in our simplified model, the CO +doesn’t transmit and reflect lights and behaves like an event horizon effectively. However, compared +to a black hole, the radius of the surface of the CO can be chosen arbitrarily while the event horizon +is fixed. Moreover, since the radius of the surface is not fixed, there might be no photon region, or +only part of the photon region remains outside the surface of the CO. As we know, the black hole +shadow curve is usually determined by the photon region. Thus, it’s fascinating to theoretically +study the shadow structures of the CO in our model. In addition, to describe the spacetime outside +1The spherical photon orbits are usually defined by r = const in a stationary and axisymmetric spacetime, where r +is the radial coordinate. In a curved spacetime as a radial parameter, r = const generally does not imply the spherical +meaning in flat space. A more strict definition can be found in [3], where authors introduced a new terminology: the +fundamental photon orbits. Some related works concerned with fundamental photon orbits can be seen in [4, 5]. +2 + +the CO, we will employ the Painlev´e-Gullstrand form of the Lense-Thirring spacetime proposed +recently in [52]. +The remaining parts of this paper are organized as follows in sec. 2, we review the Painlev´e- +Gullstrand form of the Lense-Thirring spacetime and discuss the geodesics in sec. 3, we introduce +the (possible) observational photon region and have a detailed study of the shadow curves for COs +with and without horizons. The main conclusions are summarized in sec. 4. In this work, we have +set the fundamental constants c and G, and we will work in the signature convention (−, +, +, +) +for the spacetime metric. +2 +Painlev´e-Gullstrand form of the Lense-Thirring spacetime +Since we shall use the Lense-Thirring metric to model a horizonless CO, we would like to review +the Lense-Thirring spacetime. +2.1 +Metric +In 1918, Lense and Tirring put forward an approximate solution to describe a slow rotating +large-distance stationary isolated body in the framework of the vacuum Einstein equations [53], +which takes +ds2 = +− +� +1 − 2M +r ++ O +� 1 +r2 +�� +dt2 − +�4J sin2 θ +r ++ O +� 1 +r2 +�� +dφdt ++ +� +1 + 2M +r ++ O +� 1 +r2 +�� � +dr2 + r2 � +dθ2 + sin2 θdφ2�� +, +(2.1) +where M and J are the mass and the angular momentum, respectively. +And O(r−2) denotes +the sub-dominant terms. By exquisitely regulating the specific forms of O(r−2), one can obtain +various metrics with the same asymptotic limit at large distances, which are physically different +from each other. Recently, Baines et al. constructed an explicit Painlev´e-Gullstrand variant of the +Lense–Thirring spacetime [52], whose metric reads +ds2 = −dt2 + +� +dr + +� +2M +r dt +�2 ++ r2 +� +dθ2 + sin2 θ +� +dφ − 2J +r3 dt +�2� +. +(2.2) +There are three solid advantages for this new version of the Lense–Thirring spacetime, of which the +first one is that the metric reduces to the Painlev´e–Gullstrand version of the Schwarzschild black +hole solution when J = 0; The second is that the azimuthal dependence takes in partial Painlev´e- +Gullstrand form, that is, gφφ(dφ − vφdt)2 = gφφ(dφ − ωdt)2, where vφ is minus the azimuthal +component of the shift vector in the ADM formalism denoting the “ flow ” of the space in the +3 + +azimuthal direction and ω = gtφ/gφφ is the angular velocity of the spacetime; The third is that +all the spatial dependence is in exact Painlev´e–Gullstrand type form which implies the spatial +hypersurface t = const is flat. These exciting features make the Painlev´e-Gullstrand variant much +easier to calculate the tetrads, curvature components, and the analysis of geodesics than any other +variant of the Lense–Thirring spacetime [54, 55]. +On the other hand, from the original asymptotic form in Eq. +(2.1), we can see that the +Lense–Thirring metric should only make sense in the region r > rs, where we use rs to repre- +sent the surface radius of the slow rotating isolated body. Note that the metric in Eq. (2.1) has a +coordinate singularity r = 2M when neglecting the sub-dominant terms so that the Lense-Thirring +spacetime should be valid when the condition rs > 2M holds. Moreover, for a slowly rotating +object, we must have J/r2 +s ≪ 1. Thus, we should also impose the conditions J/r2 +s ≪ 1, rs > 2M on +the Painlev´e–Gullstrand version of the Lense-Thirring spacetime when investigating the properties +of the Painlev´e–Gullstrand form. +2.2 +Geodesics +In this subsection, we would like to review the geodesics in the Painlev´e-Gullstrand form of the +Lense-Thirring spacetime, which has been carefully studied in [55]. Similar to the Kerr spacetime, +there are also four conserved quantities along the geodesics of free particles: the mass m, the energy +E, the axial angular momentum L, and the Carter constant C. For simplicity and without loss of +generality, we set m = 0 for photons and m = 1 for timelike particles. Then, the four-momentum +pa reads +pa = ˙t +� ∂ +∂t +�a ++ ˙r +� ∂ +∂r +�a ++ ˙θ +� ∂ +∂θ +�a ++ ˙φ +� ∂ +∂φ +�a +, +(2.3) +with “ ˙ ” denoting the derivative with respect to the affine parameter τ. Considering papa = 0 +for photons and papa = −1 for timelike particles, τ can be seen as the proper time for timelike +worldlines. Then the conserved quantities E, L, C can be written out +E += −pt = +� +1 − 2M +r +− 4J2 sin2 θ +r4 +� +˙t − +� +2M +r +˙r + 2J sin2 θ +r +˙φ , +L += pφ = r2 sin2 θ +� +˙φ − 2J +r3 ˙t +� +, +C = r4 ˙θ2 + +L2 +sin2 θ , +(2.4) +explicitly. Note that for timelike particles, E and L can now be treated as the energy per unit mass +and the angular momentum per unit mass. Then combining with the condition −papa = m ∈ {0, 1}, +4 + +one can obtain the exact expressions of the components of the four-momentum pa as follows +˙r += +Sr +� +R(r) , +˙t += +E − 2JL/r3 + Sr +� +(2M/r)R(r) +(1 − 2M/r) +, +˙θ += +Sθ +� +Θ(θ) +r2 +, +˙φ += +L +r2 sin2 θ + 2J E − 2JL/r3 + Sφ +� +(2M/r)R(r) +r3(1 − 2M/r) +, +(2.5) +where we define +R(r) += +� +E − 2JL +r3 +�2 +− +� +m + C +r2 +� � +1 − 2M +r +� +, +(2.6) +Θ(θ) += +C − +L2 +sin2 θ , +(2.7) +as the effective potential functions governing the radial and polar motions, and +Sr += +� ++1 outgoing geodesic +−1 ingoing geodesic +; +Sθ += +� ++1 incerasing declination geodesic +−1 decerasing declination geodesic +; +Sφ += +� ++1 prograde geodesic +−1 retrograde geodesic +; +(2.8) +following the conventions in [55]. The context for each equation in Eq. (2.8) denotes the corre- +sponding physical interpretation. Here we would like to stress that Sr and Sφ appear separately in +the t-motion and φ-motion due to the Painlev´e-Gullstrand form, however, for geodesic equations +of Kerr spacetime in Boyer-Lindquist coordinates, Sr only comes up in the radial motion, and Sφ +is not necessarily introduced. Then one can explore the properties of null and timeslike geodesics +by adequately manipulating the equations in (2.5). +3 +Observational photon region and shadow curve +This section focuses on the photon region and shadow curve in the Painlev´e-Gullstrand form of +the Lense-Thirring spacetime. Considering the null orbits are independent of photon energies, it’s +convenient to introduce the impact parameters +ξ = L +E , +η = C − L2 +E2 +. +(3.1) +5 + +to characterize the photon orbits. The conditions can determine the photon region +R(r) = ∂rR(r) = 0 , +(3.2) +which gives us the expressions of the impact parameters in terms of the radius, +˜ξ += +−3M ˜r3 + ˜r4 +2J(3M − 2˜r) , +˜η += +− ˜r3[˜r3(˜r − 3M)2 + 36J2(˜r − 2M)] +4J2(3M − 2˜r)2 +. +(3.3) +Note that we use ˜r to denote the radius of the photon orbit in the photon region, and ˜ξ, ˜η are the +corresponding impact parameters. Furthermore, from ˜η = 0 we can obtain two roots rp− < rp+ in +the region ˜r > 2M which implies the radial range of the photon region is +˜r ∈ [rp−, rp+] . +(3.4) +Note that rp± cannot be analytically given in general; however, when J → 0, one can find [55] +rp± = 3M ± +2J +√ +3M + O(J2) . +(3.5) +Considering rs > 2M for COs, in the light of rp± we would like to divide the range of rs into three +parts, that is, (1) 2M < rs < rp−, (2) rs > rp+, (3) rp− < rs < rp+, and study the shadow curve +for each case. +3.1 +Review of black hole shadows +Before we talk about the shadows of COs, we first review the shadows of ordinary black holes. +To determine the shadow of a black hole, in addition to the photon region, there is a second +condition related to the observational angle. For a certain observational angle θo, we can see that +the term under the square root Θ(θo) ≥ 0 must be satisfied in the polar motion, which gives +Θ(θo) = ηo − +ξ2 +o +sin2 θo +≥ 0 , +(3.6) +and a new function ηo(ξo) = +ξ2 +o +sin2 θo . That is to say, and the photons could reach the observer if their +impact parameters satisfy the above condition. Combing the critical impact parameters ˜η(˜ξ) with +the constraint Θ(θo) ≥ 0, one can exactly fix the photons which have critical impact parameters +and can escape to observers if they are perturbed. As a result, the shadow curve is formed by these +photons since the surface of the black hole, that is, the horizon, is always inside the photon region. +In the study of shadows of COs, including black holes, we find it convenient to define the +observational photon region (OPR) and possible observational photon region (POPR). The OPR +6 + +Figure 1: An illustration of the observational photon region for a black hole in the ξOη plane is +shown in the left panel. The right panel is borrowed from the Fig. 11 of our previous work [56], +which presents the celestial coordinates (Θ, Ψ) and standard Cartesian coordinates (x, y) in the +local rest frame of observers. +is defined as the set of impact parameters that the photons with these impact parameters precisely +determine the shadow curve for observers with a certain observational angle. And the POPR has +defined as the union of the OPRs at all possible observational angles. Thus, for the case of black +holes, the POPR is composed of the critical impact parameters ˜η(˜ξ) and the elements of the OPR +are the critical impact parameters ˜η(˜ξ) which also satisfy the condition Θ(θo) ≥ 0. In the left panel +of the Fig. 1, we show the functions of ˜η(˜ξ) and ηo(ξo) in the ξOη plane and find that the two +functions have two intersections. The OPR corresponds to the segment of ˜η(˜ξ) between the two +intersections, and the POPR corresponds to a piece of ˜η(˜ξ) above the ξ-axis. +Then one can calculate the shadow curve by standard methods, that is, introducing the celestial +coordinates and obtaining the projections on the screen of observers. In this work, we employ the +stereographic projection method, which has been used in our previous work [56]. We also bring +the Fig. 11 in work [56] to the right panel of the Fig. 1 to give a deep intuition on the celestial +coordinates and Cartesian coordinates (x, y) in the local rest frame of observers. +In terms of the metric in Eq. (2.2), the local rest frame of observers can be defined as +e0 += +ˆe(t) = ∂t − +� +2M +r ∂r + 2J +r3 ∂φ , +(3.7) +e1 += +−ˆe(r) = −∂r , +(3.8) +e2 += +ˆe(θ) = 1 +r∂θ , +(3.9) +e3 += +−ˆe(φ) = − +1 +r sin θ∂φ . +(3.10) +7 + +x +n。(。) +i() +(t)a- = Ta +0 ++ +s +e3 = -e(Φ) +(+di)? +0 +(rp-) +P +e2 += +()aIt is not hard to verify that these bases are normalized and orthogonal to each other. Moreover, +since ˆe(t) · ∂φ = 0, the observer with the 4-velocity ˆu = e0 in this local rest frame has zero angular +momentum for infinity. So this frame is usually called the ZAMO reference frame. In our model, +the relation between the celestial coordinates (Θ, Ψ) and the 4-momentum of the OPR takes +Θ = arccos +�� +2M +r0 ++ +˙˜ro +˙˜to +� +, +Ψ = − arctan +� +� +˜ξ +� +˜η csc2 θo − ˜ξ2 +� +� , +(3.11) +where “ ∼ ” denotes evaluated with critical impact parameters ˜ξ and ˜η, and the subscript “ o +” means evaluated at the observer with coordinates (0, ro, θo, 0). Then the Cartesian coordinates +(x, y) on the screen can be defined as +x = −2 tan Θ +2 sin Ψ , +y = −2 tan Θ +2 cos Ψ , +(3.12) +where we have chosen the energy of the photon observed by the ZAMOs to be unity, considering +the trajectories of photons are independent of the energies. +3.2 +Shadows of COs without horizons +In this subsection, we study the shadows of COs, which have no horizon. For simplicity, we +assume the COs are non-luminous bodies, and they neither transmit nor reflect light. Recall that +the spacetime outside a CO we consider in this work is modeled by the Painlev´e-Gullstrand form +of the Lense-Thirring spacetime, and we would like to investigate the shadows in three situations, +(1) 2M < rs < rp−, (2) rs > rp+, (3) rp− < rs < rp+. +-15 +-10 +-5 +5 +ξ +20 +40 +60 +80 +η +-6 +-4 +-2 +2 +4 +5 +10 +15 +20 +25 +30 +-6 +-4 +-2 +2 +4 +6 +ξ +5 +10 +15 +20 +25 +30 +η +ξ +(rs)) +(ξ˜(rs)， +η +˜ +rs=3.01 +rs=2.24 +rs=3.92 +η +Figure 2: Plots of the functions ˜η(˜ξ), ηs(ξs) and ηo(ξo) in the ξOη plane for rs = 2.24, rs = 3.01 +and rs = 3.92 with M = 1 and J = 0.5. In each plot, ˜η(˜ξ) is shown in the dashed line, ηs(ξs) is +shown in the solid line with downward opening, ηo(ξo) with θo = 17◦ is given by the green line and +ηo(ξo) with θo = 80◦ is given by the purple line. In addition, the POPR is shown in the red line in +each plot, while the blue one has no contribution to the shadow curve. +As mentioned above, the shadow would be clear if we find the corresponding OPR. Thus, the +main task is to look for the OPR for each case. Since the CO is regarded as a dark body in our +8 + +work, the effect on lights is equivalent to the event horizon of a black hole; that is, the photons +cannot go back if they meet the surface of the CO. As a result, the ingoing photons, which have +two turning points in the radial motion, cannot escape to infinity if the outer turning point is inside +the surface of the CO. Thus, if rs is not less than ˜rp−, the part of the photon region inside the +surface of the CO would have no contributions to the POPR. More precisely, from R(rs) = 0, we +can obtain a new relation between ξs and ηs as follows +ηs = −(rs − 2Jξs)2 +(2M − rs)r3s +− ξ2 +s , +(3.13) +where the subscript “ s ” denotes evaluated at r = rs. Considering the radius of the surfacers could +be the inner or outer turning point which corresponds to different values of (ξs, ηs), ηs(ξs) would +become the new critical parameters when rs > ˜r, where ˜r is the radius of the photon region with +˜η(˜ξ). In Fig. 2, we give examples of ˜η(˜ξ), ηs(ξs) and ηo(ξo) for three cases at the observational +angles θo = 17◦ and θ = 80◦ with the mass and the angular momentum of the CO chosen as M = 1 +and J = 0.5 here and after this. By numerically solving the equation ˜η = 0, we find +rp− ≃ 2.47 , +rp+ ≃ 3.56 . +(3.14) +rs +rs +rs =2.24 +-0.04 +-0.02 +0.00 +0.02 +0.04 +-0.04 +-0.02 +0.00 +0.02 +0.04 +x +y +-0.04 +-0.02 +0.00 +0.02 +0.04 +-0.04 +-0.02 +0.00 +0.02 +0.04 +x +y +θO=17° +θO=80° +=3.01 +=3.92 +Figure 3: Plots of shadow curves of COs. In the left plot, we set θo = 17◦, and in the right one, we +set θo = 80◦. In both plots, the green, blue and red lines denote the shadow curves with rs = 2.24, +rs = 3.01, and rs = 3.92, respectively. +In addition, implying R = ∂rR = ∂2 +rR = 0 for prograde timelike particles, we can find the +radius of the innermost stable circular orbit rI ≃ 4.29. Considering the horizon is at rh = 2, we set +rs = rh+rp− +2 +≃ 2.24 < rp−, rp− < rs = rp−+rp+ +2 +≃ 3.01 < rp+ and rs = rp++rI +2 +≃ 3.92 > rp+ for the +9 + +plots from left to right in Fig. 2. In addition, for each plot, the dashed line denotes ˜η(˜ξ), the other +curve with a downward opening indicated by a solid line denotes ηs(ξs), the curve with an upward +opening drawn in green is ηo(ξo) with θo = 17◦, and the other curve with an upward opening drawn +in purple is ηo(ξo) with θo = 80◦. For the middle plot in Fig. 2 with rp− < rs < rp+, there is an +intersection point (˜ξ(rs), ˜η(rs)) of ˜η(˜ξ) and ηs(ξs) which means the two turning points of photons +coincide with the radius r = rs. When ξ > ˜ξ(rs), we can find that rs is the outer turning point of +R(rs) = 0 and rs > ˜r. On the contrary, when ξ < ˜ξ(rs), we find that rs is the inner turning point +of R(rs) = 0 and rs < ˜r. Therefore, the red line is the POPR. And the impact parameters that +are not in POPR are shown in blue. Moreover, combined with the condition from the observer at +θo = 17◦ (θo = 80◦), the POR is the segment of the red line between the intersections of the red +and green (purple) lines. For the left plot in Fig. 2 with rs < rp−, we can see that the POPR is +still determined by ˜η(˜ξ), which is the same as that in a black hole spacetime since the surface of +the CO is always hidden in the photon region. And the OPR is the segment of ˜η(˜ξ) between the +intersections of the red line ˜η(˜ξ) and the green line ηo(ξo). While for the right plot in Fig. 2 with +rs > rp+, we can see that the POPR is determined by the solid line ηs(ξs), since the photon region +is completely encapsulated by the surface of the CO. And the OPR now is given by the segment of +the red line ηs(ξs) between the intersections of ηs(ξs) and ηo(ξo). +y +ymin +xmin +max +x max +O +x c +Figure 4: An illustration of the coordinates of the points at which the shadow curve intersects the +two axes on the screen. +Then the shadows of COs without horizons can be calculated with the help of Eqs. (3.11) +and (3.12). In Fig. 3, we show the shadow curves with dashed lines at θo = 17◦ for the left plot +10 + +and θ = 80◦ for the right plot. The red, blue and green lines correspond to rs = 3.92 > rp+, +rp− < rs = 3.01 < rp+ and rs = 2.24 < rp−, respectively. +As we have discussed above, the +shadow curve is exactly determined by the OPR, and note that in the Fig. 2, the dashed line in +each plot denotes the same photon region, that is, ˜η(˜ξ), and thus the segment of ˜η(˜ξ) between the +intersections of ˜η(˜ξ) and ηo(ξo) keeps invariable in three plots. As a result, we can find that for the +case of θo = 17◦, the blue line and the green line almost coincide in Fig. 3, since from the middle +plot in Fig. 2 one can see that the OPR with rs = 3.01 coincides with the OPR with rs = 2.24 when +ξ < ˜ξ(rs), and only has a tiny difference with the OPR with rs = 2.24 when ξ > ˜ξ(rs). Similarly, +the difference between the red and the green lines in the case of θo = 17◦ is visible in Fig. 3, since +one can see the difference of their OPRs is evident from the right plot in Fig. 2. Moreover, from +the right plot in Fig. 3, we can see that the difference between the green and blue lines becomes +significant on the right, and the three lines are very close in the left part. The reason can be easily +found in the Fig. 2 where the opening of the parabola ηo(ξo) gets bigger when θo goes from 17◦ +to 80◦. Furthermore, in the middle plot of Fig. 2, one can find that the difference of the OPRs +becomes larger at θo = 80◦, and in the right plot of Fig. 2, the red and blue lines intersect very +closely with the purple line since rs = 3.92 is near rp+ = 3.56. +Therefore, qualitatively we can conclude that when rs < rp−, the shadow of the CO is the same +as that of the black hole; when rp− < rs < rp+, the shadow of the CO is bigger than that of the +black hole, and the shadow of the CO becomes a litter bigger as θo increases from 0◦ to 90◦ with +parts of the shadow curves overlapped; and when rs > rp+ the shadow of the CO would become +larger significantly, and each point of the CO shadow curve is outside the corresponding end of the +black hole shadow curve. +3.3 +Quantitative study of the variation of the CO shadow +In this subsection, we would like to give a quantitative study of the variation of the shadow +concerning the radius of the surface of a CO. Following the work [57, 58], we use the average radius +¯R as the characteristic length of a shadow. +In Fig. 4, we give a diagram to show the coordinates of points at which the shadow curve inter- +sects two axes. O is the origin of the Cartesian coordinates on the screen. Considering the Z2 sym- +metry of the spacetime, the center of the shadow can be defined as +� +xc = xmin+xmax +2 +, ymin+ymax +2 += 0 +� +. +Then let (xc, 0) be the center, we can introduce polar coordinates (R, ψ) with R = +� +(x − xc)2 + y2. +And the parameter ¯R can be defined as +¯R = +� 2π +0 +R(ψ) +2π dψ , +(3.15) +11 + +2 +3 +4 +5 +6 +7 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +θo=80° +θo=17° +σ +rs +Figure 5: The variation of the dimensionless parameter σ = ¯R/ ¯R0 −1 of the CO shadow concerning +the radius of the surface of the CO. In the plot, we set rs = 2.07 + 0.4(i − 1), where i = 1, 2, . . . , 14 +for each point. +which denotes the average radius of the shadow curve. It is convenient to introduce a dimensionless +parameter +σ = +¯R +¯R0 +− 1 , +(3.16) +where we use ¯R0 to represent the average radius of the shadow curve when rh < rs < rp−. In +Fig. 5, we show the variation of σ concerning the radius of the CO surface, where we fix M = 1, +J = 0.5 and set rs = 2.07 + 0.4(i − 1) with i = 1, 2, . . . , 14. We can find that the average radius of +the shadow curve increases slowly as the radius of the CO surface increases from rp− to rp+, the +main reason is that rp+ − rp− = 1.09 is small. When rs > rp+, the average radius of the shadow +curve increases quickly as the radius of the CO surface increases, and the change is almost linear. +In addition, we can see that the average radius of the shadow curve at θo = 80◦ is always larger +than that at θo = 17◦ for a fixed rs in the range rs > rp− which agrees well with our analysis in +the last subsection. +4 +Summary +In this work, we studied the problem of how different of shadows of COs with and without +horizons. +For simplicity, the CO was considered not to emit or reflect any light compared to +other luminous sources in the background of the CO. In addition, we assumed that the CO is a +slowly rotating object such that the spacetime outside the surface of the CO can be described by +the Painlev´e-Gullstrand form of the Lense-Thirring metric. In terms of the photon region with +rp− ≤ ˜r ≤ rp+, we investigated three cases, that is, the radius rs of the CO is smaller than rp−, +12 + +rp− < rs < rp+ and rs > rp+. To obtain the shadow curve for different cases, we introduced OPR +and POPR in Sec. 3.1 to construct a clear correspondence between the shadow curve and the +impact parameters. Moreover, we recognized a new class of critical impact parameters ηs(ξs), with +which the photons have a turning point at rs. After a detailed analysis of the OPRs and POPRs for +COs with different rs, we found the POPR governed by the photon region ˜η(˜ξ), which is the same +as that for black holes when rh < rs < rp−, one part of the POPR is governed by the photon region +˜η(˜ξ) and the other part is controlled by ηs(ξs) when rp− < rs < rp+, and the POPR is completely +controlled by the ηs(ξs) when rs > rp+. As a result, compared with the shadow curve of a black +hole, we found that the shadow curve of a CO doesn’t change for rh < rs < rp−, partially changes +for rp− < rs < rp+ and completely changes for rs > rp+. We also gave a quantitative study on the +variation of the shadow curve concerning rs, and found the average radius of the shadow curve gets +bigger slowly when rs goes from rp− to rp+ and very quickly when rs increases after rp+. +Our results indicate that a CO with or without a horizon is not distinguished by the shadow +curve when it has a whole photon region outside its surface. +A CO without a horizon can be +distinguished from a black hole when the photon region is partially or entirely hidden in the surface +of the CO; that is to say, in this case, the EHT can be used to determine whether a CO has an +event horizon if the resolution reaches high enough. Although in the present work, our discussion +is based on an approximate metric, it seems our results should not depend on a specific metric but +reflect a universal property for a CO. 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Chen, “Shadows of Kerr black holes with a +Gaussian-distributed plasma in the polar direction,” arXiv:2206.04430 [gr-qc]. +18 + diff --git a/5dE4T4oBgHgl3EQfBQvo/content/tmp_files/load_file.txt b/5dE4T4oBgHgl3EQfBQvo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d2ac0b6c6efe39a9a5d57ebf4293f2a21e3a264f --- /dev/null +++ b/5dE4T4oBgHgl3EQfBQvo/content/tmp_files/load_file.txt @@ -0,0 +1,840 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf,len=839 +page_content='How different of shadows of compact objects with and without horizons?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Xiangyu Wang1, Yehui Hou2, Minyong Guo1∗ 1 Department of Physics, Beijing Normal University, Beijing 100875, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' China 2Department of Physics, Peking University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5 Yiheyuan Rd, Beijing 100871, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' China Abstract In this work, we theoretically assume that a compact object (CO) can have a dark surface so that the CO is simplified to have no emissions and reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Considering that the radius of the surface can be located inside or outside the photon region, which is closely related to the shadow curve, we investigate if a CO without an event horizon could produce shadow structures similar to black holes and figure out how different of shadows of COs with and without horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In particular, by introducing the (possible) observational photon region, we analytically construct an exact correspondence between the shadow curves with the impact parameters of photons and find that there are indeed several differences for shadows of COs without horizons and black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' More precisely, We found the shadow curve is still determined by the photon region when the radius of the surface is small enough to retain a whole photon region outside the shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' When only part of the photon region remains, the shadow curve is partially determined by the photon region, and the remaining portion of the shadow curve is partly controlled by the impact parameters of photons which has a turning point on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' When there’s no photon region outside the surface, the shadow curve is totally controlled by the impact parameters of photons which has a turning point on the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' ∗ Corresponding author: minyongguo@bnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='cn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04851v1 [gr-qc] 12 Jan 2023 1 Introduction It is known that due to the strong gravitational field around a black hole, lights have to bend and form a central dark area in the view of distant observers, dubbed as the black hole shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' When it comes to black hole shadows, one of the most apparent features might be the so-called shadow curve (also referred to as the critical curve in literature [1, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' And in most cases, we know that the shadow curve is closely related to the photon region, which is composed of the spherical photon orbits 1, even though the essence of a black hole shadow is the existence of an event horizon that can capture photons with specific impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In recent years, the central depression of the emission has been found in the black hole images photographed by the Event Horizon Telescope (EHT) [6–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' There have been many exciting works on shadows in terms of the EHT [13–51], among which some papers investigated whether some specific compact objects (COs) without horizons could mimic the black hole shadows [45–51], that is if the shadow is a sufficient condition for the existence of an event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Along this line, previous studies mainly focused on the boson stars, which have no hard emitting surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Considering that boson stars are illuminated by the around accretion flows which have a cut-off in the luminance at the inner edge of the accretion disk, the authors have numerically found that some boson stars, especially Proca stars, could produce images including shadow structures similar to black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In our work, we would like to consider a CO with a surface and theoretically investigate how different of shadows of COs with and without horizons are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' For simplicity, we focus on a model with two ideal assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Compared with the luminous accretion flows or other light sources in the background, we first assume the CO is a non-luminous body;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' that is, the surface of the CO has no emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Secondly, we take the CO somehow as a dark star so that few lights can reflect from the surface of the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Thus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' the reflections can be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In short, in our simplified model, the CO doesn’t transmit and reflect lights and behaves like an event horizon effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' However, compared to a black hole, the radius of the surface of the CO can be chosen arbitrarily while the event horizon is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Moreover, since the radius of the surface is not fixed, there might be no photon region, or only part of the photon region remains outside the surface of the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' As we know, the black hole shadow curve is usually determined by the photon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Thus, it’s fascinating to theoretically study the shadow structures of the CO in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In addition, to describe the spacetime outside 1The spherical photon orbits are usually defined by r = const in a stationary and axisymmetric spacetime, where r is the radial coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In a curved spacetime as a radial parameter, r = const generally does not imply the spherical meaning in flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' A more strict definition can be found in [3], where authors introduced a new terminology: the fundamental photon orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Some related works concerned with fundamental photon orbits can be seen in [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2 the CO, we will employ the Painlev´e-Gullstrand form of the Lense-Thirring spacetime proposed recently in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The remaining parts of this paper are organized as follows in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2, we review the Painlev´e- Gullstrand form of the Lense-Thirring spacetime and discuss the geodesics in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3, we introduce the (possible) observational photon region and have a detailed study of the shadow curves for COs with and without horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The main conclusions are summarized in sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In this work, we have set the fundamental constants c and G, and we will work in the signature convention (−, +, +, +) for the spacetime metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2 Painlev´e-Gullstrand form of the Lense-Thirring spacetime Since we shall use the Lense-Thirring metric to model a horizonless CO, we would like to review the Lense-Thirring spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1 Metric In 1918, Lense and Tirring put forward an approximate solution to describe a slow rotating large-distance stationary isolated body in the framework of the vacuum Einstein equations [53], which takes ds2 = − � 1 − 2M r + O � 1 r2 �� dt2 − �4J sin2 θ r + O � 1 r2 �� dφdt + � 1 + 2M r + O � 1 r2 �� � dr2 + r2 � dθ2 + sin2 θdφ2�� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1) where M and J are the mass and the angular momentum, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' And O(r−2) denotes the sub-dominant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' By exquisitely regulating the specific forms of O(r−2), one can obtain various metrics with the same asymptotic limit at large distances, which are physically different from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Recently, Baines et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' constructed an explicit Painlev´e-Gullstrand variant of the Lense–Thirring spacetime [52], whose metric reads ds2 = −dt2 + � dr + � 2M r dt �2 + r2 � dθ2 + sin2 θ � dφ − 2J r3 dt �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='2) There are three solid advantages for this new version of the Lense–Thirring spacetime, of which the first one is that the metric reduces to the Painlev´e–Gullstrand version of the Schwarzschild black hole solution when J = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The second is that the azimuthal dependence takes in partial Painlev´e- Gullstrand form, that is, gφφ(dφ − vφdt)2 = gφφ(dφ − ωdt)2, where vφ is minus the azimuthal component of the shift vector in the ADM formalism denoting the “ flow ” of the space in the 3 azimuthal direction and ω = gtφ/gφφ is the angular velocity of the spacetime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The third is that all the spatial dependence is in exact Painlev´e–Gullstrand type form which implies the spatial hypersurface t = const is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' These exciting features make the Painlev´e-Gullstrand variant much easier to calculate the tetrads, curvature components, and the analysis of geodesics than any other variant of the Lense–Thirring spacetime [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' On the other hand, from the original asymptotic form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1), we can see that the Lense–Thirring metric should only make sense in the region r > rs, where we use rs to repre- sent the surface radius of the slow rotating isolated body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Note that the metric in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1) has a coordinate singularity r = 2M when neglecting the sub-dominant terms so that the Lense-Thirring spacetime should be valid when the condition rs > 2M holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Moreover, for a slowly rotating object, we must have J/r2 s ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Thus, we should also impose the conditions J/r2 s ≪ 1, rs > 2M on the Painlev´e–Gullstrand version of the Lense-Thirring spacetime when investigating the properties of the Painlev´e–Gullstrand form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='2 Geodesics In this subsection, we would like to review the geodesics in the Painlev´e-Gullstrand form of the Lense-Thirring spacetime, which has been carefully studied in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Similar to the Kerr spacetime, there are also four conserved quantities along the geodesics of free particles: the mass m, the energy E, the axial angular momentum L, and the Carter constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' For simplicity and without loss of generality, we set m = 0 for photons and m = 1 for timelike particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then, the four-momentum pa reads pa = ˙t � ∂ ∂t �a + ˙r � ∂ ∂r �a + ˙θ � ∂ ∂θ �a + ˙φ � ∂ ∂φ �a , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='3) with “ ˙ ” denoting the derivative with respect to the affine parameter τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Considering papa = 0 for photons and papa = −1 for timelike particles, τ can be seen as the proper time for timelike worldlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then the conserved quantities E, L, C can be written out E = −pt = � 1 − 2M r − 4J2 sin2 θ r4 � ˙t − � 2M r ˙r + 2J sin2 θ r ˙φ , L = pφ = r2 sin2 θ � ˙φ − 2J r3 ˙t � , C = r4 ˙θ2 + L2 sin2 θ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='4) explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Note that for timelike particles, E and L can now be treated as the energy per unit mass and the angular momentum per unit mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then combining with the condition −papa = m ∈ {0, 1}, 4 one can obtain the exact expressions of the components of the four-momentum pa as follows ˙r = Sr � R(r) , ˙t = E − 2JL/r3 + Sr � (2M/r)R(r) (1 − 2M/r) , ˙θ = Sθ � Θ(θ) r2 , ˙φ = L r2 sin2 θ + 2J E − 2JL/r3 + Sφ � (2M/r)R(r) r3(1 − 2M/r) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5) where we define R(r) = � E − 2JL r3 �2 − � m + C r2 � � 1 − 2M r � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='6) Θ(θ) = C − L2 sin2 θ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='7) as the effective potential functions governing the radial and polar motions, and Sr = � +1 outgoing geodesic −1 ingoing geodesic ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Sθ = � +1 incerasing declination geodesic −1 decerasing declination geodesic ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Sφ = � +1 prograde geodesic −1 retrograde geodesic ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='8) following the conventions in [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The context for each equation in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='8) denotes the corre- sponding physical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Here we would like to stress that Sr and Sφ appear separately in the t-motion and φ-motion due to the Painlev´e-Gullstrand form, however, for geodesic equations of Kerr spacetime in Boyer-Lindquist coordinates, Sr only comes up in the radial motion, and Sφ is not necessarily introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then one can explore the properties of null and timeslike geodesics by adequately manipulating the equations in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3 Observational photon region and shadow curve This section focuses on the photon region and shadow curve in the Painlev´e-Gullstrand form of the Lense-Thirring spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Considering the null orbits are independent of photon energies, it’s convenient to introduce the impact parameters ξ = L E , η = C − L2 E2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1) 5 to characterize the photon orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The conditions can determine the photon region R(r) = ∂rR(r) = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='2) which gives us the expressions of the impact parameters in terms of the radius, ˜ξ = −3M ˜r3 + ˜r4 2J(3M − 2˜r) , ˜η = − ˜r3[˜r3(˜r − 3M)2 + 36J2(˜r − 2M)] 4J2(3M − 2˜r)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='3) Note that we use ˜r to denote the radius of the photon orbit in the photon region, and ˜ξ, ˜η are the corresponding impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Furthermore, from ˜η = 0 we can obtain two roots rp− < rp+ in the region ˜r > 2M which implies the radial range of the photon region is ˜r ∈ [rp−, rp+] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='4) Note that rp± cannot be analytically given in general;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' however, when J → 0, one can find [55] rp± = 3M ± 2J √ 3M + O(J2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5) Considering rs > 2M for COs, in the light of rp± we would like to divide the range of rs into three parts, that is, (1) 2M < rs < rp−, (2) rs > rp+, (3) rp− < rs < rp+, and study the shadow curve for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1 Review of black hole shadows Before we talk about the shadows of COs, we first review the shadows of ordinary black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' To determine the shadow of a black hole, in addition to the photon region, there is a second condition related to the observational angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' For a certain observational angle θo, we can see that the term under the square root Θ(θo) ≥ 0 must be satisfied in the polar motion, which gives Θ(θo) = ηo − ξ2 o sin2 θo ≥ 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='6) and a new function ηo(ξo) = ξ2 o sin2 θo .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' That is to say, and the photons could reach the observer if their impact parameters satisfy the above condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Combing the critical impact parameters ˜η(˜ξ) with the constraint Θ(θo) ≥ 0, one can exactly fix the photons which have critical impact parameters and can escape to observers if they are perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' As a result, the shadow curve is formed by these photons since the surface of the black hole, that is, the horizon, is always inside the photon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In the study of shadows of COs, including black holes, we find it convenient to define the observational photon region (OPR) and possible observational photon region (POPR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The OPR 6 Figure 1: An illustration of the observational photon region for a black hole in the ξOη plane is shown in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The right panel is borrowed from the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 11 of our previous work [56], which presents the celestial coordinates (Θ, Ψ) and standard Cartesian coordinates (x, y) in the local rest frame of observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' is defined as the set of impact parameters that the photons with these impact parameters precisely determine the shadow curve for observers with a certain observational angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' And the POPR has defined as the union of the OPRs at all possible observational angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Thus, for the case of black holes, the POPR is composed of the critical impact parameters ˜η(˜ξ) and the elements of the OPR are the critical impact parameters ˜η(˜ξ) which also satisfy the condition Θ(θo) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In the left panel of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 1, we show the functions of ˜η(˜ξ) and ηo(ξo) in the ξOη plane and find that the two functions have two intersections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The OPR corresponds to the segment of ˜η(˜ξ) between the two intersections, and the POPR corresponds to a piece of ˜η(˜ξ) above the ξ-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then one can calculate the shadow curve by standard methods, that is, introducing the celestial coordinates and obtaining the projections on the screen of observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In this work, we employ the stereographic projection method, which has been used in our previous work [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' We also bring the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 11 in work [56] to the right panel of the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 1 to give a deep intuition on the celestial coordinates and Cartesian coordinates (x, y) in the local rest frame of observers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In terms of the metric in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='2), the local rest frame of observers can be defined as e0 = ˆe(t) = ∂t − � 2M r ∂r + 2J r3 ∂φ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='7) e1 = −ˆe(r) = −∂r , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='8) e2 = ˆe(θ) = 1 r∂θ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='9) e3 = −ˆe(φ) = − 1 r sin θ∂φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='10) 7 x n。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='(。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=') i() (t)a- = Ta 0 + s e3 = -e(Φ) (+di)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 0 (rp-) P e2 = ()aIt is not hard to verify that these bases are normalized and orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Moreover, since ˆe(t) · ∂φ = 0, the observer with the 4-velocity ˆu = e0 in this local rest frame has zero angular momentum for infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' So this frame is usually called the ZAMO reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In our model, the relation between the celestial coordinates (Θ, Ψ) and the 4-momentum of the OPR takes Θ = arccos �� 2M r0 + ˙˜ro ˙˜to � , Ψ = − arctan � � ˜ξ � ˜η csc2 θo − ˜ξ2 � � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='11) where “ ∼ ” denotes evaluated with critical impact parameters ˜ξ and ˜η, and the subscript “ o ” means evaluated at the observer with coordinates (0, ro, θo, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then the Cartesian coordinates (x, y) on the screen can be defined as x = −2 tan Θ 2 sin Ψ , y = −2 tan Θ 2 cos Ψ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='12) where we have chosen the energy of the photon observed by the ZAMOs to be unity, considering the trajectories of photons are independent of the energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='2 Shadows of COs without horizons In this subsection, we study the shadows of COs, which have no horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' For simplicity, we assume the COs are non-luminous bodies, and they neither transmit nor reflect light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Recall that the spacetime outside a CO we consider in this work is modeled by the Painlev´e-Gullstrand form of the Lense-Thirring spacetime, and we would like to investigate the shadows in three situations, (1) 2M < rs < rp−, (2) rs > rp+, (3) rp− < rs < rp+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 15 10 5 5 ξ 20 40 60 80 η 6 4 2 2 4 5 10 15 20 25 30 6 4 2 2 4 6 ξ 5 10 15 20 25 30 η ξ (rs)) (ξ˜(rs)， η ˜ rs=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='01 rs=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24 rs=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='92 η Figure 2: Plots of the functions ˜η(˜ξ), ηs(ξs) and ηo(ξo) in the ξOη plane for rs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24, rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='01 and rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='92 with M = 1 and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In each plot, ˜η(˜ξ) is shown in the dashed line, ηs(ξs) is shown in the solid line with downward opening, ηo(ξo) with θo = 17◦ is given by the green line and ηo(ξo) with θo = 80◦ is given by the purple line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In addition, the POPR is shown in the red line in each plot, while the blue one has no contribution to the shadow curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' As mentioned above, the shadow would be clear if we find the corresponding OPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Thus, the main task is to look for the OPR for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Since the CO is regarded as a dark body in our 8 work, the effect on lights is equivalent to the event horizon of a black hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' that is, the photons cannot go back if they meet the surface of the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' As a result, the ingoing photons, which have two turning points in the radial motion, cannot escape to infinity if the outer turning point is inside the surface of the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Thus, if rs is not less than ˜rp−, the part of the photon region inside the surface of the CO would have no contributions to the POPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' More precisely, from R(rs) = 0, we can obtain a new relation between ξs and ηs as follows ηs = −(rs − 2Jξs)2 (2M − rs)r3s − ξ2 s , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='13) where the subscript “ s ” denotes evaluated at r = rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Considering the radius of the surfacers could be the inner or outer turning point which corresponds to different values of (ξs, ηs), ηs(ξs) would become the new critical parameters when rs > ˜r, where ˜r is the radius of the photon region with ˜η(˜ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2, we give examples of ˜η(˜ξ), ηs(ξs) and ηo(ξo) for three cases at the observational angles θo = 17◦ and θ = 80◦ with the mass and the angular momentum of the CO chosen as M = 1 and J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5 here and after this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' By numerically solving the equation ˜η = 0, we find rp− ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='47 , rp+ ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='56 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='14) rs rs rs =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 x y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='04 x y θO=17° θO=80° =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='01 =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='92 Figure 3: Plots of shadow curves of COs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In the left plot, we set θo = 17◦, and in the right one, we set θo = 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In both plots, the green, blue and red lines denote the shadow curves with rs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24, rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='01, and rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='92, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In addition, implying R = ∂rR = ∂2 rR = 0 for prograde timelike particles, we can find the radius of the innermost stable circular orbit rI ≃ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Considering the horizon is at rh = 2, we set rs = rh+rp− 2 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24 < rp−, rp− < rs = rp−+rp+ 2 ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='01 < rp+ and rs = rp++rI 2 ≃ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='92 > rp+ for the 9 plots from left to right in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In addition, for each plot, the dashed line denotes ˜η(˜ξ), the other curve with a downward opening indicated by a solid line denotes ηs(ξs), the curve with an upward opening drawn in green is ηo(ξo) with θo = 17◦, and the other curve with an upward opening drawn in purple is ηo(ξo) with θo = 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' For the middle plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2 with rp− < rs < rp+, there is an intersection point (˜ξ(rs), ˜η(rs)) of ˜η(˜ξ) and ηs(ξs) which means the two turning points of photons coincide with the radius r = rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' When ξ > ˜ξ(rs), we can find that rs is the outer turning point of R(rs) = 0 and rs > ˜r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' On the contrary, when ξ < ˜ξ(rs), we find that rs is the inner turning point of R(rs) = 0 and rs < ˜r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Therefore, the red line is the POPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' And the impact parameters that are not in POPR are shown in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Moreover, combined with the condition from the observer at θo = 17◦ (θo = 80◦), the POR is the segment of the red line between the intersections of the red and green (purple) lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' For the left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2 with rs < rp−, we can see that the POPR is still determined by ˜η(˜ξ), which is the same as that in a black hole spacetime since the surface of the CO is always hidden in the photon region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' And the OPR is the segment of ˜η(˜ξ) between the intersections of the red line ˜η(˜ξ) and the green line ηo(ξo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' While for the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2 with rs > rp+, we can see that the POPR is determined by the solid line ηs(ξs), since the photon region is completely encapsulated by the surface of the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' And the OPR now is given by the segment of the red line ηs(ξs) between the intersections of ηs(ξs) and ηo(ξo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' y ymin xmin max x max O x c Figure 4: An illustration of the coordinates of the points at which the shadow curve intersects the two axes on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then the shadows of COs without horizons can be calculated with the help of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3, we show the shadow curves with dashed lines at θo = 17◦ for the left plot 10 and θ = 80◦ for the right plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The red, blue and green lines correspond to rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='92 > rp+, rp− < rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='01 < rp+ and rs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24 < rp−, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' As we have discussed above, the shadow curve is exactly determined by the OPR, and note that in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2, the dashed line in each plot denotes the same photon region, that is, ˜η(˜ξ), and thus the segment of ˜η(˜ξ) between the intersections of ˜η(˜ξ) and ηo(ξo) keeps invariable in three plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' As a result, we can find that for the case of θo = 17◦, the blue line and the green line almost coincide in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3, since from the middle plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2 one can see that the OPR with rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='01 coincides with the OPR with rs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24 when ξ < ˜ξ(rs), and only has a tiny difference with the OPR with rs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='24 when ξ > ˜ξ(rs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Similarly, the difference between the red and the green lines in the case of θo = 17◦ is visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3, since one can see the difference of their OPRs is evident from the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Moreover, from the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3, we can see that the difference between the green and blue lines becomes significant on the right, and the three lines are very close in the left part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' The reason can be easily found in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2 where the opening of the parabola ηo(ξo) gets bigger when θo goes from 17◦ to 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Furthermore, in the middle plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2, one can find that the difference of the OPRs becomes larger at θo = 80◦, and in the right plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 2, the red and blue lines intersect very closely with the purple line since rs = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='92 is near rp+ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Therefore, qualitatively we can conclude that when rs < rp−, the shadow of the CO is the same as that of the black hole;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' when rp− < rs < rp+, the shadow of the CO is bigger than that of the black hole, and the shadow of the CO becomes a litter bigger as θo increases from 0◦ to 90◦ with parts of the shadow curves overlapped;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' and when rs > rp+ the shadow of the CO would become larger significantly, and each point of the CO shadow curve is outside the corresponding end of the black hole shadow curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='3 Quantitative study of the variation of the CO shadow In this subsection, we would like to give a quantitative study of the variation of the shadow concerning the radius of the surface of a CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Following the work [57, 58], we use the average radius ¯R as the characteristic length of a shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 4, we give a diagram to show the coordinates of points at which the shadow curve inter- sects two axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' O is the origin of the Cartesian coordinates on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Considering the Z2 sym- metry of the spacetime, the center of the shadow can be defined as � xc = xmin+xmax 2 , ymin+ymax 2 = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Then let (xc, 0) be the center, we can introduce polar coordinates (R, ψ) with R = � (x − xc)2 + y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' And the parameter ¯R can be defined as ¯R = � 2π 0 R(ψ) 2π dψ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='15) 11 2 3 4 5 6 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='7 θo=80° θo=17° σ rs Figure 5: The variation of the dimensionless parameter σ = ¯R/ ¯R0 −1 of the CO shadow concerning the radius of the surface of the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In the plot, we set rs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='07 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='4(i − 1), where i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' , 14 for each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' which denotes the average radius of the shadow curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' It is convenient to introduce a dimensionless parameter σ = ¯R ¯R0 − 1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='16) where we use ¯R0 to represent the average radius of the shadow curve when rh < rs < rp−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 5, we show the variation of σ concerning the radius of the CO surface, where we fix M = 1, J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='5 and set rs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='07 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='4(i − 1) with i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' , 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' We can find that the average radius of the shadow curve increases slowly as the radius of the CO surface increases from rp− to rp+, the main reason is that rp+ − rp− = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='09 is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' When rs > rp+, the average radius of the shadow curve increases quickly as the radius of the CO surface increases, and the change is almost linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In addition, we can see that the average radius of the shadow curve at θo = 80◦ is always larger than that at θo = 17◦ for a fixed rs in the range rs > rp− which agrees well with our analysis in the last subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 4 Summary In this work, we studied the problem of how different of shadows of COs with and without horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' For simplicity, the CO was considered not to emit or reflect any light compared to other luminous sources in the background of the CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In addition, we assumed that the CO is a slowly rotating object such that the spacetime outside the surface of the CO can be described by the Painlev´e-Gullstrand form of the Lense-Thirring metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' In terms of the photon region with rp− ≤ ˜r ≤ rp+, we investigated three cases, that is, the radius rs of the CO is smaller than rp−, 12 rp− < rs < rp+ and rs > rp+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' To obtain the shadow curve for different cases, we introduced OPR and POPR in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content='1 to construct a clear correspondence between the shadow curve and the impact parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Moreover, we recognized a new class of critical impact parameters ηs(ξs), with which the photons have a turning point at rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' After a detailed analysis of the OPRs and POPRs for COs with different rs, we found the POPR governed by the photon region ˜η(˜ξ), which is the same as that for black holes when rh < rs < rp−, one part of the POPR is governed by the photon region ˜η(˜ξ) and the other part is controlled by ηs(ξs) when rp− < rs < rp+, and the POPR is completely controlled by the ηs(ξs) when rs > rp+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' As a result, compared with the shadow curve of a black hole, we found that the shadow curve of a CO doesn’t change for rh < rs < rp−, partially changes for rp− < rs < rp+ and completely changes for rs > rp+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' We also gave a quantitative study on the variation of the shadow curve concerning rs, and found the average radius of the shadow curve gets bigger slowly when rs goes from rp− to rp+ and very quickly when rs increases after rp+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Our results indicate that a CO with or without a horizon is not distinguished by the shadow curve when it has a whole photon region outside its surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' A CO without a horizon can be distinguished from a black hole when the photon region is partially or entirely hidden in the surface of the CO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' that is to say, in this case, the EHT can be used to determine whether a CO has an event horizon if the resolution reaches high enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Although in the present work, our discussion is based on an approximate metric, it seems our results should not depend on a specific metric but reflect a universal property for a CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Obviously;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' it is fascinating to have a further study considering a more realistic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' Acknowledgments The work is partly supported by NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' 12205013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} +page_content=' MG is also endorsed by ”the Fundamental Research Funds for the Central Universities” with Grant No.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5dE4T4oBgHgl3EQfBQvo/content/2301.04851v1.pdf'} diff --git a/5tE5T4oBgHgl3EQfPQ4_/content/tmp_files/2301.05503v1.pdf.txt b/5tE5T4oBgHgl3EQfPQ4_/content/tmp_files/2301.05503v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..092e403aabe4cfa742577d388c895b31735b84cc --- /dev/null +++ b/5tE5T4oBgHgl3EQfPQ4_/content/tmp_files/2301.05503v1.pdf.txt @@ -0,0 +1,1823 @@ +arXiv:2301.05503v1 [math.NA] 13 Jan 2023 +Fractional Diffusion in the full space: decay and +regularity +Markus Faustmann∗ and Alexander Rieder† +January 16, 2023 +We consider fractional partial differential equations posed on the full space Rd. +Using the well-known Caffarelli-Silvestre extension to Rd × R+ as equivalent defini- +tion, we derive existence and uniqueness of weak solutions. We show that solutions +to a truncated extension problem on Rd × (0, Y) converge to the solution of the +original problem as Y → ∞. Moreover, we also provide an algebraic rate of decay +and derive weighted analytic-type regularity estimates for solutions to the truncated +problem. These results pave the way for a rigorous analysis of numerical methods +for the full space problem, such as FEM-BEM coupling techniques. +1 Introduction +In recent years, models using non-integer powers of differential operators garnered lots of interest +as the inherent non-locality of these operators gives a more accurate way to describe non-local +processes in physics, finance or image processing, [BV16, SZB+18]. Restricting these non-local +PDE models to some bounded domain requires one to fix values of the solution everywhere +outside of the domain, which may lead to some non-physical assumptions for the boundary +conditions. Consequently, the full-space problem is oftentimes used in analytical works. +In a similar vein, when working on a bounded domain, there are multiple non-equivalent defini- +tions of fractional differential operators such as the fractional Laplacian, [LPG+20]. The most +common ones are the integral fractional Laplacian (defined pointwise as a singular integral) +and the spectral fractional Laplacian (defined using spectral calculus). Consequently, it is of- +tentimes not obvious, which definition of the fractional Laplacian should be used in the model. +In contrast, working on the full space, one obtains a single natural definition as all different +approaches are equivalent, [Kwa17]. +In this work, we analyze fractional PDEs in the full space. Using the influential interpretation +of elliptic fractional differential operators as Dirichlet-to-Neumann operators for degenerate +elliptic PDEs, the so called Caffarelli-Silvestre extension, [CS07, ST10], defined on the half +space Rd × R+, we show well-posedness of a weak formulation of the fractional PDE. As, in +general, analytical solutions to such problems are unknown, discretizations of the equations are +usually employed to derive approximative solutions. +∗Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria, markus.faustmann@tuwien.ac.at +†Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria, alexander.rieder@tuwien.ac.at +1 + +In the case of fractional PDEs on bounded domains Ω ⊂ Rd, see e.g. [NOS15, BMN+19], a +truncation to Ω×(0, Y) is used to be able to discretize the extension problem. This induces two +natural questions: does the solution to the truncated extension problem (with homogeneous +Neumann condition on the artificial boundary) converge to the solution of the original problem +and can the rate of convergence be quantified? For bounded domains, [BMN+19] answered +both questions by showing exponential decay in Y by exploiting an explicit representation for +the y-dependence. +In this article, we employ the truncation to the full space problem, i.e., we study the extension +problem on Rd × (0, Y) and answer both questions as well. In this case, however, there is no +closed form expression for the y-dependence available. Nonetheless, we show convergence of +the truncated solution to the original solution in the full-space setting, but only with certain +algebraic rates. From a technical standpoint, the explicit representation is replaced by applying +purely variational techniques to show the decay properties. +1.1 Impact on numerical methods +Numerical methods for fractional PDEs on bounded domains are fairly developed, as can e.g. +be seen in the survey articles [BBN+18, DDG+20, LPG+20] and we especially mention approxi- +mations based on the finite element method (FEM), [AB17, BMN+19, ABH19, FKM22]. A key +limitation to the FEM is the restriction to bounded computational domains. A classical refor- +mulation for exterior problems uses boundary integral equations, which leads to the boundary +element method (BEM), [SS11]. An approach for transmission problems on unbounded do- +mains that is commonly employed is the combination of both methods, so called FEM-BEM +couplings, [Cos88, Han90]. The goal of our follow-up work, [FR22], is to formulate a fully com- +putable symmetric FEM-BEM coupling method applied to fractional transmission problems +posed in Rd. +However, before a rigorous analysis of any numerical method can be made, analytical founda- +tions regarding well-posedness and regularity of the problem at hand must be made. As a second +key result of this article, we establish analyticity of the solution in the extended direction y in +terms of certain weighted Sobolev spaces. This is achieved by deriving a small initial regularity +shift in a weighted space and then employing bootstrapping arguments to control higher-order +derivatives. Structurally, these estimates are similar to the ones for the case of bounded domains +in [BMN+19, FMMS22] and show that solutions are in certain countably normed spaces. +Combined with our follow-up work [FR22], this article establishes that the Caffarelli-Silvestre +extension approach can be combined with FEM-BEM coupling techniques to yield a good +approximation scheme. +1.2 Layout +The present paper is structured as follows: In Section 2, we introduce our model problem and +formulate assumptions on the data to be able to apply FEM-BEM techniques afterwards. Then, +the Caffarelli-Silvestre extension as well as its weak formulation and the weak formulation of +the truncated problem are introduced. Finally, we present our main results: Proposition 2.3 +shows well-posedness of both weak formulations, Proposition 2.4 provides convergence of the +truncated solution to the solution posed on Rd ×R+ and Proposition 2.5 gives the algebraic rate +of decay. In Proposition 2.6 the regularity results in weighted Sobolev spaces are presented. +Section 3 is then devoted to the proofs of the well-posedness and convergence results, where the +key step is Lemma 3.3, which shows decay properties of the full space solution as the truncation +2 + +parameter Y → ∞ by employing inf-sup theory and weighted spaces. +Finally, in Section 4 the estimates for higher order derivatives are derived. Hereby, an initial +regularity shift in Lemma 4.1 and Lemma 4.2 allows to use an induction argument to show +Proposition 2.6. +Moreover, a (finite) regularity result in the non-extended variables and a +characterization of the solution in certain countably normed spaces is presented. +1.3 Notations +Throughout the text we use the symbol a ≲ b meaning that a ≤ Cb with a generic constant +C > 0 that is independent of any crucial quantities in the analysis. Moreover, we write ≃ to +indicate that both estimates ≲ and ≳ hold. +For any multi index α = (α1, . . . , αd) ∈ Nd +0, we denote the partial derivative ∂α = ∂α1 +x1 · · · ∂αd +xd +of order |α| = �d +i=1 αi. Moreover, for k ∈ N, we employ classical integer order Sobolev spaces +Hk(Ω) on (bounded) Lipschitz domains Ω and the fractional Sobolev spaces Ht(Rd) for t ∈ (0, 1) +defined, e.g., via Fourier transformation. +2 Main results +2.1 Model problem +We consider a stationary fractional diffusion problem on the full space Rd with d = 2 or d = 3 +given by +Lβu + su = f +in Rd +(2.1) +with s ≥ 0, and β ∈ (0, 1). The self-adjoint operator L is hereby defined as +Lu := − div +� +A∇u +� +, +and, for functions u ∈ L2(Rd), the fractional differential operator Lβ is defined using spectral +calculus +Lβu := +� +σ(L) +zβdE u, +where E is the spectral measure of L and σ(L) is the spectrum of L. Using standard techniques +this definition can be extended to tempered distributions. +For the data, we assume that A : Rd → Rd×d is smooth and pointwise symmetric and positive +definite in the sense that there exists A0 > 0 such that +(A(x)y, y)2 ≥ A0 ∥y∥2 +2 +∀y ∈ Rd. +In order to avoid several additional difficulties due to decay conditions at infinity, we assume +s ≥ σ0 > 0 for the case d = 2. +Additionally, we make the following assumptions on the coefficients in the model problem: There +exists a bounded Lipschitz domain Ω ⊆ Rd such that +1. supp f ⊆ Ω, +2. A ≡ I in Rd \ Ω. +3 + +Remark 2.1. We note that adding lower order terms to the operator is also covered by our +techniques, i.e., +Lu := − div +� +A∇u +� ++ cu, +where c : Rd → R with c ≥ 0 is smooth and satisfies c ≡ c0 ∈ R in Rd \ Ω. However, in order to +make the key concepts more clear, we decided to stick to the case c = 0 in the following. +2.2 The Caffarelli-Silvestre extension +Following [ST10], we rewrite (2.1) as an extension problem in a half space in Rd+1. The extension +problem is conveniently described using weighted Sobolev spaces. +For any bounded open subset D ⊂ Rd × R and arbitrary α ∈ (−1, 1), we define L2(yα, D) as +the space of square integrable functions with respect to the weight yα. Correspondingly, the +Sobolev space H1(yα, D) ⊂ L2(yα, D) consists of functions, for which the norm +∥U∥2 +H1(yα,D) := +� � +D +yα���∇U(x, y) +��2 + +��U(x, y) +��2� +dx dy +is finite. +As our model problem is formulated on an unbounded domain, we need to take care of the +behaviour at infinity. To that end, we use appropriately weighted Sobolev spaces, as is standard +for the Poisson problem, see e.g. [AGG94]. For (x, y) ∈ Rd × R, we introduce the weight +ρ(x, y) := (1 + |x|2 + |y|2)1/2. +For a (possibly unbounded) domain D ⊂ Rd × R+, we define the space H1 +ρ(yα, D) as the space +of all square integrable functions U (with respect to the weight function yαρ−2) such that the +norm +∥U∥2 +H1ρ(yα,D) := +� � +D +yα���∇U(x, y) +��2 + ρ(x, y)−2��U(x, y) +��2� +dx dy +(2.2) +is finite. Commonly used cases are D = Rd × R+ (full space), D = Rd × (0, Y) for Y > 0 +(corresponding to truncation in y-direction), or D = ω × (0, Y) for ω ⊂ Rd and Y > 0. +Remark 2.2. For bounded sets ω ⊂ Rd and Y < ∞, we sometimes use the weighted spaces +H1 +ρ(yα, ω × (0, Y)), noting that, in this case, the weight satisfies 1 ≤ ρ(x, y) ≤ C(ω, Y) < ∞. +Consequently, the norm (2.2) defines an equivalent norm to the H1(yα, ω × (0, Y))-norm. +For functions U ∈ H1 +ρ(yα, Rd × R+), one can give meaning to their trace at y = 0, which we +denote by tr0 U. In fact, Lemma 3.1 will show that tr0 U is in a weighted fractional Sobolev +space. +Then, the extension problem reads as: find U ∈ H1 +ρ(yα, Rd × R+) such that +− div +� +yαAx∇U +� += 0 +in Rd × R+, +(2.3a) +d−1 +β ∂ναU + str0U = f +in Rd, +(2.3b) +where dβ := 21−2βΓ(1 − β)/Γ(β), α := 1 − 2β ∈ (−1, 1), ∂ναU(x) := − limy→0 yα∂yU(x, y), and +Ax = +�A +0 +0 +1 +� +∈ R(d+1)×(d+1). Then, by [ST10], the solution to (2.1) is given by u = U(·, 0). +4 + +The weak formulation of (2.3) in H1 +ρ(yα, Rd × R+) reads as finding U ∈ H1 +ρ(yα, Rd × R+) such +that +A(U, V) := +� ∞ +0 +yα +� +Rd Ax(x)∇U · ∇V dxdy + sdβ +� +Rd tr0Utr0V dx = dβ(f, tr0V)L2(Rd) +(2.4) +for all V ∈ H1 +ρ(yα, Rd × R+). If s > 0, it is natural to include the trace term into the norm. +Thus, we introduce: +∥U∥2 +H := ∥U∥2 +H1ρ(yα,Rd×R+) + s∥tr0U∥2 +L2(Rd). +The first step towards a computable formulation, before even considering any discretization +steps, is to cut the problem from the infinite cylinder Rd × R+ to a finite cylinder in the y- +direction. To do so, we fix a parameter Y > 0 to be chosen later and introduce the truncated +bilinear form +AY(U, V) := +� Y +0 +yα +� +Rd ∇U · ∇V dxdy + sdβ +� +Rd tr0Utr0V dx. +The truncated problem then reads: Find UY ∈ H1 +ρ(yα, Rd × (0, Y)) such that +AY(UY, VY) = dβ +� +f, tr0VY� +L2(Rd) +for all VY ∈ H1 +ρ(yα, Rd × (0, Y)). +(2.5) +In the following, we will often take Y ∈ (0, ∞] and refer to solutions to problem (2.5), meaning +that in the case Y = ∞ these functions actually satisfy (2.4). +We also introduce a natural norm on the truncated cylinder: +∥U∥2 +HY := ∥U∥2 +H1ρ(yα,Rd×(0,Y)) + s∥tr0U∥2 +L2(Rd). +In fact, the truncated problem (2.5) corresponds to a weak formulation of a Caffarelli-Silvestre +extension problem with an additional Neumann boundary condition at y = Y: +− div +� +yαAx∇UY� += 0 +in Rd × (0, Y), +(2.6a) +d−1 +β ∂ναUY + str0UY = f +on Rd × {0}, +(2.6b) +∂yUY = 0 +on Rd × {Y}. +(2.6c) +2.3 Main results +We are now in position to formulate the main results of the article. The proofs of the statements +are relegated to the following Sections 3 and 4. +2.3.1 Well-posedness and decay +Regarding well-posedness of our variational formulation, we have the following proposition. +Proposition 2.3. Assume that either d > 2 or s > 0. +Then, problem (2.4) has a unique +solution U ∈ H1 +ρ(yα, Rd × R+) and there is a constant C > 0 such that +∥U∥H ≤ C min(1, s−1) ∥f∥L2(Ω) . +5 + +Fix Y ∈ (0, ∞). Then, the truncated problem (2.5) has a unique solution UY ∈ H1 +ρ(yα, Rd × +(0, Y)) satisfying +��UY�� +HY ≤ C +� +1 + 1 +Y +� +min(1, s−1) ∥f∥L2(Ω) +with a constant C > 0 independent of Y. +Moreover, the bilinear forms in (2.4) and (2.5) are coercive. +By the following proposition, we also obtain that solutions to the truncated problem converge +to solutions to the non-truncated problem as the truncation parameter Y tends to infinity. +Proposition 2.4. Let U solve (2.4) and, for Y > 0, let UY solve (2.5). For any fixed 0 < �Y < Y, +it holds that UY → U in H1 +ρ(yα, Rd × (0, �Y)) as Y → ∞. If s > 0, there additionally holds +tr0UY → tr0U in L2(Rd) as Y → ∞. +Finally, we also obtain algebraic rates of convergence as Y → ∞ for the difference of the +truncated and the non-truncated full-space solutions. +Proposition 2.5. Fix Y > 0. Let U solve (2.4) and UY solve (2.5). Let µ be given by +µ := +� +1 + |α| +s > 0 +1 + α +s = 0 . +(2.7) +Then, there exists a constant C > 0 depending only on α and d such that +∥UY − U∥2 +H1ρ(yα,Rd×(0,Y)) + s∥tr0(UY − U)∥2 +L2(Rd) ≤ CY−µ ∥f∥2 +L2(Ω) . +2.3.2 Regularity +For solutions to the extension problem as well as the truncated extension problem there hold +analytic type weighted estimates for the extended variable. Estimates of that type allow to +employ hp-finite elements in the extended variable, which will be considered in [FR22]. +Proposition 2.6 (Regularity in y). Fix Y ∈ (0, ∞] and let ℓ ∈ N. Let U solve (2.5). Then, +there exists constants C, K > 0 and ε ∈ (0, 1) such that the following estimate holds: +��yℓ−ε∇∂ℓ +yU +�� +L2(yα,Rd×(0,Y)) ≤ CKℓℓ! ∥f∥L2(Ω) . +All constants are independent of ℓ, Y, and U. +In fact, the regularity results imply that solutions to our model problem are in certain countably +normed spaces. Following [BMN+19, Sec. 5.5.1], we introduce the Bochner spaces L2 +α((0, ∞); X) +of square integrable functions (with respect to the weight yα) and values in the Banach space +X as well as for constants C, K > 0, the countably normed spaces +B1 +ε,0(C, K; X) := +� +V ∈ C∞((0, ∞); X) : ∥V∥L2(yα,(0,∞);X) < C, +���yℓ+1−εV(ℓ+1)��� +L2(yα,(0,∞);X) < CKℓ+1(ℓ + 1)! ∀ℓ ∈ N0 +� +. +Proposition 2.6 provides control of yℓ−ε∂ℓ+1 +y +U, which directly gives the following Corollary. +6 + +Corollary 2.7. Fix Y ∈ (0, ∞] and let U solve (2.5). Then, there are constants C, K > 0 such +that there holds +∂yU ∈ B1 +ε,0(C, K; L2(Rd)). +(2.8) +We note that we formulated the previous corollary in terms of ∂yU, whereas the regularity +results in [BMN+19, eqn. (6.10)] are formulated for solutions U to the extension problem on +bounded domains. This is due to the fact that in the case of the full space problem the estimates +do not hold for the lowest order term as U /∈ L2(Rd × R+). Nonetheless, the regularity result +of Corollary 2.7 (together with U ∈ H1 +ρ(Rd × R+)) allows to construct interpolation operators +in a similar way as in [BMN+19, Lem. 11]. +Finally, we investigate the regularity in x. Since this will depend on the regularity of the data +A and f, we only consider the case of finite regularity. +Proposition 2.8 (Regularity in x). Assume that A ∈ Cm(Rd; Rd×d) and f ∈ Hm(Ω). Then, +for every multiindex ζ ∈ Nd +0 with |ζ| = m there holds +∥∇∂ζ +xU∥L2(yα,Rd×R+) ≤ C∥f∥Hm(Ω). +The constant C depends on Ω, A, m and d, but is independent of f and U. +3 Well-posedness and decay +In this section, we provide the proofs of Proposition 2.3 (well-posedness), Proposition 2.4 (con- +vergence) and Proposition 2.5 (algebraic rate of decay). +3.1 Trace estimate +We start with a trace estimate in a certain weighted Sobolev space. +Lemma 3.1. For all U ∈ H1 +ρ(yα, Rd × R+), there holds +|tr0U|Hβ(Rd) ≤ C ∥∇U∥L2(yα,Rd×R+) . +(3.1a) +For d = 3, we additionally have +∥(1 + |x|2)−β/2tr0U∥L2(Rd) ≤ C ∥∇U∥L2(yα,Rd×R+) . +(3.1b) +In both cases the constant C > 0 does only depend on d and α. +Proof. The estimate (3.1a) is shown in [KM19, Lem. 3.8]. To estimate the weighted L2-norm, +we use interpolation space theory. +More precisely, [Tar07, Lemma 23.1] shows that interpolation of L2-spaces with weights w0 and +w1 denoted by L2(wi, Rd) for i = 0, 1 produces an interpolation space (using the K-method) +[L2(w0, Rd), L2(w1, Rd)]θ,2 = L2(wθ, Rd) that is a weighted L2-space with weight wθ = w1−θ +0 +wθ +1. +Applying this result with θ = 1−β and w0 = ρ−2 +x +:= ρ(x, 0)−2 = (1+|x|2)−1 and w1 = 1, shows +that +∥ρ−β +x tr0U∥2 +L2(Rd) = ∥tr0U∥2 +L2(ρ−2β +x +,Rd) ≲ ∥tr0U∥2 +[L2(ρ−2 +x ,Rd),L2(Rd)]1−β,2. +7 + +Now, by [Tar07, Lemma 40.1] the interpolation spaces can be seen as trace spaces, i.e., el- +ements of the interpolation space can be seen as traces (at 0) of functions U(y) satisfying +y1−β ∥U(y)∥L2(ρ−2 +x ,Rd) ∈ L2(y−1, R+) as well as y1−β ∥∂yU(y)∥L2(Rd) ∈ L2(y−1, R+). Together +with α = 1 − 2β and the Poincar´e estimate from [AGG94, Theorem 3.3] (using the assumption +d = 3), this leads to +∥tr0U∥2 +[L2(ρ−2 +x ,Rd),L2(Rd)]1−β,2 ≲ +� ∞ +0 +yα∥ρ−1 +x U(y)∥2 +L2(Rd) dy + +� ∞ +0 +yα∥∂yU(y)∥2 +L2(Rd) dy +≲ ∥∇U∥2 +L2(yα,Rd×R+), +which produces the desired estimate. +3.2 Poincar´e inequalities and well-posedness +We now show the well-posedness of our variational formulations. +The main ingredient is a +Poincar´e type estimate. +Lemma 3.2. Let α ∈ (−1, 1). Let Y ∈ (0, ∞] and U ∈ H1 +ρ(yα, Rd × (0, Y)). There exists a +µ0 > 0 such that for all µ ∈ [0, µ0) there holds +� Y +0 +� +Rd yαρµ−2|U|2 dxdy ≤ C +�� Y +0 +� +Rd yαρµ|∇U|2 dxdy + |3 − d|∥tr0U∥2 +L2(Rd) +� +(3.2) +provided the right-hand side is finite. +Proof. For Poincar´e inequalities on the full-space without the additional weight yα, we refer +to [AGG94]. Estimate (3.2) for the case d = 3 follows directly from multiplying a full-space +Poincar´e-inequality, see for example [AGG94, Theorem 3.3], applied only in x with yα and +integrating over (0, Y). More details can also be found in our forthcoming work [FR22]. +It remains to show (3.2) for d = 2. We write U(x, y) = U(x, 0) + +� y +0 ∂yU(x, τ) dτ, which gives +� Y +0 +� +Rd yαρµ−2|U|2 dxdy ≲ +� Y +0 +� +Rd yαρµ−2|U(x, 0)|2 + yαρµ−2� � y +0 +∂yU(x, τ) dτ +�2 +dxdy. +Since +� Y +0 yαρµ−2 ≲ 1 for sufficiently small µ < µ0, with µ0 > 0 depending only on α, the first +term on the left-hand side can be bounded by C ∥tr0U∥2 +L2(Rd). For the second term, we employ +a weighted Hardy-inequality, see e.g. [Muc72], to obtain +� Y +0 +� +Rd yαρµ−2� � y +0 +∂yU(x, τ) dτ +�2 +dxdy ≲ +� +Rd +� Y +0 +yαρµ|∂yU|2 dydx, +which shows the claimed inequality. +Using this Poincar´e- type inequality, we can now look at the well-posedness of our problem. +Proof of Proposition 2.3. The boundedness of the bilinear forms A(·, ·) and AY(·, ·) follows di- +rectly from the Cauchy-Schwarz inequality and the definition of the norms ∥·∥H and ∥·∥HY +respectively. +8 + +Let Y ∈ (0, ∞]. Coercivity of the bilinear forms follows directly from the Poincar´e inequalities +in Lemma 3.2, since +��UY��2 +HY = +� Y +0 +� +Rd yαρ−2|UY|2 dxdy + +� Y +0 +� +Rd yα|∇UY|2 dxdy + s +��tr0UY��2 +L2(Rd) +(3.2) +≲ +� Y +0 +� +Rd yα|∇UY|2 dxdy + (s + (3 − d)) +��tr0UY��2 +L2(Rd) . +By assumption on s and d, the trace term is not present for the case s = 0. Therefore, the +right-hand side can be bounded by CAY(UY, UY). +Thus, the Lax-Milgram lemma shows well-posedness provided the right-hand side of the varia- +tional formulation is a bounded linear functional. For the case s > 0, we can directly use the +definition of the HY-norm together with supp f ⊂ Ω to obtain +� +Rd ftr0UY dx ≤ s−1 ∥f∥L2(Ω) s +��tr0UY�� +L2(Rd) ≤ s−1 ∥f∥L2(Ω) +��UY�� +HY . +For Y = ∞ and s = 0, which implies d = 3 by assumption, the trace estimate (3.1b) gives +� +Rd ftr0U dx ≤ +���ρ(x, 0)βf +��� +L2(Ω) +���ρ(x, 0)−βtr0U +��� +L2(Rd) ≲ ∥f∥L2(Ω) ∥∇U∥L2(yα,Rd×R+) +≤ ∥f∥L2(Ω) ∥U∥H . +For the case Y < ∞ and s = 0, we use a cut-off function χ satisfying χ ≡ 1 on (0, Y/2), +supp χ ⊂ (0, Y) and ∥∇χ∥L∞(R+) ≲ Y−1. As Ω is bounded, this gives with the trace estimate +[KM19, Lem. 3.7] +� +Rd ftr0UY dx ≤ ∥f∥L2(Ω) +��tr0(χUY) +�� +L2(Ω) +≲ ∥f∥L2(Ω) +���χUY�� +L2(yα,Ω×(0,Y)) + +��∇(χUY) +�� +L2(yα,Ω×(0,Y)) +� +≲ ∥f∥L2(Ω) +���UY�� +L2(yα,Ω×(0,Y)) + 1 +Y +��∇UY�� +L2(yα,Ω×(0,Y)) +� +≤ C +� +1 + 1 +Y +� +∥f∥L2(Ω) +��UY�� +HY , +which finishes the proof. +3.3 The truncation error +In the following subsection, we study the truncated problem (2.5). The main goal is to derive +decay estimates in the truncation parameter Y and consequently convergence of the solution of +the truncated problem to the solution of the non-truncated problem as Y → ∞. +The following lemma is the key to the main results of Proposition 2.4 and Proposition 2.5. +Using inf-sup theory we obtain that solutions to the Caffarelli-Silvestre extension problem and +the truncated problem (in y-direction) lie in certain weighted Sobolev spaces. The additional +weights then directly provide the rates of decay. In fact, we establish that the solutions are +in two different types of weighted spaces: spaces weighted with (1 + y)µ with µ given by (2.7) +(decay only in y) and spaces with weights ρε for sufficiently small ε (decay in all directions). +9 + +Lemma 3.3. Let y0 > 0. Fix Y ∈ (y0, ∞), and let µ be given by (2.7). Let UY solve (2.5). +Then, UY satisfies the estimate +� Y +0 +yα� +(1 + y)µ∥∇UY(y)∥2 +L2(Rd)+(1 + y)µ∥ρ(·, y)−1UY(y)∥2 +L2(Rd) +� +dy ≤ C min(s−1, 1)2 ∥f∥2 +L2(Ω) . +(3.3) +In addition, for Y ∈ (0, ∞], there exists ε > 0, depending only on α and Ω such that +� Y +0 +yα +� +Rd ρε|∇UY(x, y)|2dxdy ≤ C min(s−1, 1)2 ∥f∥2 +L2(Ω) . +(3.4) +In both cases, the constant C does only depend on Ω, d, α, and y0. +Proof. By the uniqueness of Proposition 2.3, it suffices to show existence of such a solution. To +that end, we use inf-sup-theory, see, e.g., [SS11, Thm. 2.1.44], i.e., we have to show +inf +U∈Xµ,Y\{0} +sup +V∈Y−µ,Y\{0} +��AY(U, V) +�� +∥U∥Xµ,Y ∥V∥Y−µ,Y +≥ γ > 0 +(inf-sup condition), +∀V ∈ Y−µ,Y\{0} : +sup +U∈Xµ,Y\{0} +��AY(U, V) +�� > 0 +(non-degeneracy condition) +with spaces Xµ,Y, Y−µ,Y specified in the following. +We define the ansatz space Xµ,Y as a subspace of H1 +ρ(yα, Rd × (0, Y)) of functions for which the +norm +∥U∥2 +Xµ,Y := +� Y +0 +yα(1 + y)µ∥∇U(y)∥2 +L2(Rd) dy + s ∥tr0U∥2 +L2(Rd) +is finite. +Step 1 (Proof of (3.11) with µ = 1 − α): We start with the simpler case s > 0 and take +µ = 1 − α. Let χ(y) := +� +1 +y ≤ 1 +y1−α +y > 1. For U ∈ Xµ,Y, we define V := (1 + δχ(y))U (for some +0 < δ < 1 to be fixed later) and calculate +� Y +0 +� +Rd yαAx∇U · ∇Vdxdy ≥ A0 +� Y +0 +yα(1 + δχ(y))∥∇U(y)∥2 +L2(Rd)dy ++ +� Y +1 +� +Rd yαδ(1 − α)y−αU∂yU dxdy += A0 +� Y +0 +yα(1 + δχ(y))∥∇U(y)∥2 +L2(Rd)dy ++ δ(1 − α) +2 +� +Rd +� Y +1 +∂ +∂y +� +U2� +dydx += A0 +� Y +0 +yα(1 + δχ(y))∥∇U(y)∥2 +L2(Rd)dy +− δ(1 − α) +2 +� +Rd U(x, 1)2 dx + δ(1 − α) +2 +� +Rd U(x, Y)2 dx +≥ A0 +� Y +0 +yα(1 + δχ(y))∥∇U(y)∥2 +L2(Rd)dy − δ(1 − α) +2 +� +Rd U(x, 1)2 dx. +10 + +In order to estimate the last term, we employ +U(1)2 ≤ 2U(0)2 + 2 +���� +� 1 +0 +∂yU(y) dy +���� +2 +≤ 2U(0)2 + 2 +� 1 +0 +yα|∂yU(y)|2dy +� 1 +0 +y−αdy += 2U(0)2 + +2 +1 − α +� 1 +0 +yα|∂yU(y)|2dy, +which gives using 1 + δχ(y) ≥ δ +4(1 + y)1−α +� Y +0 +� +Rd yαAx∇U · ∇V dxdy ≥ (A0 − δ) +� Y +0 +yα(1 + δχ(y))∥∇U(y)∥2 +L2(Rd)dy +− δ(1 − α) +� +Rd U(x, 0)2 dx +≥ δ +4(A0 − δ) +� Y +0 +yα(1 + y)1−α∥∇U(y)∥2 +L2(Rd)dy +− δ(1 − α) +� +Rd U(x, 0)2 dx. +Consequently, we obtain +AY(U, V) ≥ δ +4(A0 − δ) +� Y +0 +yα(1 + y)1−α∥∇U(y)∥2 +L2(Rd) dy ++ +� +sdβ − δ(1 − α) +� +∥tr0U∥2 +L2(Rd). +(3.5) +Choosing δ < min(A0/2, sdβ/(2 − 2α)), both terms on the right-hand side in (3.5) are non- +negative and using ∥V∥Xα−1,Y ≲ ∥U∥X1−α,Y , which follows easily from (1 + δχ(y)) ≲ (1 + y)1−α, +gives the inf-sup condition for the ansatz space X1−α,Y and the test space Xα−1,Y. Moreover, +the inf-sup constant behaves like ∼ min(1, s). +The non-degeneracy condition follows essentially with the same arguments, as, for given V, the +function U := (1 + δχ(y))−1V provides the positivity of the bilinear form. +The definition of the norm in the test-space and supp f ⊂ Ω implies +(f, tr0V)L2(Rd) ≤ ∥f∥L2(Ω) ∥V∥Xα−1,Y , +which gives a bound for the right-hand side. Now, general inf-sup theory provides the existence +of a solution that satisfies the claimed decay properties. +Step 2 (Proof of (3.11) with µ = 1 + α): Next, we show that the rate of decay µ = 1 + α is +possible for s > 0 and even for s = 0. In the following, we only discuss the harder case s = 0 +as for s > 0, we only obtain an additional non-negative term in the bilinear form. Here, we use +the test space induced by the norm +∥V∥2 +�Y−µ,Y := +� Y +0 +yα +ln(y + 2)2(1 + y)µ ∥∇V(y)∥2 +L2(Rd) dy + ∥tr0V∥2 +L2(Ω) . +For given U ∈ Xµ,Y, we choose the test function +V(x, y) := y1+αU(x, y) + (1 + α) +� Y +y +τ αU(x, τ) dτ +11 + +with the derivatives +∇xV = y1+α∇xU + (1 + α) +� Y +y +τ α∇xU(τ) dτ +and +∂yV(y) = y1+α∂yU(y). +The function V is indeed in the test space, since we can bound the norm ∥V∥ �Y−1−α,Y by +∥V∥2 +�Y−1−α,Y = +� Y +0 +yα +ln(y + 2)2(1 + y)1+α ∥∇V(y)∥2 +L2(Rd) dy + ∥tr0V∥2 +L2(Ω) +≲ +� Y +0 +yα+2(1+α) +ln(y + 2)2(1 + y)1+α ∥∇U(y)∥2 +L2(Rd) dy ++ +� +Rd +� Y +0 +yα +ln(y + 2)2(1 + y)1+α +��� +� Y +y +τ α∇xU(τ) dτ +��� +2 +dydx + ∥tr0V∥2 +L2(Ω) . +(3.6) +Since the first term is readily bounded due to U ∈ X1+α,Y and 1 + α > 0, we focus on the +second. Using a weighted Hardy inequality, see e.g. [Muc72], with the weight y−1/2/ ln(y + 2) +that is square integrable in R+ we obtain +� +Rd +� Y +0 +yα +ln(y + 2)2(1 + y)1+α +��� +� Y +y +τ α∇xU(τ) dτ +��� +2 +dydx +≤ +� +Rd +� Y +0 +��� +y−1/2 +ln(y + 2) +� Y +y +τ α∇xU(τ) dτ +��� +2 +dydx +≲ +� +Rd +� Y +0 +y1+2α|∇xU(y)|2dydx ≤ ∥U∥2 +X1+α,Y. +(3.7) +What is left is to bound the trace of V. We use a cut-off function χ satisfying χ ≡ 1 on (0, y0/2), +supp χ ⊂ (0, y0), and ∥∇χ∥L∞(R) ≤ C with a constant C depending only on y0. Then, +V(x, 0)2 = (χV)(x, 0)2 = +� � y0 +0 +∂y(χV)(x, y) dy +�2 +≤ y1−α +0 +1 − α +� y0 +0 +yα|∂y(χV)|2 dy +≲ +� y0 +0 +yα � +|∂yV|2 + |∂yχ|2V2� +dy. +Integration over Ω and using the definition of V gives +∥tr0V∥2 +L2(Ω) ≲ +� y0 +0 +yα∥∇V(y)∥2 +L2(Ω)dy ++ +� +Ω +� y0 +0 +y2+3α|∂yχ|2U2dydx + +� +Ω +� y0 +0 +yα��� +� Y +y +τ αU(x, τ) dτ +��� +2 +dydx. +On Ω×(0, y0) we can insert any appearing weights in the ansatz-space and test-space as needed, +which just adds multiplicative constants independent of Y. Moreover, we can employ standard +Poincar´e-inequalities to bound the L2-norm (here, the integrand even vanishes on (0, y0/2)). +Repeating the arguments from (3.6) and (3.7) (with slightly changed weight in the Hardy +inequality to insert the weight ρ−2), we obtain the bound +∥tr0V∥L2(Ω) ≲ ∥U∥X1+α,Y. +12 + +Thus, we have shown ∥V∥ �Y−1−α,Y ≲ ∥U∥X1+α,Y. +We continue with inserting U, V into the +truncated bilinear form AY(·, ·), which leads to +AY(U, V) = +� Y +0 +� +Rd yαAx∇U · ∇V dxdy ≥ A0 +� Y +0 +y1+2α∥∇U(y)∥2 +L2(Rd)dy ++ (1 + α) +� +Rd +� Y +0 +yαA1/2∇xU +� Y +y +τ αA1/2∇xU(τ) dτ dy dx +=: I + II. +(3.8) +We show that the term II is non-negative. To simplify notation, we write v(y) := A1/2∇xU(y) +and suppress the x-dependency. We note that by the chain rule there holds +yαv(y) · +� Y +y +τ αv(τ) dτ = −1 +2 +d +dy +��� +� Y +y +τ αv(τ) dτ +��� +2 +. +This gives for the second term in (3.8): +II = −(1 + α) +2 +� +Rd +� Y +0 +d +dy +��� +� Y +y +τ αv(τ) dτ +��� +2 +dydx += (1 + α) +2 +� +Rd +��� +� Y +0 +τ αv(τ) dτ +��� +2 +dx ≥ 0. +Overall, we get using (1 + y1+α) ≳ (1 + y)1+α +AY(U, V) + AY(U, U) ≥ A0 +� Y +0 +yα(1 + y1+α)∥∇U(y)∥2 +L2(Rd)dy ≳ ∥U∥2 +X1+α,Y +≳ ∥U∥X1+α,Y∥U + V∥ �Y−1−α,Y, +where the last inequality follows from the triangle inequality and ∥V∥ �Y−1−α,Y ≲ ∥U∥X1+α,Y. +For the non-degeneracy condition, for a given V, we can choose U = V, which is in the ansatz- +space, since due to Y < ∞ the weights in the gradient terms in the ansatz- and test-space are +equivalent. +By definition of the test-space and supp f ⊂ Ω, there holds (f, tr0V)L2(Rd) ≤ ∥f∥L2(Ω) ∥V∥ �Y−1−α,Y. +Consequently, we obtain unique solvability of our weak formulation in the ansatz-space, which +gives the decay estimate. +Step 3 (Proof of (3.4)): Again, we use inf-sup theory with a different ansatz space. Here, for +ε > 0, we choose it to be a subspace of H1 +ρ(yα, Rd×R+) such that additionally +� +Rd×(0,Y) yαρε |∇U|2 +is finite. We only work out the case s = 0 in the following, for s > 0, the same argument can +be made by additionally including a trace term in the norm. Setting z := (x, y) ∈ Rd+1 and +V(z) := ρε(z)U(z), we get with Young’s inequality and ρ−2|z|2 ≤ 1 +AY(U, V) ≥ A0 +� +Rd×(0,Y) +yαρε |∇U|2 dz + ε +� +Rd×(0,Y) +yαρε−2z · Ax∇UU dz +≥ 1 +2A0 +� +Rd×(0,Y) +yαρε |∇U|2 dz − ε2 +2 +∥Ax∥2 +L∞(Rd×R+) +A0 +� +Rd×(0,Y) +yαρε−2 |U|2 dz +≥ 1 +2A0 +� +Rd×(0,Y) +yαρε |∇U|2 dz − CPε2 +2 +∥Ax∥2 +L∞(Rd×R+) +A0 +� +Rd×(0,Y) +yαρε |∇U|2 dz, +13 + +where in the last step we applied the Poincar´e estimate from (3.2) for sufficiently small ε > 0. +If ε is sufficiently small, we can also absorb the negative term and show inf-sup stability with +the test space carrying ρ−ε as a weight. +The non-degeneracy condition and the bound on +(f, tr0V)L2(Rd) are easily checked. +Before we can proceed to quantify the cutoff error, we need the following result on the existence +of a stable extension from the cutoff domain Rd × (0, Y) to a larger set. +Lemma 3.4. Fix Y > 0. Then, there exists an extension operator E to the domain Rd ×(0, 3 +2Y) +such that: +(i) Eu = u in Rd × (0, Y). +(ii) The following stability result holds for all µ ≥ 0 and U ∈ H1 +ρ(yα, Rd × (0, Y)), if the +right-hand side is finite: +� +3 +2 Y +0 +yα+µ∥∇EU∥2 +L2(Rd) dy ≤ C +� Y +0 +yα+µ∥∇U∥2 +L2(Rd) dy. +(3.9) +The constant C > 0 depends on α, µ and d but is independent of U and Y. +Proof. We extend U by reflection along the line y = Y, i.e., we define +W(x, y) := +� +U(x, y) +0 ≤ y ≤ Y, +U(x, 2Y − y) +Y < y ≤ 3 +2Y. +By construction, the function has no jump across the line y = Y. +For the stability in the +extension domain, we compute +� +3 +2Y +Y +yα+µ∥∇W(·, y)∥2 +L2(Rd) dy ≲ Yα+µ +� +3 +2Y +Y +∥∇U(·, 2Y − y)∥2 +L2(Rd) dy += Yα+µ +� Y +Y/2 +∥∇U(·, τ)∥2 +L2(Rd) dτ +≲ +� Y +Y/2 +τ α+µ∥∇U(·, τ)∥2 +L2(Rd) dτ. +This finishes the proof. +Using this extension operator, we obtain that the sequence (U(3/2)nY)n∈N, where the cutoff point +is moved outward by a factor of 3/2 in each step, is a Cauchy sequence. +Lemma 3.5. Let UY denote the solution to (2.5) with truncation parameter Y > 0 and accord- +ingly let U3/2Y denote the solution with a cutoff at 3/2Y. Let µ be given by (2.7). Then, there +holds: +∥U3/2Y − UY∥HY ≤ CY−µ/2 ∥f∥L2(Ω) . +Iterative application of the estimate for n, m ∈ N0, n > m leads to +∥U(3/2)nY − U(3/2)mY∥HY ≤ CY−µ/2 +�2 +3 +�µ m/2 � +1 − +�2 +3 +� µ +2 (n−m) � +∥f∥L2(Ω) . +14 + +Proof. We compute using the coercivity of AY(·, ·) from Proposition 2.3 and the extension +operator from Lemma 3.4 +∥UY − U3/2Y∥2 +HY ≲ AY(UY − U3/2Y, UY − U3/2Y) += AY(UY, UY − U3/2Y) − AY(U3/2Y, UY − U3/2Y) += (f, tr0(UY − U3/2Y))L2(Rd) − A3/2Y(U3/2Y, E(UY − U3/2Y)) ++ +� +3 +2 Y +Y +yα +� +Rd Ax∇U3/2Y∇E(UY − U3/2Y) dxdy. +By definition of U3/2Y and the extension operator E, the first two terms cancel. Thus, we can +focus on bounding the remaining integral +� +3 +2Y +Y +yα +� +Rd Ax∇U3/2Y∇E(UY − U3/2Y) dxdy +≲ Y−µ/2� � +3 +2Y +Y +yα+µ ���∇U3/2Y��� +2 +dy +�1/2� � +3 +2Y +Y +yα ���∇E(UY − U3/2Y) +��� +2 +dy +�1/2 +≲ Y−µ/2� � +3 +2Y +Y +yα+µ ���∇U3/2Y��� +2 +dy +�1/2∥UY − U3/2Y∥H1ρ(yα,Rd×(0,Y)). +Using ∥UY − U3/2Y∥H1ρ(yα,Rd×(0,Y)) ≤ ∥UY − U3/2Y∥HY and canceling one such power then gives +together with the decay estimate of Lemma 3.3: +∥UY − U3/2Y∥HY ≲ Y−µ/2 ∥f∥L2(Ω) . +(3.10) +Using a telescoping sum, we can write: +U(3/2)nY − U(3/2)mY = +n−1 +� +ℓ=m +� +U(3/2)ℓ+1Y − U(3/2)ℓY� +. +With estimate (3.10) applied iteratively, this leads to +∥U(3/2)nY − U(3/2)mY∥HY ≲ +n−1 +� +ℓ=m +∥U(3/2)ℓ+1Y − U(3/2)ℓY∥HY ≲ Y−µ/2 +n−1 +� +ℓ=m +�3 +2 +�− µℓ +2 ∥f∥L2(Ω) +≃ Y−µ/2 +�2 +3 +� µ +2 m � +1 − +�2 +3 +� µ +2 (n−m) � +∥f∥L2(Ω) . +This finishes the proof. +Using the Cauchy sequence property, we can now show convergence of the truncated solution +to the full-space solution as stated in Proposition 2.4. +Proof of Proposition 2.4. We focus on the case s = 0. In the case s > 0, the same arguments can +be made including the L2-norm of of the traces, which directly gives the additional statement +regarding the convergence of tr0UY to tr0U. +Step 1: We start by fixing the half-ball B+ +Y ⊂ Rd × [0, ∞) of radius Y centered at the origin +and write z = (x, y) ∈ Rd+1. Let ε > 0 be such that the decay estimate (3.4) holds. +15 + +Defining E := U − UY and using the equations satisfied by U and UY, we use integration by +parts to obtain +� +B+ +Y +yαAx∇E · ∇E dxdy = +� +∂B+ +Y +yαAx∇E · νE dxdy += (1 + Y2)−ε/2 +� +|z|=Y +yαρεAx∇E · νE dxdy − sdβ +� +|x|≤Y +|tr0E|2 dx += (1 + Y2)−ε/2 +� +∂B+ +Y +yαρεAx∇E · νE dxdy ++ sdβ +� +|x|≤Y +�1 + |x|2 +1 + Y2 +�ε/2 +|tr0E|2 dx − sdβ +� +|x|≤Y +|tr0E|2 dx +≤ (1 + Y2)−ε/2 +� +∂B+ +Y +yαρεAx∇E · νE dxdy. +Integration by parts back (replacing ∇E by ∇(ρεE)) gives +� +∂B+ +Y +yαρεAx∇E · νE dxdy = +� +B+ +Y +yαAx∇E · (∇ρε)E dxdy + +� +B+ +Y +yαρεAx∇E · ∇E dxdy +≲ +� � +B+ +Y +yαρε |∇E|2 dz +�1/2� � +B+ +Y +yαρε−2 |E|2 dz +�1/2 ++ +� +B+ +Y +yαρε|∇E|2 dz. +We replace the half-ball B+ +Y by the cylinder Rd × (0, Y) and use the Poincar´e estimate (3.2). +Together with the decay estimate (3.4) this gives boundedness of the right-hand side with a +constant independent of Y. Consequently, we obtain +� +B+ +R +|∇E|2 dxdy ≲ +� +B+ +Y +yαAx∇E · ∇E dxdy ≤ C(1 + Y2)−ε/2 → 0 +as Y → ∞ +for all bounded half balls B+ +R with R ≤ Y, which gives UY → U in H1 +ρ(yα, B+ +R). +Step 2: As (U(3/2)nY)n∈N is a Cauchy-sequence, there exists a limit �U ∈ H1 +ρ(yα, Rd × (0, �Y)). +Assume that �U ̸= U. Then, there has to exist a half ball B+ +R such that +� +B+ +R +yα���∇(U− �U) +��� +2 +dxdy ̸= +0. For sufficiently large n, we have R ≤ (3/2)nY. This leads to +� +B+ +R +yα���∇(U − �U) +��� +2 +dxdy ≤ +� +B+ +R +yα���∇(U − U(3/2)nY) +��� +2 +dxdy + +� +B+ +R +yα���∇(U(3/2)nY − �U) +��� +2 +dxdy. +By step 1, the first term converges to zero and by definition of �U the second term converges +to zero. However, this is a contradiction to the assumption and therefore U = �U and we have +established the claimed convergence. +We can now estimate the truncation error and establish a rate of convergence as Y → ∞. +Proof of Proposition 2.5. Using a telescoping sum, we write +UY − U = +N +� +n=0 +� +UY( 3 +2 )n − UY( 3 +2)n+1� ++ UY( 3 +2 )N+1 − U. +16 + +Since we have already established that UY → U for Y → ∞ in Proposition 2.4, we can pass to +the limit N → ∞ and use Lemma 3.5 to estimate: +∥UY − U∥H1ρ(yα,Rd×(0,Y)) ≲ +∞ +� +n=0 +∥UY( 3 +2)n − UY( 3 +2 )n+1∥H1ρ(yα,Rd×(0,Y)) +≲ Y−µ/2 +∞ +� +n=0 +�3 +2 +�− µn +2 ∥f∥L2(Rd) ≤ Y−µ/2 +1 +1 − (2 +3)µ/2 ∥f∥L2(Rd) . +This finishes the proof. +We can now also close the small gap that the decay in Lemma 3.3 does not hold for the non- +truncated domain Y = ∞. +Corollary 3.6. Let µ be given by (2.7). Let U solve (2.4). Then, there exists a constant C > 0 +depending only on Ω, d, and α such that +� ∞ +0 +yα� +(1 + y)µ∥∇U(y)∥2 +L2(Rd) + (1 + y)µ∥ρ(·, y)−1U(y)∥2 +L2(Rd) +� +dy ≤ C ∥f∥2 +L2(Ω) . +(3.11) +Proof. We take a sequence (Yn)n∈N with 1 ≤ Yn → ∞ for n → ∞ and consider the correspond- +ing truncated solutions UYn to (2.5). By Lemma 3.3 and Proposition (2.5) it holds: +� Yn +0 +yα(1 + y)µ∥∇U(y)∥2 +L2(Rd) dy + +� Yn +0 +yα(1 + y)µ∥ρ−1U(y)∥2 +L2(Rd) dy +≤ (1 + Yn)µ ��U − UYn��2 +H1ρ(yα,Rd×(0,Yn)) ++ +� Yn +0 +yα(1 + y)µ∥∇UYn(y)∥2 +L2(Rd) dy + +� Yn +0 +yα(1 + y)µ∥ρ−1UYn(y)∥2 +L2(Rd) dy +≲ Yµ +nY−µ +n ∥f∥2 +L2(Ω) + min(s−1, 1)2 ∥f∥2 +L2(Ω) ≲ ∥f∥2 +L2(Ω). +Taking n → ∞ then gives the stated result. +4 Regularity and higher order decay +In this section, we derive regularity estimates for solutions to the extension problem. Assuming +sufficient differentiability of the data, we are in particular interested in weighted estimates for +higher-order y-derivatives as such estimates are needed to establish exponential approximation +estimates of hp–type. +In order to derive suitable regularity estimates around y = 0, we need to derive an initial shift +in a weighted space. +Lemma 4.1. Fix Y ∈ (0, ∞]. Let U solve (2.5). Then, there exists ε > 0 independent of Y and +U such that +� Y +0 +yα� +y−ε∥∇U(y)∥2 +L2(Rd) + y−ε∥ρ(·, y)−1U(y)∥2 +L2(Rd) +� +dy ≤ C ∥f∥2 +L2(Ω) . +(4.1) +Proof. Similar to the proof of Lemma 3.3, we use inf-sup theory to derive the stated bound. In +the following, we only work out the details for the case s = 0. The case s > 0 can be treated as +shown in Lemma 3.3 by also including a trace term in the norm of the ansatz space. +17 + +Here, for any �ε ∈ R, we define the space X�ε,Y as the space H1 +ρ(yα−�ε, Rd × (0, Y)) of functions +with finite norm +∥U∥2 +X�ε,Y := +� Y +0 +yα−�ε� +∥∇U(y)∥2 +L2(Rd) + ∥ρ(·, y)−1U(y)∥2 +L2(Rd) +� +dy. +As ansatz space, we take Xε,Y, where ε > 0 is sufficiently small. As test space we use X−ε,Y. +For fixed α ∈ (−1, 1), we actually may choose ε > 0 such that α ± ε ∈ (−1, 1) (subsequently, +we will derive an additional restriction on ε). +For given U ∈ Xε,Y, we define the test function V(x, y) := y−εU(x, y) + ε +� y +0 τ −ε−1U(x, τ)dτ. +Using Hardy’s inequality (noting that α + ε > −1), we obtain that this test-function is indeed +in the test-space +� Y +0 +yα+ε ∥∇V(y)∥2 +L2(Rd) dy ≲ +� Y +0 +yα+εy−2ε ∥∇U(y)∥2 +L2(Rd) dy ++ +� +Rd +� Y +0 +yα+ε +� +ε +� y +0 +τ −ε−1∇xU(τ)dτ +�2 +dydx +≲ (1 + ε2) +� Y +0 +yα−ε ∥∇U(y)∥2 +L2(Rd) dy < ∞. +(4.2) +The weighted L2-term in the definition of Xε,Y can be treated using the Poincar´e inequality +(3.2) replacing α with α − ε therein noting that α − ε ∈ (−1, 1) by assumption on ε. +Inserting the test function into the bilinear form gives +AY(U, V) = +� +Rd +� Y +0 +yα−εAx∇U · ∇Udydx + ε +� +Rd +� Y +0 +yαA∇xU +� y +0 +τ −ε−1∇xU(τ)dτ dydx. +Using Young’s inequality together with Hardy’s inequality (noting again that α + ε > −1), we +obtain +ε +� +Rd +� Y +0 +yαA∇xU +� y +0 +τ −ε−1∇xU(τ)dτ dydx ≤ 1 +2 +� +Rd +� Y +0 +yα−εAx∇U · ∇Udydx ++ 1 +2ε2 +� +Rd +� Y +0 +yα+ε +�� y +0 +τ −ε−1A1/2∇xU(τ)dτ +�2 +dydx +≤ 1 +2 +� +1 + CHε2� � +Rd +� Y +0 +yα−εAx∇U · ∇U dydx, +where CH indicates the constant in the Hardy inequality. Therefore, we obtain +AY(U, V) ≥ A0 +2 +� +1 − CHε2� � Y +0 +yα−ε ∥∇U∥2 +L2(Rd) dy. +Together with the Poincar´e estimate of Lemma 3.2, we obtain the inf-sup condition upon choos- +ing ε < C−1/2 +H +. +For the non-degeneracy condition, we fix V ∈ X−ε,Y and choose U = yεV − ε +� y +0 τ ε−1V(τ)dτ. +Then, essentially the same estimates as above can be made by noting that, by assumption we +have α − ε > −1, thus Hardy inequalities with the necessary modified weights can be employed +here. +The right-hand side can be bounded using the support properties of f together with a trace +estimate (in the weighted space L2(yα+ε, Ω × (0, Y)) noting that α + ε ∈ (−1, 1)) +���(f, tr0V)L2(Rd) +��� ≤ ∥f∥L2(Ω) ∥tr0V∥L2(Ω) ≤ ∥f∥L2(Ω) ∥∇V∥L2(yα+ε,Ω×(0,Y)) ≤ ∥f∥L2(Ω) ∥V∥X−ε,Y . +Now, classical inf-sup theory gives the claimed estimate. +18 + +With the initial shift in place, we can look at higher order derivatives. We first formulate the +“shift-by-one” as a separate lemma. +Lemma 4.2. Fix Y ∈ (0, ∞] and let W ∈ H1 +ρ(yα, Rd × (0, Y)) solve the problem +− div +� +yαAx∇W +� += F +in Rd × (0, Y) +with given right-hand side F. Then, for all ℓ ∈ N and ε ∈ (0, 1), the estimate +��yℓ−ε∇W +�� +L2(yα,Rd×(0,Y)) ≲ ℓ +��yℓ−1−εW +�� +L2(yα,Rd×(0,Y)) + +��yℓ+1−εF +�� +L2(y−α,Rd×(0,Y)) +holds, provided that the right-hand side is finite. The implied constant is independent of ℓ and +W. +Proof. If Y = ∞, let N ∈ N, and we fix a cutoff function ˜χN ∈ C∞ +0 (R) such that ˜χN ≡ 1 on +[0, N] and ˜χN ≡ 0 on (2N, ∞) with ∥˜χ′ +N∥L∞(R) ≤ 1/N. We define ωN(y) := yℓ−ε ˜χN. In the +easier case Y < ∞, we can skip the cutoff function altogether. For brevity, we therefore only +work out the case Y = ∞, the other case follows analogously. +We start with multiplying the equation for W with the test function V := ω2 +NW, and integrate +by parts over Rd×(0, ∞). As the weight function ωN and consequently also V vanishes at y = 0, +we do not get any boundary contributions. This gives with Young’s inequality +∥ωNA1/2 +x ∇W∥2 +L2(yα,Rd×R+) += +� +Rd×R+ ω2 +N(y)F Wdxdy − +� +Rd×R+ 2ω′ +N(y)ω(y)∂yWWdxdy +≤ ∥yωNF∥L2(y−α,Rd×R+)∥y−1ωNW∥L2(yα,Rd×R+) + 2∥ωN∂yW∥L2(yα,Rd×R+)∥ω′ +NW∥L2(yα,Rd×R+) +≤ 1 +2∥yωNF∥2 +L2(y−α,Rd×R+) + 1 +2∥y−1ωNW∥2 +L2(yα,Rd×R+) ++ 1 +2∥ωN∂yW∥2 +L2(yα,Rd×R+) + 2∥ω′ +NW∥2 +L2(yα,Rd×R+). +Absorbing the third term in the left-hand side provides +∥ωNA1/2 +x ∇W∥2 +L2(yα,Rd×R+) ≲ ∥yωNF∥2 +L2(y−α,Rd×R+) + ∥y−1ωNW∥2 +L2(yα,Rd×R+) ++ ∥ω′ +NW∥2 +L2(yα,Rd×R+). +For N → ∞, using Ax ≥ A0, the left-hand side converges to the weighted L2-norm we are +looking for. Similarly, the first two terms on the right-hand side converge to the appropriate +objects of the final estimate. Therefore we focus on the last term and show an uniform bound: +∥ω′ +NW∥L2(yα,Rd×R+) ≤ (ℓ − ε)∥yℓ−1−ε ˜χNW∥L2(yα,Rd×R+) + ∥yℓ−ε ˜χ′ +NW∥L2(yα,Rd×R+) +≲ ℓ∥yℓ−1−εW∥L2(yα,Rd×R+) + 1 +N 2 +� 2N +N +y2 +���� +≲4N2 +yα+2ℓ−2−2ε∥W(y)∥2 +L2(Rd) dy +≲ ℓ∥yℓ−1−εW∥L2(yα,Rd×R+) + +� ∞ +0 +yα+2ℓ−2−2ε∥W(y)∥2 +L2(Rd) dy, +where we used that ˜χ′ +N vanishes outside of [N, 2N]. Therefore we can pass to the limit N → ∞ +to get the stated result. +19 + +Remark 4.3. Note that U as solution of (2.3) does not fit Lemma 4.2 since it is not in +L2 +α(Rd × (0, Y)). However, the previous lemma can be applied for derivatives of the solution of +(2.3). +We are now in position to show our main result regarding weighted regularity, Proposition 2.6. +Proof of Proposition 2.6. We note that away from y = 0, we can use standard elliptic regularity +theory to show that U is C∞(Rd × R) and we can focus on the weighted estimates. We prove +this by induction, starting with ℓ = 1. By differentiating the equation in the form div(Ax∇U)+ +α +y ∂yU = 0, we get that W := ∂ℓ +yU solves: +− div(yαAx∇W) = α +ℓ−1 +� +k=0 +(−1)k ℓ! +k! +∂k+1 +y +U +yℓ−k+1−α =: Fℓ. +(4.3) +For ℓ = 1, we employ Lemma 4.2 to obtain +��y1−ε∇∂yU +�� +L2(yα,Rd×(0,Y)) ≲ +��y−ε∂yU +�� +L2(yα,Rd×(0,Y)) + ∥y2−εy−2+α∂yU∥L2(y−α,Rd×(0,Y)) +≲ +��y−ε∂yU +�� +L2(yα,Rd×(0,Y)) ≲ ∥f∥L2(Ω) , +where in the last step we used Lemma 4.1. +For ℓ > 1, we use the induction assumption valid for k < ℓ (that allows to control derivatives +up to order ℓ), which gives +��yℓ+1−εFℓ +�� +L2(y−α,Rd×(0,Y)) ≲ +ℓ−1 +� +k=0 +ℓ! +k! +��yk−ε∂k+1 +y +U +�� +L2(yα,Rd×(0,Y)) +≲ ℓ! ∥f∥L2(Rd) +ℓ−1 +� +k=0 +Kk +≲ ℓ!Kℓ ∥f∥L2(Rd) . +Using Lemma 4.2 together with the induction assumption, we get +��yℓ−ε∇∂ℓ +yU +�� +L2(yα,Rd×(0,Y)) ≲ ℓ +��yℓ−1−ε∂ℓ +yU +�� +L2(yα,Rd×(0,Y)) + +��yℓ+1−εFℓ +�� +L2(y−α,Rd×(0,Y)) +≲ ℓ +��yℓ−1−ε∇∂ℓ−1 +y +U +�� +L2(yα,Rd×(0,Y)) + ℓ!Kℓ��f +�� +L2(Rd) +≲ ℓ!Kℓ ∥f∥L2(Rd) , +which proves the lemma. +Finally, we provide the proof for the regularity estimates for the x-derivatives. +Proof of Proposition 2.8. 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Math., 15(3):733– +791, 2015. +[SS11] +S. A. Sauter and C. Schwab. Boundary element methods, volume 39 of Springer +Series in Computational Mathematics. Springer-Verlag, Berlin, 2011. Translated +and expanded from the 2004 German original. +[ST10] +P. R. Stinga and J. L. Torrea. Extension problem and Harnack’s inequality for some +fractional operators. Comm. Partial Differential Equations, 35(11):2092–2122, 2010. +[SZB+18] +H. Sun, Y. Zhang, D. Baleanu, W. Chen, and Y. Chen. A new collection of real +world applications of fractional calculus in science and engineering. Communications +in Nonlinear Science and Numerical Simulation, 64:213 – 231, 2018. +[Tar07] +L. Tartar. An introduction to Sobolev spaces and interpolation spaces, volume 3 of +Lecture Notes of the Unione Matematica Italiana. Springer, Berlin; UMI, Bologna, +2007. +22 + diff --git a/5tE5T4oBgHgl3EQfPQ4_/content/tmp_files/load_file.txt b/5tE5T4oBgHgl3EQfPQ4_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..18e836acfc3c3cb651f949ec2969acce1c095c5d --- /dev/null +++ b/5tE5T4oBgHgl3EQfPQ4_/content/tmp_files/load_file.txt @@ -0,0 +1,757 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf,len=756 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='05503v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='NA] 13 Jan 2023 Fractional Diffusion in the full space: decay and regularity Markus Faustmann∗ and Alexander Rieder† January 16, 2023 We consider fractional partial differential equations posed on the full space Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using the well-known Caffarelli-Silvestre extension to Rd × R+ as equivalent defini- tion, we derive existence and uniqueness of weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We show that solutions to a truncated extension problem on Rd × (0, Y) converge to the solution of the original problem as Y → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Moreover, we also provide an algebraic rate of decay and derive weighted analytic-type regularity estimates for solutions to the truncated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' These results pave the way for a rigorous analysis of numerical methods for the full space problem, such as FEM-BEM coupling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 1 Introduction In recent years, models using non-integer powers of differential operators garnered lots of interest as the inherent non-locality of these operators gives a more accurate way to describe non-local processes in physics, finance or image processing, [BV16, SZB+18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Restricting these non-local PDE models to some bounded domain requires one to fix values of the solution everywhere outside of the domain, which may lead to some non-physical assumptions for the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Consequently, the full-space problem is oftentimes used in analytical works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In a similar vein, when working on a bounded domain, there are multiple non-equivalent defini- tions of fractional differential operators such as the fractional Laplacian, [LPG+20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The most common ones are the integral fractional Laplacian (defined pointwise as a singular integral) and the spectral fractional Laplacian (defined using spectral calculus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Consequently, it is of- tentimes not obvious, which definition of the fractional Laplacian should be used in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In contrast, working on the full space, one obtains a single natural definition as all different approaches are equivalent, [Kwa17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In this work, we analyze fractional PDEs in the full space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using the influential interpretation of elliptic fractional differential operators as Dirichlet-to-Neumann operators for degenerate elliptic PDEs, the so called Caffarelli-Silvestre extension, [CS07, ST10], defined on the half space Rd × R+, we show well-posedness of a weak formulation of the fractional PDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As, in general, analytical solutions to such problems are unknown, discretizations of the equations are usually employed to derive approximative solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' ∗Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria, markus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='faustmann@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='at †Institute for Analysis and Scientific Computing, TU Wien, Vienna, Austria, alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='rieder@tuwien.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='at 1 In the case of fractional PDEs on bounded domains Ω ⊂ Rd, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' [NOS15, BMN+19], a truncation to Ω×(0, Y) is used to be able to discretize the extension problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This induces two natural questions: does the solution to the truncated extension problem (with homogeneous Neumann condition on the artificial boundary) converge to the solution of the original problem and can the rate of convergence be quantified?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For bounded domains, [BMN+19] answered both questions by showing exponential decay in Y by exploiting an explicit representation for the y-dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In this article, we employ the truncation to the full space problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=', we study the extension problem on Rd × (0, Y) and answer both questions as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In this case, however, there is no closed form expression for the y-dependence available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Nonetheless, we show convergence of the truncated solution to the original solution in the full-space setting, but only with certain algebraic rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' From a technical standpoint, the explicit representation is replaced by applying purely variational techniques to show the decay properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1 Impact on numerical methods Numerical methods for fractional PDEs on bounded domains are fairly developed, as can e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' be seen in the survey articles [BBN+18, DDG+20, LPG+20] and we especially mention approxi- mations based on the finite element method (FEM), [AB17, BMN+19, ABH19, FKM22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' A key limitation to the FEM is the restriction to bounded computational domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' A classical refor- mulation for exterior problems uses boundary integral equations, which leads to the boundary element method (BEM), [SS11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' An approach for transmission problems on unbounded do- mains that is commonly employed is the combination of both methods, so called FEM-BEM couplings, [Cos88, Han90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The goal of our follow-up work, [FR22], is to formulate a fully com- putable symmetric FEM-BEM coupling method applied to fractional transmission problems posed in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' However, before a rigorous analysis of any numerical method can be made, analytical founda- tions regarding well-posedness and regularity of the problem at hand must be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As a second key result of this article, we establish analyticity of the solution in the extended direction y in terms of certain weighted Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This is achieved by deriving a small initial regularity shift in a weighted space and then employing bootstrapping arguments to control higher-order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Structurally, these estimates are similar to the ones for the case of bounded domains in [BMN+19, FMMS22] and show that solutions are in certain countably normed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Combined with our follow-up work [FR22], this article establishes that the Caffarelli-Silvestre extension approach can be combined with FEM-BEM coupling techniques to yield a good approximation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 Layout The present paper is structured as follows: In Section 2, we introduce our model problem and formulate assumptions on the data to be able to apply FEM-BEM techniques afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, the Caffarelli-Silvestre extension as well as its weak formulation and the weak formulation of the truncated problem are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Finally, we present our main results: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 shows well-posedness of both weak formulations, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4 provides convergence of the truncated solution to the solution posed on Rd ×R+ and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5 gives the algebraic rate of decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6 the regularity results in weighted Sobolev spaces are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Section 3 is then devoted to the proofs of the well-posedness and convergence results, where the key step is Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3, which shows decay properties of the full space solution as the truncation 2 parameter Y → ∞ by employing inf-sup theory and weighted spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Finally, in Section 4 the estimates for higher order derivatives are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Hereby, an initial regularity shift in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 allows to use an induction argument to show Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Moreover, a (finite) regularity result in the non-extended variables and a characterization of the solution in certain countably normed spaces is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 Notations Throughout the text we use the symbol a ≲ b meaning that a ≤ Cb with a generic constant C > 0 that is independent of any crucial quantities in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Moreover, we write ≃ to indicate that both estimates ≲ and ≳ hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For any multi index α = (α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' , αd) ∈ Nd 0, we denote the partial derivative ∂α = ∂α1 x1 · · · ∂αd xd of order |α| = �d i=1 αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Moreover, for k ∈ N, we employ classical integer order Sobolev spaces Hk(Ω) on (bounded) Lipschitz domains Ω and the fractional Sobolev spaces Ht(Rd) for t ∈ (0, 1) defined, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=', via Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 2 Main results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1 Model problem We consider a stationary fractional diffusion problem on the full space Rd with d = 2 or d = 3 given by Lβu + su = f in Rd (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1) with s ≥ 0, and β ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The self-adjoint operator L is hereby defined as Lu := − div � A∇u � , and, for functions u ∈ L2(Rd), the fractional differential operator Lβ is defined using spectral calculus Lβu := � σ(L) zβdE u, where E is the spectral measure of L and σ(L) is the spectrum of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using standard techniques this definition can be extended to tempered distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For the data, we assume that A : Rd → Rd×d is smooth and pointwise symmetric and positive definite in the sense that there exists A0 > 0 such that (A(x)y, y)2 ≥ A0 ∥y∥2 2 ∀y ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In order to avoid several additional difficulties due to decay conditions at infinity, we assume s ≥ σ0 > 0 for the case d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Additionally, we make the following assumptions on the coefficients in the model problem: There exists a bounded Lipschitz domain Ω ⊆ Rd such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' supp f ⊆ Ω, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' A ≡ I in Rd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 3 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We note that adding lower order terms to the operator is also covered by our techniques, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=', Lu := − div � A∇u � + cu, where c : Rd → R with c ≥ 0 is smooth and satisfies c ≡ c0 ∈ R in Rd \\ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' However, in order to make the key concepts more clear, we decided to stick to the case c = 0 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 The Caffarelli-Silvestre extension Following [ST10], we rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1) as an extension problem in a half space in Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The extension problem is conveniently described using weighted Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For any bounded open subset D ⊂ Rd × R and arbitrary α ∈ (−1, 1), we define L2(yα, D) as the space of square integrable functions with respect to the weight yα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Correspondingly, the Sobolev space H1(yα, D) ⊂ L2(yα, D) consists of functions, for which the norm ∥U∥2 H1(yα,D) := � � D yα���∇U(x, y) ��2 + ��U(x, y) ��2� dx dy is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As our model problem is formulated on an unbounded domain, we need to take care of the behaviour at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' To that end, we use appropriately weighted Sobolev spaces, as is standard for the Poisson problem, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' [AGG94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For (x, y) ∈ Rd × R, we introduce the weight ρ(x, y) := (1 + |x|2 + |y|2)1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For a (possibly unbounded) domain D ⊂ Rd × R+, we define the space H1 ρ(yα, D) as the space of all square integrable functions U (with respect to the weight function yαρ−2) such that the norm ∥U∥2 H1ρ(yα,D) := � � D yα���∇U(x, y) ��2 + ρ(x, y)−2��U(x, y) ��2� dx dy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Commonly used cases are D = Rd × R+ (full space), D = Rd × (0, Y) for Y > 0 (corresponding to truncation in y-direction), or D = ω × (0, Y) for ω ⊂ Rd and Y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For bounded sets ω ⊂ Rd and Y < ∞, we sometimes use the weighted spaces H1 ρ(yα, ω × (0, Y)), noting that, in this case, the weight satisfies 1 ≤ ρ(x, y) ≤ C(ω, Y) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Consequently, the norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) defines an equivalent norm to the H1(yα, ω × (0, Y))-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For functions U ∈ H1 ρ(yα, Rd × R+), one can give meaning to their trace at y = 0, which we denote by tr0 U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In fact, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1 will show that tr0 U is in a weighted fractional Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, the extension problem reads as: find U ∈ H1 ρ(yα, Rd × R+) such that − div � yαAx∇U � = 0 in Rd × R+, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3a) d−1 β ∂ναU + str0U = f in Rd, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3b) where dβ := 21−2βΓ(1 − β)/Γ(β), α := 1 − 2β ∈ (−1, 1), ∂ναU(x) := − limy→0 yα∂yU(x, y), and Ax = �A 0 0 1 � ∈ R(d+1)×(d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, by [ST10], the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1) is given by u = U(·, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 4 The weak formulation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3) in H1 ρ(yα, Rd × R+) reads as finding U ∈ H1 ρ(yα, Rd × R+) such that A(U, V) := � ∞ 0 yα � Rd Ax(x)∇U · ∇V dxdy + sdβ � Rd tr0Utr0V dx = dβ(f, tr0V)L2(Rd) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) for all V ∈ H1 ρ(yα, Rd × R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' If s > 0, it is natural to include the trace term into the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Thus, we introduce: ∥U∥2 H := ∥U∥2 H1ρ(yα,Rd×R+) + s∥tr0U∥2 L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The first step towards a computable formulation, before even considering any discretization steps, is to cut the problem from the infinite cylinder Rd × R+ to a finite cylinder in the y- direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' To do so, we fix a parameter Y > 0 to be chosen later and introduce the truncated bilinear form AY(U, V) := � Y 0 yα � Rd ∇U · ∇V dxdy + sdβ � Rd tr0Utr0V dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The truncated problem then reads: Find UY ∈ H1 ρ(yα, Rd × (0, Y)) such that AY(UY, VY) = dβ � f, tr0VY� L2(Rd) for all VY ∈ H1 ρ(yα, Rd × (0, Y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) In the following, we will often take Y ∈ (0, ∞] and refer to solutions to problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5), meaning that in the case Y = ∞ these functions actually satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We also introduce a natural norm on the truncated cylinder: ∥U∥2 HY := ∥U∥2 H1ρ(yα,Rd×(0,Y)) + s∥tr0U∥2 L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In fact, the truncated problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) corresponds to a weak formulation of a Caffarelli-Silvestre extension problem with an additional Neumann boundary condition at y = Y: − div � yαAx∇UY� = 0 in Rd × (0, Y), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6a) d−1 β ∂ναUY + str0UY = f on Rd × {0}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6b) ∂yUY = 0 on Rd × {Y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6c) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 Main results We are now in position to formulate the main results of the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The proofs of the statements are relegated to the following Sections 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1 Well-posedness and decay Regarding well-posedness of our variational formulation, we have the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Assume that either d > 2 or s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) has a unique solution U ∈ H1 ρ(yα, Rd × R+) and there is a constant C > 0 such that ∥U∥H ≤ C min(1, s−1) ∥f∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 5 Fix Y ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, the truncated problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) has a unique solution UY ∈ H1 ρ(yα, Rd × (0, Y)) satisfying ��UY�� HY ≤ C � 1 + 1 Y � min(1, s−1) ∥f∥L2(Ω) with a constant C > 0 independent of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Moreover, the bilinear forms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) are coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By the following proposition, we also obtain that solutions to the truncated problem converge to solutions to the non-truncated problem as the truncation parameter Y tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let U solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) and, for Y > 0, let UY solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For any fixed 0 < �Y < Y, it holds that UY → U in H1 ρ(yα, Rd × (0, �Y)) as Y → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' If s > 0, there additionally holds tr0UY → tr0U in L2(Rd) as Y → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Finally, we also obtain algebraic rates of convergence as Y → ∞ for the difference of the truncated and the non-truncated full-space solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Fix Y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let U solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) and UY solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let µ be given by µ := � 1 + |α| s > 0 1 + α s = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7) Then, there exists a constant C > 0 depending only on α and d such that ∥UY − U∥2 H1ρ(yα,Rd×(0,Y)) + s∥tr0(UY − U)∥2 L2(Rd) ≤ CY−µ ∥f∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 Regularity For solutions to the extension problem as well as the truncated extension problem there hold analytic type weighted estimates for the extended variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Estimates of that type allow to employ hp-finite elements in the extended variable, which will be considered in [FR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6 (Regularity in y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Fix Y ∈ (0, ∞] and let ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let U solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, there exists constants C, K > 0 and ε ∈ (0, 1) such that the following estimate holds: ��yℓ−ε∇∂ℓ yU �� L2(yα,Rd×(0,Y)) ≤ CKℓℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' ∥f∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' All constants are independent of ℓ, Y, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In fact, the regularity results imply that solutions to our model problem are in certain countably normed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Following [BMN+19, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1], we introduce the Bochner spaces L2 α((0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' X) of square integrable functions (with respect to the weight yα) and values in the Banach space X as well as for constants C, K > 0, the countably normed spaces B1 ε,0(C, K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' X) := � V ∈ C∞((0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' X) : ∥V∥L2(yα,(0,∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='X) < C, ���yℓ+1−εV(ℓ+1)��� L2(yα,(0,∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='X) < CKℓ+1(ℓ + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' ∀ℓ ∈ N0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6 provides control of yℓ−ε∂ℓ+1 y U, which directly gives the following Corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 6 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Fix Y ∈ (0, ∞] and let U solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, there are constants C, K > 0 such that there holds ∂yU ∈ B1 ε,0(C, K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' L2(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='8) We note that we formulated the previous corollary in terms of ∂yU, whereas the regularity results in [BMN+19, eqn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='10)] are formulated for solutions U to the extension problem on bounded domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This is due to the fact that in the case of the full space problem the estimates do not hold for the lowest order term as U /∈ L2(Rd × R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Nonetheless, the regularity result of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7 (together with U ∈ H1 ρ(Rd × R+)) allows to construct interpolation operators in a similar way as in [BMN+19, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Finally, we investigate the regularity in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Since this will depend on the regularity of the data A and f, we only consider the case of finite regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='8 (Regularity in x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Assume that A ∈ Cm(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Rd×d) and f ∈ Hm(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, for every multiindex ζ ∈ Nd 0 with |ζ| = m there holds ∥∇∂ζ xU∥L2(yα,Rd×R+) ≤ C∥f∥Hm(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The constant C depends on Ω, A, m and d, but is independent of f and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 3 Well-posedness and decay In this section, we provide the proofs of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 (well-posedness), Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4 (con- vergence) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5 (algebraic rate of decay).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1 Trace estimate We start with a trace estimate in a certain weighted Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For all U ∈ H1 ρ(yα, Rd × R+), there holds |tr0U|Hβ(Rd) ≤ C ∥∇U∥L2(yα,Rd×R+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1a) For d = 3, we additionally have ∥(1 + |x|2)−β/2tr0U∥L2(Rd) ≤ C ∥∇U∥L2(yα,Rd×R+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1b) In both cases the constant C > 0 does only depend on d and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1a) is shown in [KM19, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' To estimate the weighted L2-norm, we use interpolation space theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' More precisely, [Tar07, Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1] shows that interpolation of L2-spaces with weights w0 and w1 denoted by L2(wi, Rd) for i = 0, 1 produces an interpolation space (using the K-method) [L2(w0, Rd), L2(w1, Rd)]θ,2 = L2(wθ, Rd) that is a weighted L2-space with weight wθ = w1−θ 0 wθ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Applying this result with θ = 1−β and w0 = ρ−2 x := ρ(x, 0)−2 = (1+|x|2)−1 and w1 = 1, shows that ∥ρ−β x tr0U∥2 L2(Rd) = ∥tr0U∥2 L2(ρ−2β x ,Rd) ≲ ∥tr0U∥2 [L2(ρ−2 x ,Rd),L2(Rd)]1−β,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 7 Now, by [Tar07, Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1] the interpolation spaces can be seen as trace spaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=', el- ements of the interpolation space can be seen as traces (at 0) of functions U(y) satisfying y1−β ∥U(y)∥L2(ρ−2 x ,Rd) ∈ L2(y−1, R+) as well as y1−β ∥∂yU(y)∥L2(Rd) ∈ L2(y−1, R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Together with α = 1 − 2β and the Poincar´e estimate from [AGG94, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3] (using the assumption d = 3), this leads to ∥tr0U∥2 [L2(ρ−2 x ,Rd),L2(Rd)]1−β,2 ≲ � ∞ 0 yα∥ρ−1 x U(y)∥2 L2(Rd) dy + � ∞ 0 yα∥∂yU(y)∥2 L2(Rd) dy ≲ ∥∇U∥2 L2(yα,Rd×R+), which produces the desired estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 Poincar´e inequalities and well-posedness We now show the well-posedness of our variational formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The main ingredient is a Poincar´e type estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let α ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let Y ∈ (0, ∞] and U ∈ H1 ρ(yα, Rd × (0, Y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' There exists a µ0 > 0 such that for all µ ∈ [0, µ0) there holds � Y 0 � Rd yαρµ−2|U|2 dxdy ≤ C �� Y 0 � Rd yαρµ|∇U|2 dxdy + |3 − d|∥tr0U∥2 L2(Rd) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) provided the right-hand side is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For Poincar´e inequalities on the full-space without the additional weight yα, we refer to [AGG94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) for the case d = 3 follows directly from multiplying a full-space Poincar´e-inequality, see for example [AGG94, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3], applied only in x with yα and integrating over (0, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' More details can also be found in our forthcoming work [FR22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' It remains to show (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We write U(x, y) = U(x, 0) + � y 0 ∂yU(x, τ) dτ, which gives � Y 0 � Rd yαρµ−2|U|2 dxdy ≲ � Y 0 � Rd yαρµ−2|U(x, 0)|2 + yαρµ−2� � y 0 ∂yU(x, τ) dτ �2 dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Since � Y 0 yαρµ−2 ≲ 1 for sufficiently small µ < µ0, with µ0 > 0 depending only on α, the first term on the left-hand side can be bounded by C ∥tr0U∥2 L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For the second term, we employ a weighted Hardy-inequality, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' [Muc72], to obtain � Y 0 � Rd yαρµ−2� � y 0 ∂yU(x, τ) dτ �2 dxdy ≲ � Rd � Y 0 yαρµ|∂yU|2 dydx, which shows the claimed inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using this Poincar´e- type inequality, we can now look at the well-posedness of our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The boundedness of the bilinear forms A(·, ·) and AY(·, ·) follows di- rectly from the Cauchy-Schwarz inequality and the definition of the norms ∥·∥H and ∥·∥HY respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 8 Let Y ∈ (0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Coercivity of the bilinear forms follows directly from the Poincar´e inequalities in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2, since ��UY��2 HY = � Y 0 � Rd yαρ−2|UY|2 dxdy + � Y 0 � Rd yα|∇UY|2 dxdy + s ��tr0UY��2 L2(Rd) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) ≲ � Y 0 � Rd yα|∇UY|2 dxdy + (s + (3 − d)) ��tr0UY��2 L2(Rd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By assumption on s and d, the trace term is not present for the case s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Therefore, the right-hand side can be bounded by CAY(UY, UY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Thus, the Lax-Milgram lemma shows well-posedness provided the right-hand side of the varia- tional formulation is a bounded linear functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For the case s > 0, we can directly use the definition of the HY-norm together with supp f ⊂ Ω to obtain � Rd ftr0UY dx ≤ s−1 ∥f∥L2(Ω) s ��tr0UY�� L2(Rd) ≤ s−1 ∥f∥L2(Ω) ��UY�� HY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For Y = ∞ and s = 0, which implies d = 3 by assumption, the trace estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1b) gives � Rd ftr0U dx ≤ ���ρ(x, 0)βf ��� L2(Ω) ���ρ(x, 0)−βtr0U ��� L2(Rd) ≲ ∥f∥L2(Ω) ∥∇U∥L2(yα,Rd×R+) ≤ ∥f∥L2(Ω) ∥U∥H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For the case Y < ∞ and s = 0, we use a cut-off function χ satisfying χ ≡ 1 on (0, Y/2), supp χ ⊂ (0, Y) and ∥∇χ∥L∞(R+) ≲ Y−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As Ω is bounded, this gives with the trace estimate [KM19, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7] � Rd ftr0UY dx ≤ ∥f∥L2(Ω) ��tr0(χUY) �� L2(Ω) ≲ ∥f∥L2(Ω) ���χUY�� L2(yα,Ω×(0,Y)) + ��∇(χUY) �� L2(yα,Ω×(0,Y)) � ≲ ∥f∥L2(Ω) ���UY�� L2(yα,Ω×(0,Y)) + 1 Y ��∇UY�� L2(yα,Ω×(0,Y)) � ≤ C � 1 + 1 Y � ∥f∥L2(Ω) ��UY�� HY , which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 The truncation error In the following subsection, we study the truncated problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The main goal is to derive decay estimates in the truncation parameter Y and consequently convergence of the solution of the truncated problem to the solution of the non-truncated problem as Y → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The following lemma is the key to the main results of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using inf-sup theory we obtain that solutions to the Caffarelli-Silvestre extension problem and the truncated problem (in y-direction) lie in certain weighted Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The additional weights then directly provide the rates of decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In fact, we establish that the solutions are in two different types of weighted spaces: spaces weighted with (1 + y)µ with µ given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7) (decay only in y) and spaces with weights ρε for sufficiently small ε (decay in all directions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let y0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Fix Y ∈ (y0, ∞), and let µ be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let UY solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, UY satisfies the estimate � Y 0 yα� (1 + y)µ∥∇UY(y)∥2 L2(Rd)+(1 + y)µ∥ρ(·, y)−1UY(y)∥2 L2(Rd) � dy ≤ C min(s−1, 1)2 ∥f∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3) In addition, for Y ∈ (0, ∞], there exists ε > 0, depending only on α and Ω such that � Y 0 yα � Rd ρε|∇UY(x, y)|2dxdy ≤ C min(s−1, 1)2 ∥f∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) In both cases, the constant C does only depend on Ω, d, α, and y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By the uniqueness of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3, it suffices to show existence of such a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' To that end, we use inf-sup-theory, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=', [SS11, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='44], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=', we have to show inf U∈Xµ,Y\\{0} sup V∈Y−µ,Y\\{0} ��AY(U, V) �� ∥U∥Xµ,Y ∥V∥Y−µ,Y ≥ γ > 0 (inf-sup condition), ∀V ∈ Y−µ,Y\\{0} : sup U∈Xµ,Y\\{0} ��AY(U, V) �� > 0 (non-degeneracy condition) with spaces Xµ,Y, Y−µ,Y specified in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We define the ansatz space Xµ,Y as a subspace of H1 ρ(yα, Rd × (0, Y)) of functions for which the norm ∥U∥2 Xµ,Y := � Y 0 yα(1 + y)µ∥∇U(y)∥2 L2(Rd) dy + s ∥tr0U∥2 L2(Rd) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Step 1 (Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='11) with µ = 1 − α): We start with the simpler case s > 0 and take µ = 1 − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let χ(y) := � 1 y ≤ 1 y1−α y > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For U ∈ Xµ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' we define V := (1 + δχ(y))U (for some 0 < δ < 1 to be fixed later) and calculate � Y 0 � Rd yαAx∇U · ∇Vdxdy ≥ A0 � Y 0 yα(1 + δχ(y))∥∇U(y)∥2 L2(Rd)dy + � Y 1 � Rd yαδ(1 − α)y−αU∂yU dxdy = A0 � Y 0 yα(1 + δχ(y))∥∇U(y)∥2 L2(Rd)dy + δ(1 − α) 2 � Rd � Y 1 ∂ ∂y � U2� dydx = A0 � Y 0 yα(1 + δχ(y))∥∇U(y)∥2 L2(Rd)dy − δ(1 − α) 2 � Rd U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 1)2 dx + δ(1 − α) 2 � Rd U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Y)2 dx ≥ A0 � Y 0 yα(1 + δχ(y))∥∇U(y)∥2 L2(Rd)dy − δ(1 − α) 2 � Rd U(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 1)2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 10 In order to estimate the last term, we employ U(1)2 ≤ 2U(0)2 + 2 ���� � 1 0 ∂yU(y) dy ���� 2 ≤ 2U(0)2 + 2 � 1 0 yα|∂yU(y)|2dy � 1 0 y−αdy = 2U(0)2 + 2 1 − α � 1 0 yα|∂yU(y)|2dy, which gives using 1 + δχ(y) ≥ δ 4(1 + y)1−α � Y 0 � Rd yαAx∇U · ∇V dxdy ≥ (A0 − δ) � Y 0 yα(1 + δχ(y))∥∇U(y)∥2 L2(Rd)dy − δ(1 − α) � Rd U(x, 0)2 dx ≥ δ 4(A0 − δ) � Y 0 yα(1 + y)1−α∥∇U(y)∥2 L2(Rd)dy − δ(1 − α) � Rd U(x, 0)2 dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Consequently, we obtain AY(U, V) ≥ δ 4(A0 − δ) � Y 0 yα(1 + y)1−α∥∇U(y)∥2 L2(Rd) dy + � sdβ − δ(1 − α) � ∥tr0U∥2 L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) Choosing δ < min(A0/2, sdβ/(2 − 2α)), both terms on the right-hand side in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) are non- negative and using ∥V∥Xα−1,Y ≲ ∥U∥X1−α,Y , which follows easily from (1 + δχ(y)) ≲ (1 + y)1−α, gives the inf-sup condition for the ansatz space X1−α,Y and the test space Xα−1,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Moreover, the inf-sup constant behaves like ∼ min(1, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The non-degeneracy condition follows essentially with the same arguments, as, for given V, the function U := (1 + δχ(y))−1V provides the positivity of the bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The definition of the norm in the test-space and supp f ⊂ Ω implies (f, tr0V)L2(Rd) ≤ ∥f∥L2(Ω) ∥V∥Xα−1,Y , which gives a bound for the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Now, general inf-sup theory provides the existence of a solution that satisfies the claimed decay properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Step 2 (Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='11) with µ = 1 + α): Next, we show that the rate of decay µ = 1 + α is possible for s > 0 and even for s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In the following, we only discuss the harder case s = 0 as for s > 0, we only obtain an additional non-negative term in the bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Here, we use the test space induced by the norm ∥V∥2 �Y−µ,Y := � Y 0 yα ln(y + 2)2(1 + y)µ ∥∇V(y)∥2 L2(Rd) dy + ∥tr0V∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For given U ∈ Xµ,Y, we choose the test function V(x, y) := y1+αU(x, y) + (1 + α) � Y y τ αU(x, τ) dτ 11 with the derivatives ∇xV = y1+α∇xU + (1 + α) � Y y τ α∇xU(τ) dτ and ∂yV(y) = y1+α∂yU(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The function V is indeed in the test space, since we can bound the norm ∥V∥ �Y−1−α,Y by ∥V∥2 �Y−1−α,Y = � Y 0 yα ln(y + 2)2(1 + y)1+α ∥∇V(y)∥2 L2(Rd) dy + ∥tr0V∥2 L2(Ω) ≲ � Y 0 yα+2(1+α) ln(y + 2)2(1 + y)1+α ∥∇U(y)∥2 L2(Rd) dy + � Rd � Y 0 yα ln(y + 2)2(1 + y)1+α ��� � Y y τ α∇xU(τ) dτ ��� 2 dydx + ∥tr0V∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6) Since the first term is readily bounded due to U ∈ X1+α,Y and 1 + α > 0, we focus on the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using a weighted Hardy inequality, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' [Muc72], with the weight y−1/2/ ln(y + 2) that is square integrable in R+ we obtain � Rd � Y 0 yα ln(y + 2)2(1 + y)1+α ��� � Y y τ α∇xU(τ) dτ ��� 2 dydx ≤ � Rd � Y 0 ��� y−1/2 ln(y + 2) � Y y τ α∇xU(τ) dτ ��� 2 dydx ≲ � Rd � Y 0 y1+2α|∇xU(y)|2dydx ≤ ∥U∥2 X1+α,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7) What is left is to bound the trace of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We use a cut-off function χ satisfying χ ≡ 1 on (0, y0/2), supp χ ⊂ (0, y0), and ∥∇χ∥L∞(R) ≤ C with a constant C depending only on y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, V(x, 0)2 = (χV)(x, 0)2 = � � y0 0 ∂y(χV)(x, y) dy �2 ≤ y1−α 0 1 − α � y0 0 yα|∂y(χV)|2 dy ≲ � y0 0 yα � |∂yV|2 + |∂yχ|2V2� dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Integration over Ω and using the definition of V gives ∥tr0V∥2 L2(Ω) ≲ � y0 0 yα∥∇V(y)∥2 L2(Ω)dy + � Ω � y0 0 y2+3α|∂yχ|2U2dydx + � Ω � y0 0 yα��� � Y y τ αU(x, τ) dτ ��� 2 dydx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' On Ω×(0, y0) we can insert any appearing weights in the ansatz-space and test-space as needed, which just adds multiplicative constants independent of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Moreover, we can employ standard Poincar´e-inequalities to bound the L2-norm (here, the integrand even vanishes on (0, y0/2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Repeating the arguments from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7) (with slightly changed weight in the Hardy inequality to insert the weight ρ−2), we obtain the bound ∥tr0V∥L2(Ω) ≲ ∥U∥X1+α,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 12 Thus, we have shown ∥V∥ �Y−1−α,Y ≲ ∥U∥X1+α,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We continue with inserting U, V into the truncated bilinear form AY(·, ·), which leads to AY(U, V) = � Y 0 � Rd yαAx∇U · ∇V dxdy ≥ A0 � Y 0 y1+2α∥∇U(y)∥2 L2(Rd)dy + (1 + α) � Rd � Y 0 yαA1/2∇xU � Y y τ αA1/2∇xU(τ) dτ dy dx =: I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='8) We show that the term II is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' To simplify notation, we write v(y) := A1/2∇xU(y) and suppress the x-dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We note that by the chain rule there holds yαv(y) · � Y y τ αv(τ) dτ = −1 2 d dy ��� � Y y τ αv(τ) dτ ��� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This gives for the second term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='8): II = −(1 + α) 2 � Rd � Y 0 d dy ��� � Y y τ αv(τ) dτ ��� 2 dydx = (1 + α) 2 � Rd ��� � Y 0 τ αv(τ) dτ ��� 2 dx ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Overall, we get using (1 + y1+α) ≳ (1 + y)1+α AY(U, V) + AY(U, U) ≥ A0 � Y 0 yα(1 + y1+α)∥∇U(y)∥2 L2(Rd)dy ≳ ∥U∥2 X1+α,Y ≳ ∥U∥X1+α,Y∥U + V∥ �Y−1−α,Y, where the last inequality follows from the triangle inequality and ∥V∥ �Y−1−α,Y ≲ ∥U∥X1+α,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For the non-degeneracy condition, for a given V, we can choose U = V, which is in the ansatz- space, since due to Y < ∞ the weights in the gradient terms in the ansatz- and test-space are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By definition of the test-space and supp f ⊂ Ω, there holds (f, tr0V)L2(Rd) ≤ ∥f∥L2(Ω) ∥V∥ �Y−1−α,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Consequently, we obtain unique solvability of our weak formulation in the ansatz-space, which gives the decay estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Step 3 (Proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4)): Again, we use inf-sup theory with a different ansatz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Here, for ε > 0, we choose it to be a subspace of H1 ρ(yα, Rd×R+) such that additionally � Rd×(0,Y) yαρε |∇U|2 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We only work out the case s = 0 in the following, for s > 0, the same argument can be made by additionally including a trace term in the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Setting z := (x, y) ∈ Rd+1 and V(z) := ρε(z)U(z), we get with Young’s inequality and ρ−2|z|2 ≤ 1 AY(U, V) ≥ A0 � Rd×(0,Y) yαρε |∇U|2 dz + ε � Rd×(0,Y) yαρε−2z · Ax∇UU dz ≥ 1 2A0 � Rd×(0,Y) yαρε |∇U|2 dz − ε2 2 ∥Ax∥2 L∞(Rd×R+) A0 � Rd×(0,Y) yαρε−2 |U|2 dz ≥ 1 2A0 � Rd×(0,Y) yαρε |∇U|2 dz − CPε2 2 ∥Ax∥2 L∞(Rd×R+) A0 � Rd×(0,Y) yαρε |∇U|2 dz, 13 where in the last step we applied the Poincar´e estimate from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) for sufficiently small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' If ε is sufficiently small, we can also absorb the negative term and show inf-sup stability with the test space carrying ρ−ε as a weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The non-degeneracy condition and the bound on (f, tr0V)L2(Rd) are easily checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Before we can proceed to quantify the cutoff error, we need the following result on the existence of a stable extension from the cutoff domain Rd × (0, Y) to a larger set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Fix Y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, there exists an extension operator E to the domain Rd ×(0, 3 2Y) such that: (i) Eu = u in Rd × (0, Y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (ii) The following stability result holds for all µ ≥ 0 and U ∈ H1 ρ(yα, Rd × (0, Y)), if the right-hand side is finite: � 3 2 Y 0 yα+µ∥∇EU∥2 L2(Rd) dy ≤ C � Y 0 yα+µ∥∇U∥2 L2(Rd) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='9) The constant C > 0 depends on α, µ and d but is independent of U and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We extend U by reflection along the line y = Y, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=', we define W(x, y) := � U(x, y) 0 ≤ y ≤ Y, U(x, 2Y − y) Y < y ≤ 3 2Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By construction, the function has no jump across the line y = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For the stability in the extension domain, we compute � 3 2Y Y yα+µ∥∇W(·, y)∥2 L2(Rd) dy ≲ Yα+µ � 3 2Y Y ∥∇U(·, 2Y − y)∥2 L2(Rd) dy = Yα+µ � Y Y/2 ∥∇U(·, τ)∥2 L2(Rd) dτ ≲ � Y Y/2 τ α+µ∥∇U(·, τ)∥2 L2(Rd) dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using this extension operator, we obtain that the sequence (U(3/2)nY)n∈N, where the cutoff point is moved outward by a factor of 3/2 in each step, is a Cauchy sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let UY denote the solution to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) with truncation parameter Y > 0 and accord- ingly let U3/2Y denote the solution with a cutoff at 3/2Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let µ be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, there holds: ∥U3/2Y − UY∥HY ≤ CY−µ/2 ∥f∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Iterative application of the estimate for n, m ∈ N0, n > m leads to ∥U(3/2)nY − U(3/2)mY∥HY ≤ CY−µ/2 �2 3 �µ m/2 � 1 − �2 3 � µ 2 (n−m) � ∥f∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We compute using the coercivity of AY(·, ·) from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 and the extension operator from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4 ∥UY − U3/2Y∥2 HY ≲ AY(UY − U3/2Y, UY − U3/2Y) = AY(UY, UY − U3/2Y) − AY(U3/2Y, UY − U3/2Y) = (f, tr0(UY − U3/2Y))L2(Rd) − A3/2Y(U3/2Y, E(UY − U3/2Y)) + � 3 2 Y Y yα � Rd Ax∇U3/2Y∇E(UY − U3/2Y) dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By definition of U3/2Y and the extension operator E, the first two terms cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Thus, we can focus on bounding the remaining integral � 3 2Y Y yα � Rd Ax∇U3/2Y∇E(UY − U3/2Y) dxdy ≲ Y−µ/2� � 3 2Y Y yα+µ ���∇U3/2Y��� 2 dy �1/2� � 3 2Y Y yα ���∇E(UY − U3/2Y) ��� 2 dy �1/2 ≲ Y−µ/2� � 3 2Y Y yα+µ ���∇U3/2Y��� 2 dy �1/2∥UY − U3/2Y∥H1ρ(yα,Rd×(0,Y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using ∥UY − U3/2Y∥H1ρ(yα,Rd×(0,Y)) ≤ ∥UY − U3/2Y∥HY and canceling one such power then gives together with the decay estimate of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3: ∥UY − U3/2Y∥HY ≲ Y−µ/2 ∥f∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='10) Using a telescoping sum, we can write: U(3/2)nY − U(3/2)mY = n−1 � ℓ=m � U(3/2)ℓ+1Y − U(3/2)ℓY� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' With estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='10) applied iteratively, this leads to ∥U(3/2)nY − U(3/2)mY∥HY ≲ n−1 � ℓ=m ∥U(3/2)ℓ+1Y − U(3/2)ℓY∥HY ≲ Y−µ/2 n−1 � ℓ=m �3 2 �− µℓ 2 ∥f∥L2(Ω) ≃ Y−µ/2 �2 3 � µ 2 m � 1 − �2 3 � µ 2 (n−m) � ∥f∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using the Cauchy sequence property, we can now show convergence of the truncated solution to the full-space solution as stated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We focus on the case s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In the case s > 0, the same arguments can be made including the L2-norm of of the traces, which directly gives the additional statement regarding the convergence of tr0UY to tr0U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Step 1: We start by fixing the half-ball B+ Y ⊂ Rd × [0, ∞) of radius Y centered at the origin and write z = (x, y) ∈ Rd+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let ε > 0 be such that the decay estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 15 Defining E := U − UY and using the equations satisfied by U and UY, we use integration by parts to obtain � B+ Y yαAx∇E · ∇E dxdy = � ∂B+ Y yαAx∇E · νE dxdy = (1 + Y2)−ε/2 � |z|=Y yαρεAx∇E · νE dxdy − sdβ � |x|≤Y |tr0E|2 dx = (1 + Y2)−ε/2 � ∂B+ Y yαρεAx∇E · νE dxdy + sdβ � |x|≤Y �1 + |x|2 1 + Y2 �ε/2 |tr0E|2 dx − sdβ � |x|≤Y |tr0E|2 dx ≤ (1 + Y2)−ε/2 � ∂B+ Y yαρεAx∇E · νE dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Integration by parts back (replacing ∇E by ∇(ρεE)) gives � ∂B+ Y yαρεAx∇E · νE dxdy = � B+ Y yαAx∇E · (∇ρε)E dxdy + � B+ Y yαρεAx∇E · ∇E dxdy ≲ � � B+ Y yαρε |∇E|2 dz �1/2� � B+ Y yαρε−2 |E|2 dz �1/2 + � B+ Y yαρε|∇E|2 dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We replace the half-ball B+ Y by the cylinder Rd × (0, Y) and use the Poincar´e estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Together with the decay estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4) this gives boundedness of the right-hand side with a constant independent of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Consequently, we obtain � B+ R |∇E|2 dxdy ≲ � B+ Y yαAx∇E · ∇E dxdy ≤ C(1 + Y2)−ε/2 → 0 as Y → ∞ for all bounded half balls B+ R with R ≤ Y, which gives UY → U in H1 ρ(yα, B+ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Step 2: As (U(3/2)nY)n∈N is a Cauchy-sequence, there exists a limit �U ∈ H1 ρ(yα, Rd × (0, �Y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Assume that �U ̸= U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, there has to exist a half ball B+ R such that � B+ R yα���∇(U− �U) ��� 2 dxdy ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For sufficiently large n, we have R ≤ (3/2)nY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This leads to � B+ R yα���∇(U − �U) ��� 2 dxdy ≤ � B+ R yα���∇(U − U(3/2)nY) ��� 2 dxdy + � B+ R yα���∇(U(3/2)nY − �U) ��� 2 dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By step 1, the first term converges to zero and by definition of �U the second term converges to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' However, this is a contradiction to the assumption and therefore U = �U and we have established the claimed convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We can now estimate the truncation error and establish a rate of convergence as Y → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using a telescoping sum, we write UY − U = N � n=0 � UY( 3 2 )n − UY( 3 2)n+1� + UY( 3 2 )N+1 − U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 16 Since we have already established that UY → U for Y → ∞ in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4, we can pass to the limit N → ∞ and use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5 to estimate: ∥UY − U∥H1ρ(yα,Rd×(0,Y)) ≲ ∞ � n=0 ∥UY( 3 2)n − UY( 3 2 )n+1∥H1ρ(yα,Rd×(0,Y)) ≲ Y−µ/2 ∞ � n=0 �3 2 �− µn 2 ∥f∥L2(Rd) ≤ Y−µ/2 1 1 − (2 3)µ/2 ∥f∥L2(Rd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We can now also close the small gap that the decay in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 does not hold for the non- truncated domain Y = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let µ be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let U solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, there exists a constant C > 0 depending only on Ω, d, and α such that � ∞ 0 yα� (1 + y)µ∥∇U(y)∥2 L2(Rd) + (1 + y)µ∥ρ(·, y)−1U(y)∥2 L2(Rd) � dy ≤ C ∥f∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='11) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We take a sequence (Yn)n∈N with 1 ≤ Yn → ∞ for n → ∞ and consider the correspond- ing truncated solutions UYn to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 and Proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5) it holds: � Yn 0 yα(1 + y)µ∥∇U(y)∥2 L2(Rd) dy + � Yn 0 yα(1 + y)µ∥ρ−1U(y)∥2 L2(Rd) dy ≤ (1 + Yn)µ ��U − UYn��2 H1ρ(yα,Rd×(0,Yn)) + � Yn 0 yα(1 + y)µ∥∇UYn(y)∥2 L2(Rd) dy + � Yn 0 yα(1 + y)µ∥ρ−1UYn(y)∥2 L2(Rd) dy ≲ Yµ nY−µ n ∥f∥2 L2(Ω) + min(s−1, 1)2 ∥f∥2 L2(Ω) ≲ ∥f∥2 L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Taking n → ∞ then gives the stated result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 4 Regularity and higher order decay In this section, we derive regularity estimates for solutions to the extension problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Assuming sufficient differentiability of the data, we are in particular interested in weighted estimates for higher-order y-derivatives as such estimates are needed to establish exponential approximation estimates of hp–type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In order to derive suitable regularity estimates around y = 0, we need to derive an initial shift in a weighted space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Fix Y ∈ (0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Let U solve (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, there exists ε > 0 independent of Y and U such that � Y 0 yα� y−ε∥∇U(y)∥2 L2(Rd) + y−ε∥ρ(·, y)−1U(y)∥2 L2(Rd) � dy ≤ C ∥f∥2 L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Similar to the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3, we use inf-sup theory to derive the stated bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In the following, we only work out the details for the case s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The case s > 0 can be treated as shown in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 by also including a trace term in the norm of the ansatz space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 17 Here, for any �ε ∈ R, we define the space X�ε,Y as the space H1 ρ(yα−�ε, Rd × (0, Y)) of functions with finite norm ∥U∥2 X�ε,Y := � Y 0 yα−�ε� ∥∇U(y)∥2 L2(Rd) + ∥ρ(·, y)−1U(y)∥2 L2(Rd) � dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As ansatz space, we take Xε,Y, where ε > 0 is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As test space we use X−ε,Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For fixed α ∈ (−1, 1), we actually may choose ε > 0 such that α ± ε ∈ (−1, 1) (subsequently, we will derive an additional restriction on ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For given U ∈ Xε,Y, we define the test function V(x, y) := y−εU(x, y) + ε � y 0 τ −ε−1U(x, τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using Hardy’s inequality (noting that α + ε > −1), we obtain that this test-function is indeed in the test-space � Y 0 yα+ε ∥∇V(y)∥2 L2(Rd) dy ≲ � Y 0 yα+εy−2ε ∥∇U(y)∥2 L2(Rd) dy + � Rd � Y 0 yα+ε � ε � y 0 τ −ε−1∇xU(τ)dτ �2 dydx ≲ (1 + ε2) � Y 0 yα−ε ∥∇U(y)∥2 L2(Rd) dy < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) The weighted L2-term in the definition of Xε,Y can be treated using the Poincar´e inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2) replacing α with α − ε therein noting that α − ε ∈ (−1, 1) by assumption on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Inserting the test function into the bilinear form gives AY(U, V) = � Rd � Y 0 yα−εAx∇U · ∇Udydx + ε � Rd � Y 0 yαA∇xU � y 0 τ −ε−1∇xU(τ)dτ dydx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using Young’s inequality together with Hardy’s inequality (noting again that α + ε > −1), we obtain ε � Rd � Y 0 yαA∇xU � y 0 τ −ε−1∇xU(τ)dτ dydx ≤ 1 2 � Rd � Y 0 yα−εAx∇U · ∇Udydx + 1 2ε2 � Rd � Y 0 yα+ε �� y 0 τ −ε−1A1/2∇xU(τ)dτ �2 dydx ≤ 1 2 � 1 + CHε2� � Rd � Y 0 yα−εAx∇U · ∇U dydx, where CH indicates the constant in the Hardy inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Therefore, we obtain AY(U, V) ≥ A0 2 � 1 − CHε2� � Y 0 yα−ε ∥∇U∥2 L2(Rd) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Together with the Poincar´e estimate of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2, we obtain the inf-sup condition upon choos- ing ε < C−1/2 H .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For the non-degeneracy condition, we fix V ∈ X−ε,Y and choose U = yεV − ε � y 0 τ ε−1V(τ)dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, essentially the same estimates as above can be made by noting that, by assumption we have α − ε > −1, thus Hardy inequalities with the necessary modified weights can be employed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The right-hand side can be bounded using the support properties of f together with a trace estimate (in the weighted space L2(yα+ε, Ω × (0, Y)) noting that α + ε ∈ (−1, 1)) ���(f, tr0V)L2(Rd) ��� ≤ ∥f∥L2(Ω) ∥tr0V∥L2(Ω) ≤ ∥f∥L2(Ω) ∥∇V∥L2(yα+ε,Ω×(0,Y)) ≤ ∥f∥L2(Ω) ∥V∥X−ε,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Now, classical inf-sup theory gives the claimed estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 18 With the initial shift in place, we can look at higher order derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We first formulate the “shift-by-one” as a separate lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Fix Y ∈ (0, ∞] and let W ∈ H1 ρ(yα, Rd × (0, Y)) solve the problem − div � yαAx∇W � = F in Rd × (0, Y) with given right-hand side F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Then, for all ℓ ∈ N and ε ∈ (0, 1), the estimate ��yℓ−ε∇W �� L2(yα,Rd×(0,Y)) ≲ ℓ ��yℓ−1−εW �� L2(yα,Rd×(0,Y)) + ��yℓ+1−εF �� L2(y−α,Rd×(0,Y)) holds, provided that the right-hand side is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' The implied constant is independent of ℓ and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' If Y = ∞, let N ∈ N, and we fix a cutoff function ˜χN ∈ C∞ 0 (R) such that ˜χN ≡ 1 on [0, N] and ˜χN ≡ 0 on (2N, ∞) with ∥˜χ′ N∥L∞(R) ≤ 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We define ωN(y) := yℓ−ε ˜χN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In the easier case Y < ∞, we can skip the cutoff function altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For brevity, we therefore only work out the case Y = ∞, the other case follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We start with multiplying the equation for W with the test function V := ω2 NW, and integrate by parts over Rd×(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As the weight function ωN and consequently also V vanishes at y = 0, we do not get any boundary contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This gives with Young’s inequality ∥ωNA1/2 x ∇W∥2 L2(yα,Rd×R+) = � Rd×R+ ω2 N(y)F Wdxdy − � Rd×R+ 2ω′ N(y)ω(y)∂yWWdxdy ≤ ∥yωNF∥L2(y−α,Rd×R+)∥y−1ωNW∥L2(yα,Rd×R+) + 2∥ωN∂yW∥L2(yα,Rd×R+)∥ω′ NW∥L2(yα,Rd×R+) ≤ 1 2∥yωNF∥2 L2(y−α,Rd×R+) + 1 2∥y−1ωNW∥2 L2(yα,Rd×R+) + 1 2∥ωN∂yW∥2 L2(yα,Rd×R+) + 2∥ω′ NW∥2 L2(yα,Rd×R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Absorbing the third term in the left-hand side provides ∥ωNA1/2 x ∇W∥2 L2(yα,Rd×R+) ≲ ∥yωNF∥2 L2(y−α,Rd×R+) + ∥y−1ωNW∥2 L2(yα,Rd×R+) + ∥ω′ NW∥2 L2(yα,Rd×R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For N → ∞, using Ax ≥ A0, the left-hand side converges to the weighted L2-norm we are looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Similarly, the first two terms on the right-hand side converge to the appropriate objects of the final estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Therefore we focus on the last term and show an uniform bound: ∥ω′ NW∥L2(yα,Rd×R+) ≤ (ℓ − ε)∥yℓ−1−ε ˜χNW∥L2(yα,Rd×R+) + ∥yℓ−ε ˜χ′ NW∥L2(yα,Rd×R+) ≲ ℓ∥yℓ−1−εW∥L2(yα,Rd×R+) + 1 N 2 � 2N N y2 ���� ≲4N2 yα+2ℓ−2−2ε∥W(y)∥2 L2(Rd) dy ≲ ℓ∥yℓ−1−εW∥L2(yα,Rd×R+) + � ∞ 0 yα+2ℓ−2−2ε∥W(y)∥2 L2(Rd) dy, where we used that ˜χ′ N vanishes outside of [N, 2N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Therefore we can pass to the limit N → ∞ to get the stated result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 19 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Note that U as solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3) does not fit Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 since it is not in L2 α(Rd × (0, Y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' However, the previous lemma can be applied for derivatives of the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We are now in position to show our main result regarding weighted regularity, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We note that away from y = 0, we can use standard elliptic regularity theory to show that U is C∞(Rd × R) and we can focus on the weighted estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' We prove this by induction, starting with ℓ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' By differentiating the equation in the form div(Ax∇U)+ α y ∂yU = 0, we get that W := ∂ℓ yU solves: − div(yαAx∇W) = α ℓ−1 � k=0 (−1)k ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' ∂k+1 y U yℓ−k+1−α =: Fℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3) For ℓ = 1, we employ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 to obtain ��y1−ε∇∂yU �� L2(yα,Rd×(0,Y)) ≲ ��y−ε∂yU �� L2(yα,Rd×(0,Y)) + ∥y2−εy−2+α∂yU∥L2(y−α,Rd×(0,Y)) ≲ ��y−ε∂yU �� L2(yα,Rd×(0,Y)) ≲ ∥f∥L2(Ω) , where in the last step we used Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' For ℓ > 1, we use the induction assumption valid for k < ℓ (that allows to control derivatives up to order ℓ), which gives ��yℓ+1−εFℓ �� L2(y−α,Rd×(0,Y)) ≲ ℓ−1 � k=0 ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' ��yk−ε∂k+1 y U �� L2(yα,Rd×(0,Y)) ≲ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' ∥f∥L2(Rd) ℓ−1 � k=0 Kk ≲ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='Kℓ ∥f∥L2(Rd) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='2 together with the induction assumption, we get ��yℓ−ε∇∂ℓ yU �� L2(yα,Rd×(0,Y)) ≲ ℓ ��yℓ−1−ε∂ℓ yU �� L2(yα,Rd×(0,Y)) + ��yℓ+1−εFℓ �� L2(y−α,Rd×(0,Y)) ≲ ℓ ��yℓ−1−ε∇∂ℓ−1 y U �� L2(yα,Rd×(0,Y)) + ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='Kℓ��f �� L2(Rd) ≲ ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='Kℓ ∥f∥L2(Rd) , which proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Finally, we provide the proof for the regularity estimates for the x-derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' In order to obtain estimates for the x-derivatives, for a given multi- index ζ, we differentiate the equation with respect to ∂ζ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' As the weight yα remains unchanged, we see that W := ∂ζ xU solves the extension problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3) − div � yαAx∇W � = Fζ in Rd × R+, d−1 β ∂ναW + str0W = fζ in Rd, 20 with data fζ := ∂ζ xf and right-hand side Fζ := − div � yα � ζ′<ζ �ζ ζ′ � (∂ζ−ζ′ x Ax)∂ζ′ x ∇U � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' One can modify the arguments of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='3 to also include the source term (Fζ, W)L2(Rd×R+), which can be estimated using ���(Fζ, W)L2(Rd×R+) ��� ≤ ∥Fζ∥L2(y−α,Rd×R+) ∥W∥L2(yα,Rd×R+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' This gives ∥∇W∥L2(yα,Rd×R+) ≲ ∥Fζ∥L2(y−α,Rd×R+) + ∥fζ∥L2(Ω) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' Now, an induction argument can be set up as in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content='6 to control ∥Fζ∥L2(y−α,Rd×R+) by L2-norms of derivatives of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE5T4oBgHgl3EQfPQ4_/content/2301.05503v1.pdf'} diff --git a/7NE2T4oBgHgl3EQfPQZw/content/tmp_files/2301.03757v1.pdf.txt b/7NE2T4oBgHgl3EQfPQZw/content/tmp_files/2301.03757v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0af4084f26fd09ed000b290ac726490b9f435f7f --- /dev/null +++ b/7NE2T4oBgHgl3EQfPQZw/content/tmp_files/2301.03757v1.pdf.txt @@ -0,0 +1,3717 @@ +Constructions of Delaunay-type solutions for the +spinorial Yamabe equation on spheres +Ali Maalaoui +Yannick Sire +Tian Xu +Abstract +In this paper we construct singular solutions to the critical Dirac equation on spheres. +More precisely, first we construct solutions admitting two points singularities that we call +Delaunay-type solutions because of their similarities with the Delaunay solutions con- +structed for the singular Yamabe problem in [32, 35]. Then we construct another kind of +singular solutions admitting a great circle as a singular set. These solutions are the building +blocks for singular solutions on a general Spin manifold. +Keywords. Spinorial Yamabe; Singular Solutions; Delaunay-type Solutions. +Contents +1 +Introduction and statement of the main result +2 +2 +Geometric preliminaries +8 +2.1 +General preliminaries about spin geometry . . . . . . . . . . . . . . . . . . . . +8 +2.2 +Spinor bundle and the Dirac operator on product manifolds . . . . . . . . . . . +9 +2.3 +A particular ansatz in Euclidean spaces . . . . . . . . . . . . . . . . . . . . . . +10 +3 +Set up of the problems +14 +3.1 +The singular set is a pair of antipodal points . . . . . . . . . . . . . . . . . . . +15 +3.2 +The singular set is an equatorial circle . . . . . . . . . . . . . . . . . . . . . . +16 +4 +Analysis of the ODE systems +18 +4.1 +The nondissipative case: Bifurcation of the positive periodic orbits . . . . . . . +19 +4.2 +The dissipative case: Shooting method . . . . . . . . . . . . . . . . . . . . . . +24 +Mathematics Subject Classification (2010): Primary 53C27; Secondary 35R01 +1 +arXiv:2301.03757v1 [math.AP] 10 Jan 2023 + +2 +1 +Introduction and statement of the main result +Since the resolution of the Yamabe problem, much has been clarified about the behavior of +solutions of the semilinear elliptic equation relating the scalar curvature functions of two con- +formally related metrics. One of the starting points for several recent developments was R. +Schoen’s construction of complete metrics with constant positive scalar curvature on the sphere +Sm, conformal to the standard round metric, and with prescribed isolated singularities (see [36]). +In analytical terms, it is equivalent to seeking for a function u > 0 satisfying +− ∆gSmu + m(m − 2) +4 +u = m(m − 2) +4 +u +m+2 +m−2 +on Sm \ Σ, m ≥ 3 +(1.1) +in the distributional sense with u singular at every point of Σ ⊂ Sm. Here we denote by gSm the +standard Riemannian metric on Sm. +Eq. (1.1) and its counterpart on a general manifold (M, g) are known as the singular Yamabe +problem, and has been extensively studied. Just as the classical Yamabe problem in the com- +pact setting, the questions concerning metrics of constant positive scalar curvature are consid- +erably more involved. Remarkable breakthroughs and geometrically appealing examples were +obtained by Schoen and Yau [37] and Schoen [36] when the ambient manifold is the m-sphere +Sm. The former established that if Sm \ Σ admits a complete metric with scalar curvature +bounded below by a positive constant, then the Hausdorff dimension of Σ is at most (m − 2)/2, +and the latter constructed several examples of domains Sm \Σ that admit complete conformally +flat metrics with constant positive scalar curvature, including the case where Σ is any finite set +with at least two points. Subsequently, Mazzeo and Smale [34] and Mazzeo and Pacard [32,33] +generalized the existence results, allowing Σ to be a disjoint union of submanifolds with di- +mensions between 1 and (m − 2)/2 when the ambient manifold (M, g) is a general compact +manifold with constant nonnegative scalar curvature, and between 0 and (m − 2)/2 in the case +(M, g) = (Sm, gSm). +In the past two decades, it has been realized that the conformal Laplacian, namely the op- +erator appearing as the linear part of (1.1), falls into a particular family of operators. These +operators are called conformally covariant elliptic operators of order k and of bidegree ((m − +k)/2, (m + k)/2), acting on manifolds (M, g) of dimension m > k. Many important geometric +operators are in this class, for instance, the conformal Laplacian, the Paneitz operator, the Dirac +operator, see also [10, 13, 20] for more examples. All such operators share several analytical +properties, in particular, they are associated to the non-compact embedding of Sobolev space +Hk/2 �→ L2m/(m−k). And often, they have a central role in conformal geometry. +Let (M, g, σ) be an m-dimensional spin manifold, m ≥ 2, with a fixed Riemannian metric +g and a fixed spin structure σ : PSpin(M) → PSO(M). The Dirac operator Dg is defined in +terms of a representation ρ : Spin(m) → Aut(Sm) of the spin group which is compatible with +Clifford multiplication. Let S(M) := PSpin(M) ×ρ Sm be the associated bundle, which we +call the spinor bundle over M. Then the Dirac operator Dg acts on smooth sections of S(M), +i.e. Dg : C∞(M, S(M)) → C∞(M, S(M)), is a first order conformally covariant operator of +bidegree ((m − 1)/2, (m + 1)/2). We point out here that the spinor bundle S(M) has complex +dimension 2[ m +2 ]. +Analogously to the conformal Laplacian, where the scalar curvature is involved, the Dirac +operator on a spin manifold has close relations with the mean curvature function associated to + +3 +conformal immersions of the universal covering into Euclidean spaces. This theory is referred +as the spinorial Weierstraß representation, and we refer to [2,3,17,25–27,31,41–43] and refer- +ences therein for more details in this direction. In a similar way as in the Yamabe problem, the +spinorial analogue of the Yamabe equation (related with a normalized positive constant mean +curvature) reads as +Dgψ = |ψ| +2 +m−1 +g +ψ +on (M, g) +(1.2) +where | · |g stands for the induced hermitian metric on fibers of the spinor bundle. One may also +consider the equation with an opposite sign +Dgψ = −|ψ| +2 +m−1 +g +ψ +on (M, g) +(1.3) +which corresponds to negative constant mean curvature surfaces. However, since the spectrum +of Dg is unbounded on both sides of R and is symmetric about the origin on many manifolds +(say, for instance dim M ̸≡ 3(mod 4)), the two problems (1.2) and (1.3) are of the same struc- +ture from analytical point of view. +Although conformally covariant operators share many properties, only few statements can be +proven simultaneously for all of them. Particularly, the behavior of solutions of the conformally +invariant equation (1.2) or (1.3) is still unclear. From the analytic perspective, some of the +conformally covariant operators are bounded from below (e.g. the Yamabe and the Paneitz +operator), whereas others are not (e.g. the Dirac operator). Some of them act on functions, +while others on sections of vector bundles. For the Dirac operators, additional structure (e.g. +spin structure) is used for defining it, and hence, more attention needs to be payed on such an +exceptional case. +In this paper we initiate an investigation into the singular solutions of the nonlinear Dirac +equation (1.2) when the ambient manifold is Sm, which is perhaps the most geometrically ap- +pealing instance of this problem. As was described earlier, for a given closed subset Σ ⊂ Sm, +it is to find metrics g = |ψ|4/(m−1) +gSm +gSm which are complete on Sm \ Σ and such that ψ satisfies +Eq. (1.2) with (M, g) = (Sm \ Σ, gSm). This is the singular spinorial Yamabe problem. Let us +mention that, up until now, no existence examples have been known for the singular solutions +of Eq. (1.2). Our first main result is follows: +Theorem 1.1. Let Σ ⊂ Sm be a pair of antipodal points, for m ≥ 2, or an equatorial circle for +m ≥ 3. There is a one-parameter family Sm of spinors ψ solving the problem +DgSmψ = |ψ| +2 +m−1 +gSm ψ +on Sm \ Σ +(1.4) +such that g = |ψ| +4 +m−1 +gSm gSm is a complete metric on Sm \ Σ. Moreover, +(1) if Σ is a pair of antipodal points, the family Sm is parameterized by µ ∈ [− (m−1)m +2m+1m , +∞)\ +{0}. +(2) if Σ is an equatorial circle, the family Sm is parameterized by O = ∪k∈NOk, where each +Ok ⊂ (0, +∞) is a bounded open set, Ok ∩ Oj = ∅ for k ̸= j and O is unbounded. +Remark 1.2. Let us remark that Eq. (1.4), or more generally Eq. (1.2), is invariant under several +Lie group actions. For instance, the canonical action of S1 = {eiθ ∈ C : θ ∈ [0, 2π]} on spinors + +4 +keeps the equation invariant (i.e. if ψ is a solution of Eq. (1.4) then eiθψ is also a solution, for +every fixed θ). Moreover, for the case m ≡ 2, 3, 4(mod 8), the spinor bundle has a quaternionic +structure which commutes with Clifford multiplication, see for instance the construction in [18, +Section 1.7] or [28, Page 33, Table III]. In these cases, Eq. (1.4) is invariant under the action +of the unit quaternions S3 = {q = H : |q| = 1} on spinors. Therefore, in general, it is +crucial to distinguish solutions of Dirac equations under various group actions. For instance, +these symmetries were exploited in [29] to construct families of solutions on the sphere and +the S1 symmetry was used in [30] to exhibit also non-trivial solutions for the sub-critical Dirac +equation. Thanks to our constructions, the solutions in the family Sm obtained in Theorem 1.1 +are distinguished via their parameterizations. And if G is a group that keeps Eq. (1.4) invariant, +our construction shows a larger family G × Sm of singular solutions. +As we will see in Section 3, via a conformal change of the metric gSm, problem (1.4) can be +transformed to +DgRmψ = |ψ| +2 +m−1 +gRm ψ +on Rm \ {0} +(1.5) +when Σ is a pair of antipodal points and +DgRm−1ψ = f(x) +1 +m−1|ψ| +2 +m−1 +gRm−1ψ +on Rm−1 \ {0} +(1.6) +when Σ is an equatorial circle, where f(x) = +2 +1+|x|2. To obtain the results for Eq. (1.4) in +consistence with similar results for the classical Yamabe equation, a fundamental idea is to +express the equation (1.5) and (1.6) on the cylinder R×Sl, l = m−1 or m−2. By introducing +the cylindrical coordinates (t, θ) ∈ R × Sl: +t = − ln |x|, +θ = x +|x| +for x ∈ Rl+1, one may be expecting that the ansatz +ϕ(t, θ) = |x| +l +2ψ(x) +could turn Eq. (1.5) into a more manageable problem via a separation of variables process +leading to a ”radial” solution ψ(x) = ψ(|x|). This is the very case for many elliptic problems +(with a corresponding change of the exponent on |x|), including the Yamabe equation, fractional +Yamabe equation [12] and the Q-curvature problem [24]. However, we point out that in the +scalar case, there is a symmetrization process that behaves well with elliptic operators, reducing +the problem to the study of an ODE. But when dealing with differential operators acting on +vector bundles (spinor bundle in our case), one does not have a general symmetrization process. +In particular, even on the Euclidean spaces Rm, one cannot use the radial ansatz ψ = ψ(r), +r = |x| for x ∈ Rm, to reduce a Dirac equation to an ODE system in terms of r. +Notice that the spinorial Yamabe equation (1.5) (resp. (1.6)) contains 2[ m +2 ] (resp. 2[ m−1 +2 +]) un- +known complex-functions, which is a considerably large number as m grows. Instead of blindly +“guessing” a particular ansatz, our starting point is the spin structure, or more precisely the +spin representation. In fact, we use the matrix representation of Clifford multiplication to con- +struct a “nice” function space E(Rm) for spinor fields which is invariant under the action of the +Dirac operator DgRm, see in Section 2.3 for the definition. We find that the space E(Rm) is of + +5 +particular interest from two perspectives (see Remark 2.1 below): First of all, when the dimen- +sion m = 2, 3, 4, E(Rm) encapsulates several important and special formulations of spinors +which are of interest to particle physicists when they study quantum electrodynamic systems. +Many important physical simulations have been obtained by using these special spinors, see for +instance [11,14,40,45]. The second perspective is that, spinors in E(Rm) reduce the equation +(1.5) significantly in the sense that, for any dimension m ≥ 2, Eq. (1.5) and (1.6) can be reduced +to the following ODE systems of only two unknown functions +� +� +� +− f ′ +2 − m − 1 +r +f2 = (f 2 +1 + f 2 +2) +1 +m−1f1 +f ′ +1 = (f 2 +1 + f 2 +2) +1 +m−1f2 +for r > 0 +(1.7) +and +� +� +� +� +� +� +� +− f ′ +2 − m − 2 +r +f2 = +� +2 +1 + r2 +� +1 +m−1(f 2 +1 + f 2 +2) +1 +m−1f1 +f ′ +1 = +� +2 +1 + r2 +� +1 +m−1(f 2 +1 + f 2 +2) +1 +m−1f2 +for r > 0 +(1.8) +where f1, f2 ∈ C1(0, +∞). After using the Emden-Fowler change of variable r = e−t and +writing f1(r) = −u(t)e +m−1 +2 +t, f2(r) = v(t)e +m−1 +2 +t in (1.7), we get a nondissipative Hamiltonian +system of (u, v) +� +� +� +� +� +u′ + m − 1 +2 +u = (u2 + v2) +1 +m−1v, +−v′ + m − 1 +2 +v = (u2 + v2) +1 +m−1u. +(1.9) +And, by writing f1(r) = −u(t)e +m−2 +2 +t and f2(r) = v(t)e +m−2 +2 +t, we can transform (1.8) into +� +� +� +� +� +u′ + m − 2 +2 +u = cosh(t)− +1 +m−1(u2 + v2) +1 +m−1v +−v′ + m − 2 +2 +v = cosh(t)− +1 +m−1(u2 + v2) +1 +m−1u +(1.10) +which is a dissipative Hamiltonian system. +Let us denote by +H(u, v) = −m − 1 +2 +uv + m − 1 +2m (u2 + v2) +m +m−1 +the corresponding Hamiltonian energy for the systems (1.9). Notice that H is constant along +trajectories of (1.9). Moreover, the equilibrium points of H are +(0, 0) +and +± +�(m − 1)(m−1)/2 +2m/2 +, (m − 1)(m−1)/2 +2m/2 +� +, +(1.11) +where (0, 0) is a saddle point and the other two are center points; then it follows easily that for +µ ∈ [− (m−1)m +2m+1m , +∞) \ {0} there is a periodic solution of (1.9) at the level {H = µ}. We set +D1 +m for these periodic solutions, parameterized by their Hamiltonian energies. We distinguish + +6 +a dichotomy within the set D1 +m based on the sign of the Hamiltonian energy µ. Indeed, D1 +m = +D1,+ +m ∪ D1,− +m , where +D1,+ +m := {(u, v) ∈ D1 +m; H(u, v) > 0} and D1,− +m := {(u, v) ∈ D1 +m; H(u, v) < 0}. +We will call elements of D1,− +m , positive Delaunay-type solutions and elements of D1,+ +m , sign- +changing Delaunay-type solutions for Eq. (1.5). This terminology is based on the similarities +between D1,− +m +and the classical Delaunay solutions for the Yamabe problem. We will clarify +more these similarities along the paper. Since any (u, v) ∈ D1 +m will not reach the rest point +(0, 0), we have u(t)2 + v(t)2 is bounded away from 0 for all t ∈ R. Besides the above existence +results, we have the following bifurcation phenomenon for the solutions (u, v) ∈ D1,− +m . +Theorem 1.3. Let m ≥ 2, the following facts hold for the system (1.9): +(1) For every T > 0, (1.9) has the constant 2T-periodic solutions +± +�(m − 1)(m−1)/2 +2m/2 +, (m − 1)(m−1)/2 +2m/2 +� +. +Moreover, for T ≤ +√m−1 +2 +π, these are the only solutions to (1.9). +(2) Let T > +√m−1 +2 +π and d ∈ N such that d +√m−1 +2 +π < T ≤ (d+1) +√m−1 +2 +π. Then (1.9) has d+1 +inequivalent solutions. Particularly, these solutions are given by the constant solution and +k periods of a solution (uT,k, vT,k) with fundamental period 2T/k. +(3) The Hamiltonian energy H(uT,1, vT,1) ↗ 0 as T → +∞ and (uT,1, vT,1) is (locally) com- +pact in the sense that (uT,1, vT,1) converges in C1 +loc(R, R2) to the nontrivial homoclinic +solution of (1.9). That is, there exists t0 ∈ R such that (uT,1, vT,1) converges in C1 +loc to +(u0(· − t0), v0(· − t0)), where +u0(t) = +m(m−1)/2et/2 +2m/2 cosh(t)m/2 +and +v0(t) = m(m−1)/2e−t/2 +2m/2 cosh(t)m/2. +By translating the above results to system (1.7) (hence Eq. (1.5)), we have +Corollary 1.4. Let m ≥ 2, Eq. (1.5) has a one-parameter family S1 +m of singular solutions on +Rm\{0}, parameterized by [− (m−1)m +2m+1m , +∞)\{0}. Moreover, the following asymptotic estimates +hold +• |ψ(x)| ̸= 0, +• |ψ(x)| = O(|x|− m−1 +2 ) as |x| → +∞, +• |ψ(x)| = O(|x|− m−1 +2 ) as |x| → 0, +for each ψ ∈ S1 +m. Moreover, if ψµ is the solution corresponding to µ ∈ [− (m−1)m +2m+1m , 0), then +there exists λ > 0 such that ψµ converges in C1 +loc(Rm) to ψ∞ = +� +2λ +λ2+|x|2 +� m +2 � +1 − x +λ +� +· γ0 as +µ → 0, where γ0 is a constant spinor with |γ0| = +1 +√ +2 +� m +2 +� m−1 +2 +and “·” stands for the Clifford +multiplication on spinors. + +7 +It is important here to notice the difference between the decay rate of singular solutions that +we found in the previous Corollary and the one of regular solutions of (1.5), studied in [8]. +Indeed, the decay rate of a regular solution is O(|x|−m+1) but the one of a singular solution is +O(|x|− m−1 +2 ). +For the system (1.10) we have +Theorem 1.5. Let m ≥ 3, the system (1.10) with initial datum u(0) = v(0) = µ > 0 has a +solution (uµ, vµ) globally defined on R. Moreover, there are exactly two types of initial data, +which can be characterized by: +Ak = +� +µ > 0 : vµ changes sign k times on (0, +∞) and +lim +|t|→+∞ Hµ(t) < 0 +� +, +and +Ik = +� +µ > 0 : vµ changes sign k times on (0, +∞) and Hµ(t) > 0 for all t ∈ R +� +for k ∈ N ∪ {0}, where +Hµ(t) := −m − 2 +2 +uµvµ + m − 1 +2m +cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +m +m−1. +In particular, +(1) Ak ̸= ∅ is a bounded open set for all k; +(2) if we set µk = sup Ak, then µk ∈ Ik and µ0 < µ1 < · · · < µj < µj+1 < · · · → +∞; +(3) if we set νk = sup Ik, then νk < +∞ and (νk, νk + ε) ⊂ Ak+1 for some small ε > 0; +(4) if µ ∈ Ik, then (uµ(t), vµ(t)) → (0, 0) as |t| → ∞. To be more precise, we have +uµ(t)2 + vµ(t)2 = O(e−(m−2)t) +as |t| → +∞; +(5) if µ ∈ Ak, then uµ(t)2 + vµ(t)2 is bounded from below by a positive constant for all +t ∈ R and is unbounded as |t| → +∞; furthermore, up to a multiplication by constant, +uµ(t)2 + vµ(t)2 is upper bounded by cosh(t) for all |t| large. +By setting D2 +m = {(uµ, vµ) : µ ∈ ∪k≥0Ak}, we call these unbounded solution the Delaunay- +type solution for Eq. (1.6). As a direct consequence of Theorem 1.5, we have a characterization +of singular solutions for Eq. (1.6) on Rm−1 \ {0}. +Corollary 1.6. Let m ≥ 3, Eq. (1.5) has a one-parameter family S2 +m of singular solutions on +Rm−1 \ {0}, parameterized by ∪k≥0Ak. Moreover, the following asymptotic estimates hold +|x|− m−2 +2 +< |ψ(x)| ≲ |x|− m−1 +2 +as |x| → 0 +and +|x|− m−2 +2 +< |ψ(x)| ≲ |x|− m−3 +2 +as |x| → +∞ +for each ψ ∈ S2 +m + +8 +This paper is organized as follows. First, in Section 2, we lay down the necessary geometric +preliminaries that we will need to formulate our problem, including the main ansatz that will +be adopted to find our families of singular solutions. Next, in Section 3, we use the ansatz +to formulate the problem as a Hamiltonian system in R2 (autonomous in the case of a point +singularity and non-autonomous in the case of a one dimensional singularity). In section 4, we +study the properties of the solutions of the Hamiltonian system in the two cases. This allows us +to prove Theorems 1.3 and 1.5. +2 +Geometric preliminaries +2.1 +General preliminaries about spin geometry +Let (M, g) be an m-dimensional Riemannian manifold (not necessarily compact) with a chosen +orientation. Let PSO(M) be the set of positively oriented orthonormal frames on (M, g). This is +a SO(m)-principal bundle over M. A spin structure on M is a pair σ = (PSpin(M), ϑ) where +PSpin(M) is a Spin(m)-principal bundle over M and ϑ : PSpin(M) → PSO(M) is a map such +that the diagram +PSpin(M) × Spin(m) +� +ϑ × Θ +� +PSpin(M) +ϑ +� +� M +PSO(M) × SO(m) +� PSO(M) +� +commutes, where Θ : Spin(m) → SO(m) is the nontrivial double covering of SO(m). There is +a topological condition for the existence of a spin structure, namely, the vanishing of the second +Stiefel-Whitney class ω2(M) ∈ H2(M, Z2). Furthermore, if a spin structure exists, it need not +be unique. For these results we refer to [18,28]. +In order to introduce the spinor bundle, we recall that the Clifford algebra Cl(Rm) is the +associative R-algebra with unit, generated by Rm satisfying the relation x · y − y · x = −2(x, y) +for x, y ∈ Rm (here (·, ·) is the Euclidean scalar product on Rm). It turns out that Cl(Rm) has +a smallest representation ρ : Spin(m) ⊂ Cl(Rm) → End(Sm) of dimension dimC(Sm) = 2[ m +2 ] +such that Cl(Rm) := Cl(Rm)⊗C ∼= EndC(Sm) as C-algebra. In case m is even, this irreducible +representations is uniquely determined, but it splits into non-equivalent sub-representations S+ +m +and S− +m as Spin(m)-representations. If m is odd, there are two irreducible Clm-representations +S0 +m and S1 +m. Both of them coincide if considered as Spin(m)-representations. +Define the chirality operator ωRm +C += i[ m+1 +2 +]e1 · e2 · · · em ∈ Clm with {e1, . . . , em} being a +positively oriented orthonormal frame on Rm. In case m is even, we have ωRm +C +act as ±1 on S± +m, +and sections of S+ +m (resp. S− +m) are called positive (resp. negative) spinors. While if m is odd, the +chirality operator acts on Sj +m as (−1)j, j = 0, 1. Hence, for m odd, it will cause no confusion +if we simply identify S0 +m and S1 +m as the same vector space, that is Sm = S0 +m = S1 +m, and equip +them with Clifford multiplication of opposite sign. +Associated to the above observations, the spinor bundle is then defined as +S(M) := PSpin(M) ×ρ Sm. +Note that the spinor bundle carries a natural Clifford multiplication, a natural hermitian metric + +9 +and a metric connection induced from the Levi-Civita connection on TM (see [18, 28]), this +bundle satisfies the axioms of Dirac bundle in the sense that +(i) for any x ∈ M, X, Y ∈ TxM and ψ ∈ Sx(M) +X · Y · ψ + Y · X · ψ + 2g(X, Y )ψ = 0; +(ii) for any X ∈ TxM and ψ1, ψ2 ∈ Sx(M), +(X · ψ1, ψ2)g = −(ψ1, X · ψ2)g, +where (·, ·)g is the hermitian metric on S(M); +(iii) for any X, Y ∈ Γ(TM) and ψ ∈ Γ(S(M)), +∇S +X(Y · ψ) = (∇XY ) · ψ + Y · ∇S +Xψ, +where ∇S is the metric connection on S(M). +The Dirac operator is then defined on the spinor bundle S(M) as the composition +Dg : Γ(S(M)) +∇S +−→ +Γ(T ∗M ⊗ S(M)) +−→ +Γ(TM ⊗ S(M)) +m +−→ +Γ(S(M)) +where m denotes the Clifford multiplication m : X ⊗ ψ �→ X · ψ. +Let us remark that there is an implicit g-dependence in the Clifford multiplication “m” or +“·”. In fact, considering a simple case where we replace g with a conformal metric ˜g = e2ug, +the isometry X �→ e−uX from (TM, g) onto (TM, ˜g) defines a principal bundle isomorphism +SO(TM, g) → SO(TM, ˜g) lifting to the spin level. Then it induces a bundle isomorphism +S(M, g) → S(M, ˜g), ψ �→ ˜ψ, fiberwisely preserving the Hermitian inner product and sending +X · ψ to e−uX˜· ˜ψ. In the sequel, when necessary, we shall write DM +g and ·g, etc., to precise the +underlying manifold M and the metric g. +2.2 +Spinor bundle and the Dirac operator on product manifolds +In this subsection our notation is close to [38]. Let (N = M1 × M2, gN = gM1 ⊕ gM2) be a +product of Riemannian spin mj-manifolds (Mj, gMj, σMj), j = 1, 2. We have +PSpin(N) = (PSpin(M1) × PSpin(M2)) ×ζ Sm1+m2 +where ζ : Spin(m1) × Spin(m2) → Spin(m1 + m2) is the Lie group homomorphism lifting the +standard embedding SO(m1) × SO(m2) → SO(m1 + m2). +The spinor bundle over N can be identified with +S(N) = +� +(S(M1) ⊕ S(M1)) ⊗ S(M2) +both m1 and m2 are odd, +S(M1) ⊗ S(M2) +m1 is even. + +10 +That is, we always put the even dimensional factor in the place of M1. And the Clifford multi- +plication on S(N) can be explicitly given in terms of the Clifford multiplications on its factors. +In fact, for X ∈ TM1, Y ∈ TM2, ϕ ∈ Γ(S(M2)) and +ψ = +� +ψ1 ⊕ ψ2 ∈ Γ(S(M1) ⊕ S(M1)) +for both m1 and m2 odd +ψ ∈ Γ(S(M1)) +for m1 even +we have +(X ⊕ Y ) ·gN (ψ ⊗ ϕ) = (X ·gM1 ψ) ⊗ ϕ + (ωM1 +C +·gM1 ψ) ⊗ (Y ·gM2 ϕ) +(2.1) +where in case m1 and m2 odd we set X ·gM1 ψ = (X ·gM1 ψ1) ⊕ (−X ·gM1 ψ2) and ωM1 +C ·gM1 ψ = +i(ψ2⊕−ψ1). Let us remark that there are different ways to formulate the Clifford multiplication +(2.1), but such changes are equivalent. Indeed, due to the uniqueness of Cl(TM1 ⊕ TM2), any +definition of the Clifford multiplication on S(N) can be identified with (2.1) via a vector bundle +isomorphism (see the examples in the next subsection). +Let ∇S(M1) and ∇S(M2) be the Levi-Civita connections on S(M1) and S(M2). By +∇S(M1)⊗S(M2) = ∇S(M1) ⊗ IdS(M2) + IdS(M1) ⊗ ∇S(M2) +we mean the tensor product connection on S(M1) ⊗ S(M2). Then, by (2.1), the Dirac operator +on N is given by +DN +g = ˜DM1 +gM1 ⊗ IdS(M2) + (ωM1 +C +·gM1 IdS(M1)) ⊗ DM2 +gM2 +(2.2) +where ˜DM1 +gM1 = DM1 +gM1 ⊕ −DM1 +gM1 if both m1 and m2 are odd and ˜DM1 +gM1 = DM1 +gM1 if m1 is even. +For the case m1 + m2 even, we have the decomposition S(N) = S(N)+ ⊕ S(N)− and, +moreover, when restrict DN +g on those half-spinor spaces we get DN +g : Γ(S(N)±) → Γ(S(N)∓). +2.3 +A particular ansatz in Euclidean spaces +Let M = Rm be equipped with the Euclidean metric, then the spinor bundle is given by +S(Rm) = Rm × Sm ∼= Rm × C2[ m +2 ]. Although, from the abstract setting, the Dirac operator +can be given by +DgRmψ = +m +� +k=1 +ek ·gRm ∇ekψ, +ψ ∈ S(Rm) +where {e1, . . . , em} is a orthonormal base of Rm, we can have a more explicit representation of +this operator. In fact the Dirac operator can be formulated as a constant coefficient differential +operator of the form +DgRm = +m +� +k=1 +α(m) +k +∂ +∂xk +(2.3) +where α(m) +k +is a linear map α(m) +k +: C2[ m +2 ] → C2[ m +2 ] satisfying the relation +α(m) +j +α(m) +k ++ α(m) +k +α(m) +j += −2δij +(2.4) + +11 +for all j, k. +Let us give a possible construction of these {α(m) +j +} by using 2[ m +2 ] × 2[ m +2 ] complex matrices +with a block structure. We start with m = 1 and the 1-dimensional Dirac operator DgR = i d +dx, +that is we have α(1) +1 += i the pure imaginary unit. For m is even, we define +α(m) +j += +� +0 +−iα(m−1) +j +iα(m−1) +j +0 +� +for j = 1, . . . , m − 1 +and +α(m) +m += +� +0 +i Id +i Id +0 +� +where “Id” is understood to be the identity on C2[ m−1 +2 +]. And, if m is odd, we define +α(m) +j += α(m−1) +j +for j = 1, . . . , m − 1 +and +α(m) +m += i +m+1 +2 α(m−1) +1 +· · · α(m−1) +m−1 . +It is illuminating to consider this construction in low dimensions: +Example 1. For m = 2, we have +α(2) +1 += +� 0 +1 +−1 +0 +� +and +α(2) +2 += +�0 +i +i +0 +� +. +Writing a spinor field ψ : R2 → S(R2) in components as +�ψ1 +ψ2 +� +∈ C2, we then have +DgR2ψ = +� 0 +1 +−1 +0 +� � ∂ψ1 +∂x1 +∂ψ2 +∂x1 +� ++ +�0 +i +i +0 +� � ∂ψ1 +∂x2 +∂ψ2 +∂x2 +� += +� ∂ψ2 +∂x1 + i ∂ψ2 +∂x2 +− ∂ψ1 +∂x1 + i ∂ψ1 +∂x2 +� +. +(2.5) +Thus, in this case, the Dirac operator is simply the Cauchy-Riemann operator. +Consider the product R2 = R × R and the identification S(R2) = (S(R) ⊕ S(R)) ⊗ S(R). +We see that the fiberwise isomorphism is given explicitly by +(S(R) ⊕ S(R)) ⊗ S(R) ∋ +�u1v +u2v +� +←→ 1 +√ +2 +�(u1 + u2)v +(u1 − u2)v +� +∈ S(R2) +(2.6) +for u1, u2, v ∈ Γ(S(R)). In particular, by (2.2), we see that +�i d +dx +0 +0 +−i d +dx +� �u1v +u2v +� +− d +dy +� u2v +−u1v +� += +� iu′ +1v − u2v′ +−iu′ +2v + u1v′ +� +which coincides with (2.5) (under the action of the isomorphism in (2.6)). +Example 2. For m = 3, we have +α(3) +1 += +� 0 +1 +−1 +0 +� +, +α(3) +2 += +�0 +i +i +0 +� +and +α(3) +3 += +�−i +0 +0 +i +� +which are exactly the classical Pauli matrices. And for the product R3 = R2 × R, it is easy to +obtain from (2.3) that +DgR3 = DgR2 ⊗ IdS(R) + +�−1 +0 +0 +1 +� +⊗ DgR +fitting into (2.2). + +12 +Example 3. For m = 4, we have +α(4) +1 += +� +� +� +� +0 +−i +i +i +−i +0 +� +� +� +� , +α(4) +2 += +� +� +� +� +0 +1 +1 +−1 +−1 +0 +� +� +� +� , +α(4) +3 += +� +� +� +� +0 +−1 +0 +0 +1 +1 +0 +0 +−1 +0 +� +� +� +� +and +α(4) +4 += +� +� +� +� +0 +i +0 +0 +i +i +0 +0 +i +0 +� +� +� +� +And for the product R4 = R2 × R2, we have S(R4) = S(R2) ⊗ S(R2). By considering a bundle +isomorphism +S(R2) ⊗ S(R2) ∋ +�u1 +u2 +� +⊗ +�v1 +v2 +� +←→ +� +� +� +� +−iu1v1 +−iu2v2 +iu1v2 +iu2v1 +� +� +� +� ∈ S(R4) +for u1, u2, v1, v2 ∈ Γ(S(R2)), one easily verifies the correspondence +DgR4 = DgR2 ⊗ IdS(R2) + +�−1 +0 +0 +1 +� +⊗ DgR2 +which justifies (2.2). Meanwhile, for the product R4 = R3 ×R and the associated spinor bundle +S(R4) = (S(R3) ⊕ S(R3)) ⊗ S(R), we have the fiberwise isomorphism +(S(R3) ⊕ S(R3)) ⊗ S(R) ∋ +� +� +� +� +ψ1ϕ +ψ2ϕ +ψ3ϕ +ψ4ϕ +� +� +� +� ←→ 1 +√ +2 +� +� +� +� +(ψ4 − ψ2)ϕ +(ψ3 − ψ1)ϕ +(ψ2 + ψ4)ϕ +−(ψ1 + ψ3)ϕ +� +� +� +� ∈ S(R4) +for +�ψ1 +ψ2 +� +, +�ψ3 +ψ4 +� +∈ S(R3) and ϕ ∈ S(R) such that the action of +�DgR3 +0 +0 +−DgR3 +� +⊗ IdS(R) + i +� +0 +IdS(R3) +−IdS(R3) +0 +� +⊗ DgR +on (S(R3)⊕S(R3))⊗S(R) coincides with the action of DgR4 on S(R4). This verifies (2.2). Note +the analogy with dimension two. +We could continue this analysis. For general m, one can compute the matrices {α(m) +j +}, the +chirality operator ωRm +C +and, particularly when m is even, the corresponding bundle isomorphism +to decompose the Dirac operator in a product structure. However, these explicit formulas are +seldom. It is always simpler to use the abstract setting of the Clifford module. + +13 +It is interesting to note that the aforementioned explicit formula for the Dirac operator mo- +tivates a “nice” function space which is invariant under the actions of the Dirac operator. More +precisely, let us set +E(Rm) := +� +ψ(x) = f1(|x|)γ0 + f2(|x|) +|x| +x · γ0 : x ∈ Rm, f1, f2 ∈ C∞(0, ∞) and γ0 ∈ S2[ m +2 ] +C +� += +� +ψ(x) = f1(|x|)γ0 + f2(|x|) +|x| +m +� +k=1 +xkα(m) +k +γ0 : f1, f2 ∈ C∞(0, ∞) and γ0 ∈ S2[ m +2 ] +C +� +. +where S2[ m +2 ] +C +stands for the complex unit sphere in the spin-module Sm ∼= C2[ m +2 ]. Then, following +the rule of the Clifford multiplication or the relation (2.4), it is easy to check that +DgRmψ = − +� +f ′ +2(|x|) + (m − 1)f2(|x|) +|x| +� +γ0 + f ′ +1(|x|) +|x| +x · γ0 ∈ E(Rm) +∀ψ ∈ E(Rm). +Moreover, in order to make sure that ψ is continuous at the origin, one may consider a further +restriction to the subspace +E0(Rm) = +� +ψ(x) = f1(|x|)γ0+f2(|x|) +|x| +x·γ0 ∈ E : f ′ +1(t) = O(t) and f2(t) = O(t) as t ↘ 0 +� +. +Remark 2.1. +(1) It is interesting to see that the specific ansatz provided in E(Rm) contains +some important formulations of spinors, which are of interest to many physicists when +they are dealing with spinor fields in quantum electrodynamics. In fact, to the best of our +knowledge, it can be traced back to R. Finkelstein, R. LeLevier and M. Ruderman [14] +in 1951 when they investigated a nonlinear Dirac equation in R3 × R. By separating the +time variable, the authors introduced a very special formulation of a spinor field, i.e. +ψ(r, θ1, θ2) = +� +� +� +� +� +f1(r) +0 +if2(r) cos θ1 +if2(r) sin θ1eiθ2 +� +� +� +� +� or +� +� +� +� +� +if2(r) cos θ1 +if2(r) sin θ1eiθ2 +f1(r) +0 +� +� +� +� +� +(2.7) +where (r, θ1, θ2) ∈ (0, +∞) × [0, π] × [0, 2π] is the spherical coordinates on R3. And +subsequently, this ansatz has been commonly used in particle physics where spinors play +a crucial role, see for instance [40, 45] and [11] for a 2-dimensional analogue. Now, in +our setting, we understand that the above spinor field belongs to the sub-bundle S(R3) ⊕ +S(R3). Consider the standard spherical coordinates +x1 = r cos θ1, +x2 = r sin θ1 cos θ2, +x3 = r sin θ1 sin θ2 cos θ3 +and +x4 = r sin θ1 sin θ2 sin θ3 +for r > 0, θ1, θ2 ∈ [0, π] and θ3 ∈ [0, 2π], if we restrict to θ2 = π +2 (i.e. the variable x2 is +separated out, treated as the time variable) and take +γ0 = +� +� +� +� +1 +0 +0 +0 +� +� +� +� ∈ S4 +C, + +14 +we soon derive that +f1(|x|)γ0 + f2(|x|) +|x| +4 +� +k=1 +xkα(4) +k γ0 = +� +� +� +� +� +if2(r) cos θ1 +if2(r) sin θ1eiθ3 +f1(r) +0 +� +� +� +� +� +which is exactly the latter one in (2.7). +(2) Although the special ansatz (2.7) for a spinor has been known for over half a century, it +is still new and important to have the family E(Rm) for general dimensions. Particularly, +the ansatz in E(Rm) reduces the Dirac equation significantly. Indeed, for the semilinear +equations of the form +DgRmψ = h(|x|, |ψ|)ψ, +ψ : Rm → Sm ∼= C2[ m +2 ] +(2.8) +where h : [0, +∞) × [0, ∞) → R is a given function, the ansatz in E(Rm) transforms it +equivalently to +� +� +� +� +� +− f ′ +2 − m − 1 +r +f2 = h +� +r, +� +f 2 +1 + f 2 +2 +� +f1, +f ′ +1 = h +� +r, +� +f 2 +1 + f 2 +2 +� +f2, +for r > 0 +making the problem much easier to deal with. +(3) This ansatz was also used to study several mathematical physics models. We cite for +instance [6–8] for the study of Dirac-type equation, [15,39] for the study of particle like +solutions of coupled Dirac type equations. +(4) The space E(Rm) is somehow natural within spinor fields. Indeed, if one looks at the +parallel spinors on Rm and the Dirac bubbles [9] (corresponding to Killing spinors on +the sphere), then one notices that they all belong to E(Rm). Hence, we can think about +E(Rm) as a generalized special class of spinors. +3 +Set up of the problems +Let us consider the m-sphere Sm to be Rm ∪ {∞}, where the coordinates x ∈ Rm is given +by the standard stereographic projection from the north pole αm : Sm \ {P m+1 +N +} → Rm (here +P m+1 +N += (0, . . . , 0, 1) ∈ Sm ⊂ Rm+1 stands for the north pole). For clarity, we use the sub- +or superscripts to indicate the underlying dimensions. By setting P m+1 +S += (0, . . . , 0, −1) for +the south pole, we can see that the manifold R × Sm−1 is conformally equivalent to Sm \ +{P m+1 +N +, P m+1 +S +}. The conformal diffeomorphism can be explicitly formulated by +Sm \ {P m+1 +N +, P m+1 +S +} +αm +−→ +Rm \ {0} +βm +−→ +R × Sm−1 +ξ = (ξ1, . . . , ξm+1) +�−→ +x = (x1, . . . , xm) +�−→ +(ln |x|, x/|x|) +(3.1) + +15 +where we have (α−1 +m )∗gSm = +4 +(1+|x|2)2gRm and (βm)∗(gR ⊕ gSm−1) = +1 +|x|2gRm. +This observation leads to some further considerations. Typical examples arise from the (con- +nected) domain Ω ⊂ Sn whose complement is an equatorial circle. Without loss of generality, +we may consider the domain +Sm \ S1 = +� +(ξ1, . . . , ξm+1) ∈ Rm+1 : +� +k +ξ2 +k = 1, ξ2 +1 + ξ2 +m+1 < 1 +� +. +Then we have the following conformal equivalence +Ω = Sm \ S1 +αm +−→ +Rm \ {(R, 0, . . . , 0)} +βm +−→ +R × (Sm−1 \ {P m +N , P m +S }) +(3.2) +We now consider the solutions of the spinorial Yamabe equation on the sphere (Sm, gSm), +that are singular at a prescribed closed set Σ ⊂ Sm. More specifically, we will consider the +problem +DgSmφ = |φ| +2 +m−1 +gSm φ +on Ω = Sm \ Σ +(3.3) +when Σ is given by a pair of antipodal points, say {P m+1 +N +, P m+1 +S +}, or an equatorial circle S1. +Before discussing the Delaunay family of solutions to Eq. (3.3), let us recall the transforma- +tion formula of the Dirac operator under conformal changes (see [21,23]): +Proposition 3.1. Let g0 and g = f 2g0 be two conformal metrics on a Riemannian spin m- +manifold M. Then, there exists an isomorphism of vector bundles F : S(M, g0) → S(M, g) +which is a fiberwise isometry such that +Dg +� +F(ψ) +� += F +� +f − m+1 +2 Dg0 +� +f +m−1 +2 ψ +�� +, +where Dg0 and Dg are the Dirac operators on M with respect to the metrics g0 and g, respec- +tively. +In what follows, our discussions will be build upon this formula. +3.1 +The singular set is a pair of antipodal points +In this setting, without loss of generality, we assume Σ = {P m+1 +N +, P m+1 +S +} ⊂ Sm. Then, as a +direct consequence of Proposition 3.1, we have that if ψ is a solution to the equation +DgRmψ = |ψ| +2 +m−1 +gRm ψ +on Rm \ {0} +(3.4) +then φ = F(f − m−1 +2 ψ) (f(x) = +2 +1+|x|2) is a solution to Eq. (3.3). Notice that since Eq. (3.4) has +the same structure as (2.8), we shall look at solutions of the form +ψ(x) = f1(|x|)γ0 + f2(|x|) +|x| +x · γ0 ∈ E(Rm). +(3.5) +Then, applying the Emden-Fowler change of variable r = e−t and write f1(r) = −u(t)e +m−1 +2 +t +and f2(r) = v(t)e +m−1 +2 +t, we are led to consider the following system +� +� +� +� +� +u′ + m − 1 +2 +u = (u2 + v2) +1 +m−1v, +−v′ + m − 1 +2 +v = (u2 + v2) +1 +m−1u. +(3.6) + +16 +This system is easily integrated and is nondissipative, in particular, the Hamiltonian energy +H(u, v) = −m − 1 +2 +uv + m − 1 +2m +� +u2 + v2� +m +m−1 +is constant along solutions of (3.6). +The equilibrium points for system (3.6) are +(0, 0) +and +± +�(m − 1)(m−1)/2 +2m/2 +, (m − 1)(m−1)/2 +2m/2 +� +. +And there is a special homoclinic orbit +u0(t) = +m(m−1)/2et/2 +2m/2 cosh(t)m/2, +v0(t) = m(m−1)/2e−t/2 +2m/2 cosh(t)m/2 +(3.7) +corresponding to the level set H = 0; it limits on the origin as t tends to ±∞, and encloses +a bounded set Λ in the first quadrant of the (u, v)-plane, given by {H ≤ 0}. It is easy to see +that orbits not enclosed by this level set, i.e. those orbits in {H > 0}, must pass across the +u-axis and v-axis. That is u and v must change sign. Observe that the equilibrium point (0, 0) +is contained exactly in two orbits: the homoclinic one and the stationary orbit (0, 0). Hence, for +orbits (u(t), v(t)) in {H ̸= 0}, we must have that u2 + v2 ̸= 0 for all t. And thus, we have an +unbounded one parameter family of periodic solutions +D1 +m = +� +(u, v) is a solution to Eq. (3.6) : u(0) = v(0) = µ > 0, µ ̸= m(m−1)/2 +2m/2 +� +, +which induces correspondingly a family of singular solutions S1 +m to Eq. (3.4) via (3.5). Remark +that |ψ(x)| → +∞ as |x| → 0 and |ψ(x)| = O(|x|− m−1 +2 ) as |x| → +∞ for each ψ ∈ S1 +m. +Therefore, these solutions give rise to distinguished singular solutions of Eq. (3.3). +If we take into account just the periodic solutions in D1 +m, we will call them the Delaunay-type +solutions of the spinorial Yamabe problem (3.4). Although we do not know them explicitly, in +Section 4, we will study the bifurcation phenomenon for solution in the first quadrant of (u, v)- +plane. +3.2 +The singular set is an equatorial circle +First of all, we need to observe that Eq. (3.3) can be interpreted as an equation on R × (Sm−1 \ +{P m +N , P m +S }) by a conformal change of the Riemannian metric gSm on Sm \ S1. Consider the +product metric on R × Sm−1, given in (τ, ϑ)-coordinates by ¯g = dτ 2 + dϑ2, where ϑ = +(ϑ1, . . . , ϑm−1) parameterizes the unit sphere Sm−1. Then it follows from the conformal equiv- +alence (3.2) that +(α−1 +m ◦ β−1 +m )∗gSm = +4e2τ +(1 + e2τ)2¯g = +1 +cosh(τ)2¯g. +And as a direct consequence of Proposition 3.1, we have that if ϕ is a solution to the equation +D¯gϕ = |ϕ| +2 +m−1 +¯g +ϕ +on R × (Sm−1 \ {P m +N , P m +S }) +(3.8) + +17 +then φ = F(cosh(τ) +m−1 +2 ϕ) is a solution to Eq. (3.3) with F being a bundle isomorphism. +Let us remark that the formula (2.2) on product manifolds indicates a way to construct +singular solutions for Eq. (3.8). In fact, if m is odd (hence m ≥ 3), then m − 1 is even and we +can consider a special spinor of the form ϕ = 1 ⊗ ˜ψ so that Eq. (3.8) is reduced to +DgSm−1 ˜ψ = | ˜ψ| +2 +m−1 +gSm−1 ˜ψ +(3.9) +where ˜ψ = ˜ψ(ϑ) is a spinor on Sm−1 \ {P m +N , P m +S }. And once again, by using the conformal +formula in Proposition 3.1, Eq. (3.9) can be equivalently transformed to +DgRm−1ψ = f(x) +1 +m−1|ψ| +2 +m−1 +gRm−1ψ +on Rm−1 \ {0} +(3.10) +where f(x) = +2 +1+|x|2 for x ∈ Rm−1. And the solutions of (3.9) and (3.10) are in one-to-one +correspondence via the identification ˜ψ ↔ f − m−2 +2 ψ for spinors. +Now, by considering the ansatz +ψ(x) = f1(|x|)γ0 + f2(|x|) +|x| +x · γ0 ∈ E(Rm−1). +and applying the change of variable r = e−t, we can reduce Eq. (3.10) to the system +� +� +� +� +� +u′ + m − 2 +2 +u = cosh(t)− +1 +m−1(u2 + v2) +1 +m−1v +−v′ + m − 2 +2 +v = cosh(t)− +1 +m−1(u2 + v2) +1 +m−1u +(3.11) +where f1(r) = −u(t)e +m−2 +2 +t and f2(r) = v(t)e +m−2 +2 +t. +If m is even, then the spinor bundle on R × (Sm−1 \ {P m +N , P m +S }) can be identified with +S(R) ⊗ (S(Sm−1) ⊕ S(Sm−1)) and the Dirac operator can be formulated as +D¯g = +�DgSm−1 +0 +0 +−DgSm−1 +� +⊗ IdS(R) + i +� +0 +IdS(Sm−1) +−IdS(Sm−1) +0 +� +⊗ DgR. +Hence, considering a spinor of the form ϕ = 1 ⊗ ( ˜ψ1 ⊕ ˜ψ1) for ˜ψ1, ˜ψ1 ∈ Γ(S(Sm−1)), we may +reduce Eq. (3.8) to the following Dirac system +� +DgSm−1 ˜ψ1 +−DgSm−1 ˜ψ2 +� += +� +| ˜ψ1|2 +gSm−1 + | ˜ψ2|2 +gSm−1 +� +1 +m−1 +� ˜ψ1 +˜ψ2 +� +on Sm−1 \ {P m +N , P m +S }. Similar to Eq. (3.10), we can transform the above system to +� +DgRm−1ψ1 +−DgRm−1ψ2 +� += f(x) +1 +m−1� +|ψ1|2 +gRm−1 + |ψ2|2 +gRm−1 +� +1 +m−1 +� +ψ1 +ψ2 +� +(3.12) +on Rm−1 \ {0}. + +18 +Now, using the ansatz +ψ1(x) = f1(|x|)γ0 + f2(|x|) +|x| +x · γ0 +and +ψ2(x) = f3(|x|)γ0 + f4(|x|) +|x| +x · γ0 +in E(Rm−1) and applying the change of variable r = e−t, we then get the following system +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +u′ +1 + m − 2 +2 +u1 = cosh(t)− +1 +m−1� +u2 +1 + u2 +2 + v2 +1 + v2 +2 +� +1 +m−1v1 +−v′ +1 + m − 2 +2 +v1 = cosh(t)− +1 +m−1� +u2 +1 + u2 +2 + v2 +1 + v2 +2 +� +1 +m−1u1 +u′ +2 + m − 2 +2 +u2 = cosh(t)− +1 +m−1� +u2 +1 + u2 +2 + v2 +1 + v2 +2 +� +1 +m−1v2 +−v′ +2 + m − 2 +2 +v2 = cosh(t)− +1 +m−1� +u2 +1 + u2 +2 + v2 +1 + v2 +2 +� +1 +m−1u2 +(3.13) +where we have substituted f1(r) = −u1(t)e +m−2 +2 +t, f2(r) = v1(t)e +m−2 +2 +t, f3(r) = u2(t)e +m−2 +2 +t and +f4(r) = v2(t)e +m−2 +2 +t. Therefore, we can consider the solutions for which u1 = u2 and v1 = v2; +these are the solutions having the simplest and clearest structure. By writing u = +√ +2u1 and +v = +√ +2v1, we can turn (3.13) into +� +� +� +� +� +u′ + m − 2 +2 +u = cosh(t)− +1 +m−1� +u2 + v2� +1 +m−1v +−v′ + m − 2 +2 +v = cosh(t)− +1 +m−1� +u2 + v2� +1 +m−1u +which exactly coincides with (3.11). +Clearly, the system (3.11) has an Hamiltonian structure, where the Hamiltonian energy is +given by +H(t, u, v) = −m − 2 +2 +uv + m − 1 +2m +cosh(t)− +1 +m−1(u2 + v2) +m +m−1. +It is evident that this system is dissipative and there is no periodic solution. However, one +may consider solutions that are not converging to (0, 0) as t → ±∞. More precisely, we will +characterize the following family of solutions +D2 +m = +� +(u, v) is a solution to Eq. (3.11) : u2(t) + v2(t) → +∞ as t → ±∞ +� +which induces a family of singular solutions S2 +m to Eq. (3.9). Hence these solutions gives rise +to singular solutions of Eq. (3.3). In this setting, we shall call the family D2 +m the Delaunay-type +solutions. +4 +Analysis of the ODE systems +This section contains our main study of the dynamical systems (3.6) and (3.11). We point out +that both systems have a variational structure. In fact, if we denote z = (u, v) ∈ R2, systems +(3.6) and (3.11) can be rewritten as +˙z = dz +dt = J∇zH(t, z) +(4.1) + +19 +where +J = +� 0 +1 +−1 +0 +� +and H stands for the corresponding Hamiltonian energy. The functionals +ΦT(z) = 1 +2 +� T +−T +(−J ˙z, z)dt − +� T +−T +H(t, z)dt +and +Φ(z) = 1 +2 +� +R +(−J ˙z, z)dt − +� +R +H(t, z)dt +can be used to obtain periodic solutions and homoclinic solutions for (4.1) respectively. In par- +ticular, there is one-to-one correspondence between 2T-periodic solutions of (4.1) and critical +points of ΦT (as long as H(t, z) is periodic in the t-variable or independent of t). Similarly, +critical points of Φ correspond to homoclinic solutions of (4.1), i.e., z(t) → (0, 0) as t → ±∞. +For the autonomous system, i.e. (3.6), we point out that the existence of a 2T-periodic solu- +tion for every T > T0, some T0 > 0, and the asymptotic behavior of these solutions as T ↗ +∞ +have been already investigated in [1,44]. By summarizing their results, we have +Proposition 4.1. There exists T0 > 0 such that for every T > T0 the Hamiltonian system (3.6) +has a non-constant 2T-periodic solution zT. The family {zT : T > T0} is compact in the +following sense: for any sequence Tn ↗ +∞, up to a subsequence if necessary, zTn converges +in C1 +loc(R, R2) to a nontrivial solution z∞ of the system (3.6) on R satisfying +lim +|t|→+∞ z∞(t) = +lim +|t|→+∞ ˙z∞(t) = 0, +i.e., z∞ is a homoclinic orbit. +Notice that the previous proposition does not provide a clear description of the behavior +of the solutions zT as T ↘ T0 or a characterization of z∞. For instance, from the arguments +in [1, 44], we do not have an estimate of T0 and we do not know if there are non-constant so- +lutions below T0. In fact, if H has a “good” structure around its equilibrium points, then one +can use Lyapunov’s center theorem to exhibit a family of small amplitude periodic solutions +bifurcating from the equilibrium solution and also have an estimate on T0. Nevertheless, this +does not provide uniqueness of the family of non-constant solutions. +In the sequel, we will perform different approaches to characterize the Delaunay-type fam- +ilies D1 +m and D2 +m. We also want to point out that an alternative method can be used to find +periodic solutions of family D1,− +m using variational analysis and by tracking the least energy so- +lution, we can characterize the homoclinic energy z∞, corresponding to the least energy solution +for the functional Φ. This procedure was used in a more general setting of product manifolds +in [5]. +4.1 +The nondissipative case: Bifurcation of the positive periodic orbits +In order to analyse the dynamical system (3.6), we recall that +H(u, v) = −m − 1 +2 +uv + m − 1 +2m +� +u2 + v2� +m +m−1 + +20 +for u, v ∈ R and m ≥ 2, which is independent of t. We will focus on the periodic solu- +tions/orbits of (3.6) in the first quadrant of the (u, v)-plane, that is u, v : R/2TZ → (0, +∞) +for all T > 0. Such solutions will be referred as positive solutions. +System (3.6) has an “obvious” constant solution u = v ≡ (m−1)(m−1)/2 +2m/2 +for all T > 0. From +now on, we intend to look at non-constant solutions. By setting z = u2 + v2 and w = u2 − v2, +we have uv = +√ +z2−w2 +2 +and (3.6) becomes +� +� +� +z′ = −2λw +zz′ − ww′ = 1 +λzp−1z′√ +z2 − w2 +(4.2) +where we denote λ = m−1 +2 +> 0 and p = +m +m−1 ∈ (1, 2] for simplicity. After multiplication by +(z2 − w2)−1/2 in the second equation, we obtain +d +dt +�√ +z2 − w2 � += d +dt +� 1 +λpzp� +. +Thus, for any solution z and w, there exists a constant K such that +√ +z2 − w2 = +1 +λpzp + K, that +is, +w2 = z2 − +� 1 +λpzp + K +�2 +and +1 +λpzp + K ≥ 0. +(4.3) +For K ∈ R, let us denote +FK(s) = s2 − +� 1 +λpsp + K +�2 +for s ≥ 0. +Remark that, if (z, w) is a non-constant 2T-periodic solution of (4.2), then z must achieve the +maximum and minimum in one period. Hence z′ has at least two zeros. This, together with the +first equation in (4.2), implies that FK should vanish at least twice. Therefore, the conditions +on K are particularly restrictive. In fact, for K = 0, we can combine the first equation in (4.2) +and (4.3) together to obtain (z′)2 = 4λ2z2 − +4 +p2z2p. Then, if there exist t0 and t1 such that +z(t0) < z(t1) and z′(t0) = z′(t1) = 0, we have z(t0) = 0 and z(t1) = (m +2 )m−1. Clearly, +this should corresponds to the homoclinic solution (3.7) and can not be periodic. For K < 0, +by analyzing the algebraic equation FK(s) = 0, we can see that Fk has exactly two zeros +0 < s0 < s1 on (0, +∞) given by the relations +� +� +� +� +� +� +� +s0 = − 1 +λpsp +0 − K, +s1 = 1 +λpsp +1 + K. +But we find 1 +λpsp +0+K < 0, which fails to satisfy the second inequality in (4.3). So the remaining +range for K is (0, +∞). However, it is obvious that K can not be large. +Lemma 4.2. If K > 0 is small, FK has exactly two zeros on (0, +∞). + +21 +Proof. We only prove the case p = +m +m−1 ∈ (1, 2), i.e. m > 2, since p = 2 is much easier. Notice +that +F ′ +K(s) = 2s − 2 +λ +� 1 +λpsp + K +� +sp−1 +for s ≥ 0 and p ∈ (1, 2], we have F ′ +K(0) = 0 and F ′ +K(s) < 0 in (0, δ1) for some δ1 > 0 small. +Observe that the two maps s �→ λs2−p and s �→ +1 +λpsp + K have exactly two intersections +for K > 0 small enough. We denote the horizontal coordinates of these two intersections by +0 < s0,1 < s0,2. Then we have F ′ +K < 0 on (0, s0,1) ∪ (s0,2, +∞) and F ′ +K > 0 on (s0,1, s0,2). +Therefore, FK(s0,1) < 0 is a strict local minimum, whereas FK(s0,2) is a strict local maximum. +Since F0(1) = 1 − +1 +λ2p2 > 0 (we used the facts λ = m−1 +2 , p = +m +m−1 and m > 2), we have +FK(1) > 0 for all small K. Hence FK(s0,2) > 0. This implies FK has exactly two zeros on +(0, +∞). +Let +K0 := sup +� +K > 0 : FK has two zeros +� +. +We remark that, for K > 0, FK can not have a third zero in (0, +∞) since F ′ +K changes sign at +most twice and FK(0) < 0. +Lemma 4.3. K0 < +∞ and FK0 has only one zero, which is the global maximum. Furthermore, +FK(s) < 0 for all K > K0 and s ≥ 0. +Proof. Since K0 < +∞ is obvious, we only need to check the remaining statements. To begin +with, we mention that +∂ +∂K FK(s) = −2 +� 1 +λpsp + K +� +< 0 +(4.4) +provided that K > 0 and s ≥ 0. Hence, if F ˆ +K(s ˆ +K) > 0 for some ˆK > 0 and s ˆ +K > 0, we +have FK(s ˆ +K) > 0 for all K ∈ (0, ˆK]. Moreover, due to the continuity of FK with respect to +K, there exists ε > 0 such that FK(s ˆ +K) > 0 for K ∈ ( ˆK, ˆK + ε). Therefore, we can see that +� +K > 0 : FK has two zeros +� += (0, K0) is an open interval and that max FK0 ≤ 0 (otherwise +FK0 will have two zeros). By choosing a sequence Kn ↗ K0 and sn > 0 such that FKn(sn) > 0, +we have {sn} is bounded and FKn(sn) → 0 as n → ∞. Therefore FK0 has only one zero, which +is the global maximum. The last assertion comes from the fact (4.4). +Remark 4.4. The value of K0 can be explicitly computed. Precisely, we have +K0 = +� +1 − 1 +p +� +λ +1 +p−1 = 1 +m +�m − 1 +2 +�m−1 +. +In fact, K = K0 is the largest positive number such that the equation s = +1 +λpsp + K has a +solution. +In the sequel, let K ∈ (0, K0), we set 0 < s0 < s1 the points such that FK vanishes. It is +worth pointing out that s0 and s1 are functions of K. Then FK is positive on the interval (s0, s1). +And Eq. (4.3) is now equivalent to +dz +2λ +� +FK(z) += ±dt, + +22 +which can be solved by ηK(z) = ±t + C, where +ηK(z) = +� s +s0 +dz +2λ +� +FK(z) +and C ∈ R is a constant. +Of course, ηK is defined on the interval (s0, s1). By noting that s0 and s1 are simple roots +of FK (that is F ′ +K(sj) ̸= 0 for j = 0, 1), we have ηK(s1) is well-defined. Moreover, we have +η′ +K(s) > 0 and η′ +K(s) → +∞ as s → s0 or s1. Therefore, ηK has an inverse η−1 +K which +increases from s0 to s1 on the interval [0, ηK(s1)]. Now, solutions to (4.3) can be represented as +z(t) = η−1 +K (±t + C) for C ∈ R. +Setting +zK(t) = +� +η−1 +K (t) +t ∈ [0, ηK(s1)], +η−1 +K (−t) +t ∈ [−ηK(s1), 0], +(4.5) +it follows that zK is a 2ηK(s1)-periodic solution of Eq. (4.2) and can not have smaller period. +Moreover, this zK (jointly with the corresponding wK from Eq. (4.2)) gives rise to a positive +solution (uK, vk) of Eq. (3.6) with H(uK, vK) = − λK +2 < 0. +Lemma 4.5. The mapping K �→ ηK(s1) is continuous. Particularly, +lim +K↘0 ηK(s1) = +∞ +and +lim +K↗K0 ηK(s1) = +√m − 1 +2 +π +Proof. For starters, we shall write s0 = s0(K) and s1 = s1(K) to emphasize that s0 and s1 +are functions of K. Notice that s0 and s1 are solutions to the equation s = +1 +λpsp + K. By the +implicit function theorem, we have s0 and s1 are C1 functions, in particular, +� +� +� +� +� +� +1 − 1 +λs0(K)p−1� +s′ +0(K) = 1, +� +1 − 1 +λs1(K)p−1� +s′ +1(K) = 1. +Since we have assumed s0 < s1, we have +� +1 − 1 +λs0(K)p−1� +> 0 +and +� +1 − 1 +λs1(K)p−1� +< 0 +which implies that s′ +0(K) > 0 and s′ +1(K) < 0. +The continuity of ηK(s1) is obvious and, without digging out very much from the function +ηK(s1), we can evaluate the asymptotic behavior of ηK(s1) as K goes to the end points 0 and K0. +In fact, to see the limiting behavior of ηK(s1) as K ↘ 0, we first observe that FK(0) < 0 and +FK(2K) > 0 for all small K. Hence we have 0 < s0(K) < 2K. Moreover λ1/(p−1) < s1(K) +since s1(K) is the larger solution to the equation s = +1 +λpsp + K. Then +ηK(s1) ≥ +� λ1/(p−1) +2K +dz +2λ +� +FK(z) +≥ 1 +2λ +� λ1/(p−1) +2K +dz +z = 1 +2λ +� +ln λ1/(p−1) − ln 2K +� +. + +23 +Thus, by taking K → 0, we have limK↘0 ηK(s1) = +∞. +For K close to K0, we set GK(t) = FK(tm−1), that is +GK(t) = t2(m−1) − +� 2 +mtm + K +�2 +. +By writing t0 = s1/(m−1) +0 +and t1 = s1/(m−1) +1 +, we can write GK in its factorization +GK(t) = 4 +m2(t − t0)(t1 − t)PK(t) +with +PK(t) = +� +tm + m +2 tm−1 + m +2 K +�� +a0tm−2 + a1tm−3 + · · · + am−3t + am−2 +� +, +where +a0 = 1, +a1 = t0 + t1 − m +2 +and +aj = −t0t1aj−2 + (t0 + t1)aj−1 +for j = 2, . . . , m − 2. +From elementary computations, we can simply write +aj = tj+1 +1 +− tj+1 +0 +t1 − t0 +− m +2 +tj +1 − tj +0 +t1 − t0 +(4.6) +for j = 0, 1, . . . , m − 2. Then we can reformulate ηK(s1) as +ηK(s1) = +� t1 +t0 +tm−2dt +� +GK(t) += m +2 +� 1 +0 +(t0 + (t1 − t0)τ)m−1dτ +� +τ(1 − τ)PK(t0 + (t1 − t0)τ) +. +(4.7) +Notice that, as K approaches K0, we have t0, t1 → m−1 +2 . By the continuity of ηK(s1), we +have +lim +K→K0 ηK(s1) = cm +� 1 +0 +dτ +� +τ(1 − τ) += cmπ +where +cm = m(m − 1)m−1 +2m +� +PK0( m−1 +2 ) += +√m − 1 +2 +. +This completes the proof. +Remark 4.6. Recall that we are looking at the 2ηK(s1)-periodic solutions of Eq. (4.2), then +Lemma 4.5 implies: +(1) For every T > 0, Eq. (4.2) has the constant solution z0 ≡ (m−1)m−1 +2m−1 +and w0 ≡ 0, which +gives the nontrivial constant solution of Eq. (3.6). And, for T ≤ +√m−1 +2 +π, this is the only +possible solution of Eq. (4.2). +(2) Let d ∈ N with d +√m−1 +2 +π < T ≤ (d + 1) +√m−1 +2 +π. Then for any k = 1, . . . , d, we have +T +k ≥ T +d > +√m−1 +2 +π and there exists K = K(T/k) ∈ (0, K0) such that ηK(s1) = T/k. + +24 +(3) The solutions given by (4.5) corresponds to the solutions obtained in Proposition 4.1, +since the Hamiltonian energy H(uK, vK) → 0 and the minimal period ηK(s1) → +∞ as +K → 0. Moreover, we have T0 = +√m−1 +2 +π. +We end this section by comparing the classical Delaunay solutions that appear in the study of +the singular Yamabe problem and the solutions that we have just studied above. Let us recall the +classical Delaunay solutions for the singular Yamabe problem as in [32, 35], that are obtained +by solving the ODE +u′′ − (m − 2)2 +4 +u + m(m − 2) +4 +u +m+2 +m−2 = 0, +u > 0. +(4.8) +This equation is clearly nondissipative, and the corresponding Hamiltonian energy is +�H(u, u′) = 1 +2|u′|2 − (m − 2)2 +8 +u2 + (m − 2)2 +8 +u +2m +m−2. +By examining the level sets of �H, we see that all bounded positive solutions of Eq. (4.8) lie in +the region of the (u, u′)-plane where �H is non-positive. In the figures below, we show a few +orbits for both the Hamitonians for the systems (3.6) and (4.8) when m = 3. +Figure 1: The orbits for the spinorial Yam- +abe equation +Figure 2: The orbits for the classical Yam- +abe equation +4.2 +The dissipative case: Shooting method +In this subsection, we investigate the system (3.11). In particular, since we are looking for +singular solutions of the spinorial Yamabe equation, we are interested in solutions of (3.11) +such that +(u(t), v(t)) ̸→ (0, 0) +as t → ±∞. + +1.0 +0.5 +1.0 +0.6 +0.5 +1.0 +0.5 +1.00 2 +0 1 +上 +0.2 +0 4 +08 +0 1 +上 +0 2 +E D25 +In order to avoid unnecessary complexity and to get non-trivial solutions, we choose as +initial conditions +u(0) = v(0) = µ ∈ R \ {0}. +Moreover, the symmetry of the system allows us to consider only the case µ > 0. +Recall that the Hamiltonian energy associated to (3.11) is given by +H(t, u, v) = −m − 2 +2 +uv + m − 1 +2m +cosh(t)− +1 +m−1(u2 + v2) +m +m−1. +We begin with: +Lemma 4.7. For any µ > 0, there is (uµ, vµ) ∈ C1(R, R2), unique solution of (3.11) satisfying +uµ(0) = vµ(0) = µ. Furthermore, (uµ, vµ) depends continuously on µ, uniformly on [−T, T], +for any T > 0. +Proof. To begin with, we may write the system (3.11) in integral form as +� +� +� +� +� +� +� +u(t) = µ + +� t +0 +� +cosh(s)− +1 +m−1� +u(s)2 + v(s)2� +1 +m−1v(s) − m − 2 +2 +u(s) +� +ds +v(t) = µ − +� t +0 +� +cosh(s)− +1 +m−1� +u(s)2 + v(s)2� +1 +m−1u(s) − m − 2 +2 +v(s) +� +ds +for t ≥ 0. Since the right-hand side of the above equation is a Lipschitz continuous function +of (u, v), the classical contraction mapping argument gives us a local existence of (uµ, vµ) on +[0, δ). Let [0, Tµ) be the maximal interval of existence for (uµ, vµ). +Clearly, if we define uµ(t) := vµ(−t) and vµ(t) := uµ(−t) for t < 0, we have (uµ, vµ) is +a solution on (−Tµ, Tµ). Suppose that Tµ < +∞. Then we have |uµ(t)| + |vµ(t)| → +∞ as +|t| → Tµ. +Let us denote +Hµ(t) = H(t, uµ(t), vµ(t)), +t ∈ (−Tµ, Tµ). +A simple computation implies +d +dtHµ(t) = d +dt +� +cosh(t)− +1 +m−1 +�m − 1 +2m (u2 +µ + v2 +µ) +m +m−1 ≤ 0, +∀t ≥ 0 +so that the energy Hµ is non-increasing along the solution (uµ, vµ), on [0, Tµ). However, since +we have |uµ(t)| + |vµ(t)| → +∞ as t → Tµ, we find +Hµ(t) ≥ −m − 2 +2 +uµ(t)vµ(t) + m − 1 +2m +cosh(Tµ)− +1 +m−1(uµ(t)2 + vµ(t)2) +m +m−1 → +∞ +as t → Tµ, which is absurd. Hence we have uµ and vµ are globally defined on R. +In what follows, we state some basic properties for solutions of (3.11). +Lemma 4.8. Given µ > 0, then the following holds: +• If, for some t0 ̸= 0, we have uµ(t0) = 0, then vµ(t0) ̸= 0 and u′ +µ(t0) ̸= 0. + +26 +• If, for some t0 > 0, we have vµ(t0) = 0, then uµ(t0) ̸= 0 and v′ +µ(t0) ̸= 0. +Moreover, both uµ and vµ can not change sign infinitely many times in a bounded interval +[−T, T]. +Proof. Observe that the only rest point of system (3.11) is (0, 0). Furthermore, for t0 ̸= 0, the +Cauchy problem for (3.11) is locally well-posed for any initial datum (u(t0), v(t0)) ∈ R2, for +both t > t0 and t < t0. Thus, a rest point cannot be reached in a finite time. +In order to see that both uµ and vµ can only change sign a finite number of times in a bounded +interval [−T, T], we assume by contradiction that there exists {tu +j } and {tv +j} in [−T, T] such that +tu +j → Tu and tv +j → Tv as j → ∞, uµ(tu +j ) = vµ(tv +j) = 0 for all j, and uµ (resp. vµ) changes sign +a finite number of times on [−|Tu| + δ, |Tu| − δ] (resp. [−|Tv| + δ, |Tv| − δ]) for any δ > 0. +If |Tu| < |Tv|, then vµ will not change sign in a left neighborhood of |Tu| and in a right +neighborhood of −|Tu|. Then the first equation in (3.11) implies that u′ +µ(tu +j ) has the same sign +as vµ, which is impossible. Hence |Tu| ≥ |Tv|. Similarly, one obtains |Tv| ≥ |Tu|. Therefore +|Tu| = |Tv|. Moreover, it can not happen that Tu = −Tv while uµ (resp. vµ) keeps a definite +sign around Tv (resp. Tu). Therefore, we must have Tu = Tv = T0. In particular, we have +uµ(T0) = vµ(T0) = 0, which is also impossible. +Lemma 4.9. Given µ > 0. If (uµ, vµ) is a bounded solution, i.e., |uµ(t)| + |vµ(t)| ≤ M for all +t ∈ R and some M > 0, then (uµ, vµ) → (0, 0) as |t| → +∞. +Proof. By symmetry, we only need to prove the result for t → +∞. Multiplying by uµ (resp. +vµ) the equations in (3.11), we have +� +� +� +� +� +uu′ = cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +1 +m−1uµvµ − m − 2 +2 +u2 +µ, +−vv′ = cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +1 +m−1uµvµ − m − 2 +2 +v2 +µ. +Thus we need to show that uµ(t)2 + vµ(t)2 → 0 as t → +∞. +Suppose by contradiction that, for arbitrary small ε > 0, there exists t0 > 0 large such that +cosh(t0)− +1 +m−1M +m +m−1 ≤ 2ε +and +uµ(t0)2 + vµ(t0)2 ≥ 2δ0, +for some δ0 > 0. Since +1 +2(u2 +µ)′ ≤ ε − m − 2 +2 +u2 +µ, +we find +uµ(t)2 ≤ +2ε +m − 2 − +2ε +m − 2e(m−2)(t0−t) + uµ(t0)2e(m−2)(t0−t). +Therefore, by enlarging t0, we can assume without loss of generality that vµ(t0)2 > δ0. And +hence, we obtain +−1 +2(v2 +µ)′ ≤ ε − m − 2 +2 +v2 +µ, +which implies +vµ(t)2 ≥ +2ε +m − 2 − +2ε +m − 2e(m−2)(t−t0) + vµ(t0)2e(m−2)(t−t0). +By taking ε < m−2 +2 δ0, we have vµ(t)2 → +∞ as t → +∞. This contradicts the boundedness +of vµ. + +27 +Remark 4.10. From the above result, we can conclude that, if there exists t0 > 0 such that +Hµ(t0) ≤ 0, the corresponding solution (uµ, vµ) must be unbounded as t → ±∞ (since the +energy Hµ(t) = H(t, uµ(t), vµ(t)) is decreasing). +Lemma 4.11. Let µ > 0. If (uµ, vµ) is a solution such that lim|t|→+∞ Hµ(t) ∈ [−∞, 0). Then +uµ(t)2 + vµ(t)2 = O(cosh(t)) as |t| → +∞. +Proof. Since Hµ(t) is decreasing, we can take t0 > 0 such that Hµ(t0) ≤ 0 and +0 ≥ Hµ(t) ≥ m − 1 +2m +cosh(t)− +1 +m−1(uµ(t)2 + vµ(t)2) +m +m−1 − m − 2 +4 +(uµ(t)2 + vµ(t)2) +for all t ≥ t0 Notice that uµ(t)2 + vµ(t)2 can not reach 0 in a finite time, we soon have +uµ(t)2 + vµ(t)2 ≤ cm cosh(t) +for all t ≥ t0 and cm > 0 depends only on m. +Lemma 4.12. Let (uµ, vµ) be a solution of (3.11) such that vµ changes sign a finite number of +times on R, then there exists T > 0 such that uµ(t)vµ(t) > 0 for all |t| ≥ T. +Proof. Since vµ changes sign a finite number of times on R, we suppose without loss of gener- +ality that vµ(t) > 0 for all t ≥ T1, some T1 > 0. +Assume, by contradiction, that uµ(t) < 0 for all t > T1. Then the second equation of (3.11) +implies that v′ +µ(t) > 0 for t > T1, that is, vµ(t) is increasing for t > T1. Hence we have +lim +t→+∞ vµ(t) = v∞ ∈ (0, +∞]. +Notice that, by the second equation again, we have +v′ ≥ m − 2 +2 +v +for t ≥ T1. +We deduce that +vµ(t) ≥ vµ(T1)e +m−2 +2 +(t−T1) +for t ≥ T1. +Hence v∞ = +∞. However, since uµ and vµ have opposite sign, we find +Hµ(t) ≥ m − 1 +2m +cosh(t)− +1 +m−1(uµ(t)2 + vµ(t)2) +m +m−1 > m − 1 +2m +cosh(t)− +1 +m−1vµ(t) +2m +m−1 → +∞ +as t → +∞, which is impossible. +Let t0 ≥ T1 be such that uµ(t0) = 0. Then, it follow from the first equation of (3.11) that +u′ +µ(t0) > 0. If there exists ˆt0 > t0 such that uµ(ˆt0) = 0 and uµ(t) > 0 on (t0, ˆt0), we soon derive +that u′ +µ(t) < 0 in a left neighborhood of ˆt0. Thus, by the the first equation of (3.11) again, we +get vµ(ˆt0) ≤ 0. This is impossible since we have assumed vµ(t) > 0 for all t > T1. Therefore, +by taking T > t0, we conclude uµ(t) > 0 for all t ≥ T. +Corollary 4.13. Let (uµ, vµ) be a solution of (3.11) such that uµ changes sign a finite number +of times on R, then there exists T > 0 such that uµ(t)vµ(t) > 0 for all |t| ≥ T. + +28 +Proof. Suppose that we have uµ(t) > 0 for all t ≥ T, some T > 0. By Lemma 4.8 and 4.12, +we can not have vµ(t) < 0 for all t > T. +Suppose that there exists t0 > T1 such that vµ(t0) = 0. Then v′ +µ(t0) < 0 and vµ enters to +negative values, and can not have further zeros. In fact, if there is ˆt0 > t0 such that vµ(ˆt0) = 0 +and vµ(t) < 0 on (t0, ˆt0). We will have v′ +µ(ˆt0) ≥ 0, which is impossible. Then we obtain a +contradiction with Lemma 4.12. +Corollary 4.14. Let (uµ, vµ) be a bounded solution of (3.11) such that vµ (or uµ) changes sign +a finite number of times on R, then +uµ(t)2 + vµ(t)2 = O(e−(m−2)t) +as |t| → +∞. +Proof. By virtue of Lemma 4.12 and Corollary 4.13, we can take T > 1 large enough such that +uµ(t)vµ(t) > 0 for all t ≥ T. Then, it can be derived from (3.11) that +−(u2 +µ + v2 +µ)′′ + (m − 2)2(u2 +µ + v2 +µ) = 4(m − 2) cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +1 +m−1uµvµ. +Hence, from the boundedness of uµ and vµ, we have +� +− (u2 +µ + v2 +µ)′′ + (m − 2)2(u2 +µ + v2 +µ) > 0 +− (u2 +µ + v2 +µ)′′ + (m − 2)2(u2 +µ + v2 +µ) ≤ δe− +1 +m−1 t(u2 +µ + v2 +µ) +(4.9) +for t sufficiently large, where δ > 0 is a constant. +Let Γ1(t) = e−(m−2)t and Γ2(t) = arctan(t)e−(m−2)t, for t > 0. One checks easily that +−Γ′′ +1 + (m − 2)2Γ1 = 0 +and +− Γ′′ +2 + (m − 2)2Γ2 ≥ 2(m − 2) +1 + t2 +e−(m−2)t. +By taking C1, C2 > 0 such that +C1Γ1(T0) ≤ uµ(T0)2 + vµ(T0)2 ≤ C2Γ2(T0), +for some T0 > T, we find +� +� +� +− (u2 +µ + v2 +µ − C1Γ1)′′ + (m − 2)2(u2 +µ + v2 +µ − C1Γ1) > 0, +− (u2 +µ + v2 +µ − C2Γ2)′′ + +� +(m − 2)2 − +2(m − 2) +(1 + t2) arctan(t) +� +(u2 +µ + v2 +µ − C2Γ2) < 0, +for all t > T0. Then, by the comparison principle, we have +C1Γ1(t) ≤ uµ(t)2 + vµ(t)2 ≤ C2Γ2(t), +for all t > T0, which completes the proof. +Lemma 4.15. Let (uµ, vµ) be a solution of (3.11) such that vµ changes sign a finite number of +times on R. If Hµ(t) = H(t, uµ(t), vµ(t)) > 0 for all t > 0, then Hµ(t) ≤ Ce−c|t| as t → ±∞, +for some constants C, c > 0 possibly depending on µ. + +29 +Proof. We only prove the result for t → +∞. Note that +d +dtHµ(t) = d +dt +� +cosh(t)− +1 +m−1 +�m − 1 +2m (u2 +µ + v2 +µ) +m +m−1 += − 1 +2m cosh(t)− +1 +m−1 et − e−t +et + e−t (u2 +µ + v2 +µ) +m +m−1 +≤ −1 − δ +2m cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +m +m−1 +≤ − 1 − δ +m − 1Hµ(t), +for t ≥ Tδ, +where δ > 0 can be fixed arbitrarily small and the last inequality comes from Lemma 4.12. +Therefore, we have +Hµ(t) ≤ Hµ(Tδ)e− 1−δ +m−1 t +for all t ≥ Tδ, which completes the proof. +Now, for µ > 0 and (uµ, vµ) the corresponding solution of (3.11), we introduce the sets Ak, +Bk and Ik defined for k ∈ N ∪ {0} by +Ak = +� +µ > 0 : vµ changes sign k times on (0, +∞) and +lim +|t|→+∞ Hµ(t) < 0 +� +, +Bk = +� +µ > 0 : vµ changes sign k times on (0, +∞), Hµ(t) > 0 and (uµ, vµ) is unbounded +� +, +Ik = +� +µ > 0 : vµ changes sign k times on (0, +∞), Hµ(t) > 0 and (uµ, vµ) is bounded +� +. +Notice that (0, 0) is a hyperbolic equilibrium point of the Hamiltonian energy H(t, ·, ·) for any +t ∈ R. It is, then, immediate to see that A0 ̸= ∅ as it includes the interval (0, +√ +2 +2 ], since +H(0, µ, µ) < 0 +for all µ ∈ +� +0, +√ +2 +2 +� +. +As we will see later, tracking the sign changes of the solutions is crucial for the proof of Theo- +rem 1.5. The main idea is to study the stratified structure of the solutions. This will be done by +checking their topology and boundedness. The boundedness, allows us to track the sup of Ak +and Ik allowing us to prove that all the sets Ak are not empty. As we will see below, the idea of +tracking the signs coming from a limiting problem with explicit solutions and infinitely many +sign changes. This property will allow us to prove boundedness of the desired sets. +Let us start first by discarding the sets Bk: +Lemma 4.16. Bk = ∅ for all k ∈ N ∪ {0}. +Proof. Suppose to the contrary that Bk ̸= ∅ for some k. Let µ ∈ Bk and (uµ, vµ) be the +corresponding solution. Then, by substituting (uµ, vµ) into Eq. (3.11), we obtain +� +� +� +� +� +u′ +µvµ = cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +1 +m−1v2 +µ − m − 2 +2 +uµvµ, +−uµv′ +µ = cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +1 +m−1u2 +µ − m − 2 +2 +uµvµ. +(4.10) + +30 +This gives +u′ +µvµ − uµv′ +µ = cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +m +m−1 − (m − 2)uµvµ += +2m +m − 1Hµ(t) + m − 2 +m − 1uµvµ > m − 2 +m − 1uµvµ, +for all t. By Lemma 4.12, for t large enough, we can divide the above inequality by uµvµ to get +(ln uµ − ln vµ)′ > m − 2 +m − 1, +where we have assumed without loss of generality that uµ(t) > 0 and vµ(t) > 0 for t large. +Hence we have +uµ(t) +vµ(t) ≥ Ce +m−2 +m−1 t +(4.11) +for some constant C > 0. And therefore, there exists T > 0 such that uµ(t) > vµ(t) for all +t > T. Now, by (4.10), we have +u′ +µvµ + uµv′ +µ = cosh(t)− +1 +m−1(u2 +µ + v2 +µ) +1 +m−1(v2 +µ − u2 +µ) < 0 +for t > T, that is, uµvµ is decreasing for all large t. +Assume that uµ(t)vµ(t) → a∞ ∈ [0, +∞) as t → ∞. By Lemma 4.12 and 4.15, we have +m − 1 +2m +cosh(t)− +1 +m−1(uµ(t)2 + vµ(t)2) +m +m−1 → m − 2 +2 +a∞ +as t → ∞. Therefore, for arbitrary small ε > 0, there exists Tε > 0 such that +� +� +� +� +� +u′ +µ ≤ ε − m − 2 +2 +uµ +−v′ +µ ≤ ε − m − 2 +2 +vµ +for all t ≥ Tε. This implies +uµ(t) ≤ +2ε +m − 2 − +2ε +m − 2e +m−2 +2 +(Tε−t) + uµ(Tε)e +m−2 +2 +(Tε−t) +and +vµ(t) ≥ +2ε +m − 2 − +2ε +m − 2e +m−2 +2 +(t−Tε) + vµ(Tε)e +m−2 +2 +(t−Tε) +for all t ≥ Tε. Since µ ∈ Bk, we have |uµ(t)| + |vµ(t)| is unbounded as |t| → +∞. Hence, by +fixing ε > 0 suitably small, we find +vµ(t) ∼ e +m−2 +2 +t +and +uµ(t) → 0 +as t → +∞, this contradicts (4.11). +Lemma 4.17. There exists constants C0 > 0 such that, if for some T > 1, +(1) Hµ(T) ≤ C0; + +31 +(2) uµ(T)vµ(T) > 0; +(3) vµ changes sign k times on [0, T]; +then µ ∈ Ak ∪ Ik ∪ Ak+1. +Proof. Suppose that µ ̸∈ Ak ∪ Ik, it remains to show that µ ∈ Ak+1. Without loss of generality, +let us assume that uµ(T) > 0 and vµ(T) > 0. Set +�T = inf +� +t > T : uµ(t) ≤ 0 +� +∈ (T, +∞]. +If �T = +∞, we have vµ changes sign at most once in (T, +∞). Indeed, as long as uµ > 0, +the second equation of (3.11) implies that v′ +µ < 0 whenever vµ vanishes. Therefore, vµ can not +change sign more than once. If vµ does not change sign on (T, +∞), we have µ ∈ Ak ∪ Ik, +which is absurd. However, if vµ does change sign once in (T, +∞), we have uµ(t)vµ(t) < 0 for +all large t. This contradicts Lemma 4.12. Therefore, we have �T < +∞ and uµ( �T) = 0. +Claim 1. vµ changes sign exactly once in (T, �T). +In fact, by rewriting the second equation of (3.11), we have +� +vµ(t)e− m−2 +2 +t�′ += − cosh(t)− +1 +m−1(uµ(t)2 + vµ(t)2) +1 +m−1uµ(t)e− m−2 +2 +t < 0 +for t ∈ (T, �T). If vµ stays positive on (T, �T), by Lemma 4.8, we have u′ +µ ≥ 0 on a left neigh- +borhood of �T, which is impossible. +To proceed, let us set fµ = (uµ − vµ)/ +√ +2 and gµ = (uµ + vµ)/ +√ +2. Then (fµ, gµ) satisfies +the following system +� +� +� +� +� +f ′ = cosh(t)− +1 +m−1(f 2 + g2) +1 +m−1g − m − 2 +2 +g, +−g′ = cosh(t)− +1 +m−1(f 2 + g2) +1 +m−1f + m − 2 +2 +f, +(4.12) +with Hamiltonian energy +�H(t, f, g) = m − 2 +4 +f 2 − m − 2 +4 +g2 + m − 1 +2m +cosh(t)− +1 +m−1(f 2 + g2) +m +m−1. +Clearly, we have Hµ(t) = �H(t, fµ, gµ) for t ∈ R. And, by Claim 1, we can make T slightly +larger so that uµ > vµ on [T, �T]. That is, we have fµ > 0 on [T, �T], gµ(T) > 0, gµ( �T) < 0 and +gµ changes sign once in (T, �T). +In what follows, we are going to prove that fµ stays positive on [T, +∞). Then the second +equation in (4.12) shows that g′ +µ < 0 for all t ≥ T. And hence µ ̸∈ Ij for any j ∈ N ∪ {0}. +In this case, we have fµ(t) > 0 and gµ(t) < 0 for all t ≥ �T, which implies vµ(t) < 0 for +t ∈ [ �T, +∞). That is, vµ changes sign exactly once on (T, +∞). Therefore µ ∈ Ak+1. +Suppose, by contradiction, that there exists �T > �T such that fµ( �T) = 0 and fµ > 0 on +[T, �T). Then, the second equation in (4.12) implies that gµ is decreasing on [T, �T]. And hence, + +32 +gµ( �T) < gµ( �T) < 0. Then, we only need to consider the situation Hµ( �T) > 0, since the +condition Hµ( �T) ≤ 0 will immediately trap the solution (uµ, vµ) in the third quadrant of (u, v)- +plane for t > �T, and leads us to have µ ∈ Ak+1. +In the case Hµ( �T) > 0, by fµ( �T) = 0 and gµ( �T) < 0, we have +gµ( �T) < − +�m(m − 2) +2(m − 1) +� m−1 +2 +cosh( �T) +1 +2. +Let T < T1 < T2 < �T be such that +m − 1 +2m +cosh( �T)− +1 +m−1gµ(T1) +2m +m−1 − m − 2 +4 +gµ(T1)2 = −C0 +and +m − 1 +2m +cosh( �T)− +1 +m−1gµ(T2) +2m +m−1 − m − 2 +4 +gµ(T2)2 = 0. +By assuming C0 suitably small, such T1 and T2 always exist, and we can have that gµ( �T) < +gµ(T2) < gµ(T1) < gµ(T2)/2 < 0. In fact, by setting +F(s) = m − 1 +2m +cosh( �T)− +1 +m−1|s| +2m +m−1 − m − 2 +4 +|s|2, +s ∈ R +we have gµ(T2) is nothing but the vanishing point of F in the negative line, i.e., +gµ(T2) = − +�m(m − 2) +2(m − 1) +� m−1 +2 +cosh( �T) +1 +2, +(4.13) +and gµ(T1) is the smallest point such that F = −C0. Then, use the fact Hµ(t) ≤ C0 for all +t > T, we have +m − 2 +4 +fµ(t)2 ≤ C0 − F(gµ(t)) ≤ 2C0 +for t ∈ [T1, T2]. Hence, we deduce +0 < fµ(t) ≤ δ0 := +� +8C0 +m − 2 +(4.14) +for t ∈ [T1, T2]. Notice that +F ′(gµ(T2)) = − +1 +m − 1 +� +m +m − 1 +� m−1 +2 �m − 2 +2 +� m+1 +2 +cosh( �T) +1 +2 < 0 +and +F ′′(gµ(T2)) = m − 2 +2 +�m(m + 1) +(m − 1)2 − 1 +� +> 0. +By using the second equation in (4.12) and (4.14), we find +C0 +F ′(gµ(T2)) > gµ(T2) − gµ(T1) = +� T2 +T1 +g′ +µ(t)dt +≥ − +� T2 +T1 +�� +δ2 +0 + gµ(T2)2� +1 +m−1δ0 + m − 2 +2 +δ0 +� +dt +≥ −Cmgµ(T2) +2 +m−1δ0(T2 − T1) +(4.15) + +33 +where Cm > 0 depends only on m (since we have assumed C0 is small). On the other hand, we +have +d +dtHµ(t) = − 1 +2m cosh(t)− +1 +m−1 et − e−t +et + e−t (fµ(t)2 + gµ(t)2) +m +m−1 +≤ − 1 +2m +e − e−1 +e + e−1 cosh( �T)− +1 +m−1gµ(T1) +2m +m−1 +≤ −cm cosh( �T)− +1 +m−1gµ(T2) +2m +m−1 +for t ∈ [T1, T2], where in the last inequality we used |gµ(T1)| > 1 +2|gµ(T2)| and +cm = +1 +2m +�1 +2 +� 2m +m−1 e − e−1 +e + e−1. +Hence, by (4.15), we obtain +Hµ(T2) − Hµ(T1) = +� T2 +T1 +d +dtHµ(t)dt ≤ −cm cosh( �T)− +1 +m−1gµ(T2) +2m +m−1(T2 − T1) +≤ cm cosh( �T)− +1 +m−1gµ(T2) +2m +m−1C0 +CmF ′(gµ(T2))gµ(T2) +2 +m−1δ0 += − �Cm cosh( �T) +1 +2 − +1 +m−1� +C0 < −C0 +provided that m ≥ 3 and C0 is small enough. This implies Hµ(T2) ≤ 0 reaching a contradiction, +and the proof is hereby completed. +The next lemma provides the main properties of the sets Ak and Ik. +Lemma 4.18. For all k ∈ N ∪ {0}, we have +(1) Ak is an open set; +(2) if µ ∈ Ik, then there exists ε > 0 such that (µ − ε, µ + ε) ⊂ Ak ∪ Ik ∪ Ak+1; +(3) if Ak ̸= ∅ and is bounded, then sup Ak ∈ Ik; +(4) if both Ak and Ik are bounded, set µ = sup Ik, then there exists ε > 0 such that (µ, µ + +ε) ⊂ Ak+1. +Proof. (1) is quite obvious, since it comes from the continuity of the solutions (uµ, vµ) with +respect to the initial datum. +To see (2), we fix µ ∈ Ik. Then we have Hµ(t) → 0 as |t| → +∞. Given C0 as in Lemma +4.17, there exists T > 1 such that Hµ(T) < C0, uµ(T)vµ(T) > 0 and vµ changes sign k times +on [0, T]. The continuity of the solution (uµ, vµ) with respect to µ implies that the same holds +for an initial datum ˜µ ∈ (µ − ε, µ + ε) for ε > 0 small. Then the conclusion follows by Lemma +4.17. +To check (3), let us set µ = sup Ak and take a sequence {µj} ⊂ Ak such that µj ↗ µ as +j → +∞. If we suppose that µ ∈ Al for some l, then (1) suggests that µj ∈ Al for j large. +Hence we have l = k. This implies µ ∈ Ak which is absurd since Ak is an open set. Notice that, +by the continuity property of the solutions, the corresponding vµ can change sign only a finite + +34 +number of times on (0, +∞). Therefore we must have that µ ∈ Is for some s. By (2), we have +(µ − ε, µ + ε) ⊂ As ∪ Is ∪ As+1. This implies s = k. +Finally, to see (4), we first observe that µ = sup Ik ∈ Ik. Indeed, let {µj} ∈ Ik be such +that µj ↗ µ as j → +∞, we have µ ̸∈ Al for any l ∈ N ∪ {0}. This is because Al is an open +set. Then, arguing similarly as in (3), we get that µ ∈ Ik as claimed. Now, by (2), we have +(µ, µ + ε) ⊂ Ak ∪ Ak+1 for some ε > 0. Since we have assumed the boundedness of Ak, we +find sup Ak ≤ µ. Thus (µ, µ + ε) ⊂ Ak+1. +Our next result is the boundedness property of the sets Ak and Ik. +Proposition 4.19. Ak ∪ Ik is bounded for each k ∈ N ∪ {0}. +Before prove Proposition 4.19, let us do some preparations. Denoted by ε = µ−1 > 0, we +consider the following rescaling +� +Uε(t) = εuµ +� +ε +2 +m−1t +� +, +Vε(t) = εvµ +� +ε +2 +m−1t +� +. +We find the system for (Uε, Vε) is +� +� +� +� +� +U ′ +ε = cosh +� +ε +2 +m−1t +�− +1 +m−1(U 2 +ε + V 2 +ε ) +1 +m−1Vε − ε +2 +m−1 m − 2 +2 +Uε +−V ′ +ε = cosh +� +ε +2 +m−1t +�− +1 +m−1(U 2 +ε + V 2 +ε ) +1 +m−1Uε − ε +2 +m−1 m − 2 +2 +Vε +(4.16) +together with the initial datum Uε(0) = Vε(0) = 1. The limiting problem associated to Eq. (4.16) +is +� +U ′ +0 = (U 2 +0 + V 2 +0 ) +1 +m−1V0 +−V ′ +0 = (U 2 +0 + V 2 +0 ) +1 +m−1U0 +(4.17) +with U0(0) = V0(0) = 1. +Lemma 4.20. There holds +(Uε, Vε) → (U0, V0) +as ε → 0 +uniformly on [0, T], for all T > 0, where (U0, V0) is the solution to Eq. (4.17). +Proof. First of all, we have (4.16) is equivalent to +� +� +� +� +� +� +� +Uε(t) = 1 + +� t +0 +� +cosh +� +ε +2 +m−1s +�− +1 +m−1(U 2 +ε + V 2 +ε ) +1 +m−1Vε − ε +2 +m−1 m − 2 +2 +Uε +� +ds +Vε(t) = 1 − +� t +0 +� +cosh +� +ε +2 +m−1s +�− +1 +m−1(U 2 +ε + V 2 +ε ) +1 +m−1Uε − ε +2 +m−1 m − 2 +2 +Vε +� +ds +(4.18) +and, similarly, (4.17) is equivalent to +� +� +� +� +� +� +� +U0(t) = 1 + +� t +0 +(U 2 +0 + V 2 +0 ) +1 +m−1V0 ds, +V0(t) = 1 − +� t +0 +(U 2 +0 + V 2 +0 ) +1 +m−1U0 ds. +(4.19) + +35 +The Hamiltonian energy associated to (4.16) is given by +Hε(t, U, V ) = −ε +2 +m−1 m − 2 +2 +UV + m − 1 +2m +cosh +� +ε +2 +m−1t +�− +1 +m−1(U 2 + V 2) +m +m−1. +And it is easy to see that Hε is decreasing along the flow, so that +Hε(t, Uε(t), Vε(t)) ≤ Hε(0, 1, 1) < m − 2 +2m 2 +m +m−1. +This implies that +Uε(t)2 + Vε(t)2 ≤ Cm cosh +� +ε +2 +m−1t +� +(4.20) +for some constant Cm > 0 independent of ε. +Fix T > 0 and consider t ∈ [0, T], we have +|Uε(t) − U0(t)| + |Vε(t) − V0(t)| +≤ +� t +0 +cosh +� +ε +2 +m−1t +�− +1 +m−1 +���(U 2 +ε + V 2 +ε ) +1 +m−1Vε − (U 2 +0 + V 2 +0 ) +1 +m−1V0 +���ds ++ +� t +0 +cosh +� +ε +2 +m−1t +�− +1 +m−1 +���(U 2 +ε + V 2 +ε ) +1 +m−1Uε − (U 2 +0 + V 2 +0 ) +1 +m−1U0 +���ds ++ +� t +0 +� +1 − cosh +� +ε +2 +m−1t +�− +1 +m−1� +(U 2 +0 + V 2 +0 ) +1 +m−1� +|U0| + |V0| +� +ds ++ Cmε +2 +m−1 cosh +� +ε +2 +m−1T +� 1 +2. +(4.21) +Since the first two integrands in the right-hand-side of (4.21) are locally Lipschitz, by (4.20) +and the boundedness of U0 and V0, we have +|Uε(t) − U0(t)| + |Vε(t) − V0(t)| ≲ +� t +0 +� +|Uε − U0| + |Vε − V0| +� +ds + ε +2 +m−1 cosh +� +ε +2 +m−1T +� 1 +2. +Now, using the Gronwall inequality, we have +|Uε(t) − U0(t)| + |Vε(t) − V0(t)| ≲ ε +2 +m−1 +for t ∈ [0, T], proving the lemma. +Proof of Proposition 4.19. Suppose the contrary, that Ak ∪ Ik is unbounded for some k. Then +we can find a sequence µj ∈ Ak ∪ Ik such that µj → +∞ as j → +∞. +By taking εj = µ−1 +j , Lemma 4.20 implies that Vεj → V0 uniformly on [0, T] as j → ∞, for +any fixed T > 0. Notice that the solution (U0, V0) of Eq. (4.17) can be explicitly formulated: +U0(t) = +√ +2 sin +� +2 +1 +m−1t + π +4 +� +and +V0(t) = +√ +2 cos +� +2 +1 +m−1t + π +4 +� +. +We can take T > 0 large enough so that V0 changes sign k +1 times on [0, T]. Then, by Lemma +4.20, we have Vεj changes k + 1 times on [0, T] for all large j. However, due to µj ∈ Ak ∪ Ik +and Vεj(t) = εjvµj +� +ε2/(m−1) +j +t +� +, we have Vεj should change sign only k times on (0, +∞). And +thus, we get a contradiction. + +36 +Proof of Theorem 1.5. Let µ0 = sup A0. By Lemma 4.18, we have µ0 ∈ I0. Let now ν0 = +sup I0. Applying Proposition 4.19 and Lemma 4.18, we have (ν0, ν0 + ε0) ⊂ A1 for some +ε0 > 0. Thus A1 ̸= ∅. Let µ1 = sup A1. We have µ1 > ν0 ≥ µ0; and so, by Lemma 4.18, +µ1 ∈ I1, and then ν1 = sup I1 ∈ I1 and (ν1, ν1 + ε1) ⊂ A2, for some ε1 > 0. Iterating this +argument, we construct two increasing sequences {µj} and {νj}, νj+1 ≥ µj+1 > νj ≥ µj, with +µj ∈ Ij and (νj, νj + εj) ⊂ Aj+1, for some {εj} ⊂ (0, +∞). +Next, we will show that µj → +∞ as j → +∞. Suppose, by contradiction, that µj is +bounded and µj → µ∞. We can see that Hµ∞(t) > 0 for all t ∈ R. Indeed, if Hµ∞(t0) ≤ 0 +for some finite t0 > 0, it follows that (uµ∞(t), vµ∞(t)) will be trapped in one of the connected +components of {(u, v) ∈ R2 : H(t, u, v < 0)}, for all t > t0. Since Lemma 4.8 implies that vµ∞ +changes sign a finite number of times in [0, t0], we have µ∞ ∈ Ak0 for some k0. This contradicts +the definition of µ∞ as Ak0 is open. Moreover, vµ∞ must change sign infinite many times on +(0, +∞). +Using the facts Hµ∞ is decreasing on (0, +∞) and bounded from below, we have H′ +µ∞ ∈ +L1(0, +∞). In particular, +cosh(·)− +1 +m−1(u2 +µ∞ + v2 +µ∞) +m +m−1 ∈ L1(0, +∞). +(4.22) +Multiplying by vµ∞ (resp. uµ∞) the equations in (3.11), we have +� +� +� +� +� +vµ∞u′ +µ∞ = cosh(t)− +1 +m−1(u2 +µ∞ + v2 +µ∞) +1 +m−1v2 +µ∞ − m − 2 +2 +uµ∞vµ∞, +−uµ∞v′ +µ∞ = cosh(t)− +1 +m−1(u2 +µ∞ + v2 +µ∞) +1 +m−1u2 +µ∞ − m − 2 +2 +uµ∞vµ∞. +This implies +vµ∞u′ +µ∞ + uµ∞v′ +µ∞ = cosh(t)− +1 +m−1(u2 +µ∞ + v2 +µ∞) +1 +m−1(v2 +µ∞ − u2 +µ∞). +Hence we have (uµ∞vµ∞)′ ∈ L1(0, +∞), which shows that uµ∞(t)vµ∞(t) → C∞ ∈ R as +t → ∞. Since vµ∞(t) changes sign infinitely many times as t → ∞, we have C∞ = 0. This, +together with (4.22), implies that Hµ∞(t) → 0 as t → +∞. +Therefore, one may take T > 0 sufficiently large such that Hµ∞(T) < C0 (where C0 > 0 +is given by Lemma 4.17), uµ∞(T)vµ∞(T) > 0 and vµ∞ changes sign kT times on [0, T]. By +Lemma 4.17, we have µ∞ ∈ AkT ∪ IkT ∪ AkT +1, reaching another contradiction. +Finally, in order to see that lim inft→+∞ |uµ(t)| + |vµ(t)| = +∞ for µ ∈ Ak, let us consider +two possibilities: Hµ(t) → −∞ and Hµ(t) → H∞ ∈ (−∞, 0). In the first case, we must have +that uµ(t)vµ(t) → +∞ as t → +∞, which directly implies the assertion. In the latter case, we +deduce that uµ(t)vµ(t) → C > 0 as t → +∞. And hence cosh(·)− +1 +m−1(u2 +µ + v2 +µ) +m +m−1 converges +to a positive constant. This shows that |uµ(t)| + |vµ(t)| grows as cosh(t)1/2m for t large. +The upper bound of (uµ, vµ), µ ∈ Ak follows from Lemma 4.11, and the exponential decay +of (uµ, vµ), µ ∈ Ik, follows from Corollary 4.14. Thus, the proof of Theorem 1.5 is complete. +Remark 4.21. The numerical simulations performed on system (3.11) indicate the following. +For each k ∈ N∪{0}, starting from µ larger than some µ∗ +k ∈ Ak, the solution orbits will make a +circle around a particular point (in either the first quadrant or the third quadrant) before going to + +37 +infinity. As µ grows, the circle is becoming larger; and once the circle touches the origin, we will +have a homoclinic solution of (3.11), which implies µ ∈ Ik. The set Ik seems to have only one +point, and hence Ak are just open intervals. In particular, we conjecture that ∪k≥0Ik is simply +a countable set of discrete points. This is illustrated in the following Fig. 3, where numerical +experiments are performed on a 3-dimensional system. The first row shows the solution orbits +(uµ, vµ) on R with three different initial datum in A0, and specifically µ = 0.1, 0.6 and 0.7. The +second and third rows show the solutions with initial datum µ ∈ A1 and A2, respectively +Figure 3: Unbounded trajectories with initial datum µ ∈ Ak, k = 0, 1, 2. +Acknowledgements +Y.S. is partly supported by NSF grant DMS 2154219, ” Regularity vs singularity formation in +elliptic and parabolic equations”. +References +[1] A. Abbondandolo, J. Molina, Index estimates for strongly indefinite functionals, periodic +orbits and homoclinic solutions of first order Hamiltonian systems, Cal. Var. PDEs, 11 +(2000), 395-430. + +2.0 H +3.5 +3F +3.0 +1.5 +2.5 +2 +2.0 F +1.0 +1.5F +1.0 +0.5 +9.5 +2 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +0.5 +1.0 +1.5 +2.0 +2卜 +-3 +- +-1 +-1 +-3 +3 +8 +ef +-2 +2 +-2 +2 +8 +438 +[2] B. Ammann, A variational problem in conformal spin geometry, Habilitationsschrift, Uni- +versit¨at Hamburg 2003. +[3] B. Ammann, The smallest Dirac eigenvalue in a spin-conformal class and cmc immer- +sions, Comm. Anal. Geom. 17 (2009), no. 3, 429-479. +[4] C. 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Steklov. 225 (1999), 339-361; English transl., Proc. +Steklov Inst. Math. 225 (1999), 225-243. +[44] K. Tanaka, Homoclinic orbits in a first order superquadratic Hamiltonian system: conver- +gence of subharmonic orbits, J. Diff. Equ. 94 (1991), 315-339. +[45] M. Wakano, Intensely localized solutions of the classical Dirac-Maxwell field equations, +Progr. Theoret. Phys. 35:6 (1966), 1117-1141. + +41 +ALI MAALAOUI +DEPARTMENT OF MATHEMATICS, +CLARK UNIVERSITY, +WORCESTER, MA 01610-1477 +amaalaoui@clarku.edu +YANNICK SIRE +DEPARTMENT OF MATHEMATICS, JOHNS HOPKINS UNIVERSITY, +3400 N. CHARLES STREET, BALTIMORE, MARYLAND 21218 +ysire1@jhu.edu +TIAN XU +CENTER FOR APPLIED MATHEMATICS, TIANJIN UNIVERSITY, +300072, TIANJIN, CHINA +xutian@amss.ac.cn + diff --git a/7NE2T4oBgHgl3EQfPQZw/content/tmp_files/load_file.txt b/7NE2T4oBgHgl3EQfPQZw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9939647d4ec000dd120fd684367cc179d86749a8 --- /dev/null +++ b/7NE2T4oBgHgl3EQfPQZw/content/tmp_files/load_file.txt @@ -0,0 +1,1521 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf,len=1520 +page_content='Constructions of Delaunay-type solutions for the spinorial Yamabe equation on spheres Ali Maalaoui Yannick Sire Tian Xu Abstract In this paper we construct singular solutions to the critical Dirac equation on spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' More precisely, first we construct solutions admitting two points singularities that we call Delaunay-type solutions because of their similarities with the Delaunay solutions con- structed for the singular Yamabe problem in [32, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we construct another kind of singular solutions admitting a great circle as a singular set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' These solutions are the building blocks for singular solutions on a general Spin manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Spinorial Yamabe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Singular Solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Delaunay-type Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Contents 1 Introduction and statement of the main result 2 2 Geometric preliminaries 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1 General preliminaries about spin geometry .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2 Spinor bundle and the Dirac operator on product manifolds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 10 3 Set up of the problems 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1 The singular set is a pair of antipodal points .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2 The dissipative case: Shooting method .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 24 Mathematics Subject Classification (2010): Primary 53C27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Secondary 35R01 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='03757v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='AP] 10 Jan 2023 2 1 Introduction and statement of the main result Since the resolution of the Yamabe problem, much has been clarified about the behavior of solutions of the semilinear elliptic equation relating the scalar curvature functions of two con- formally related metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' One of the starting points for several recent developments was R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Schoen’s construction of complete metrics with constant positive scalar curvature on the sphere Sm, conformal to the standard round metric, and with prescribed isolated singularities (see [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In analytical terms, it is equivalent to seeking for a function u > 0 satisfying − ∆gSmu + m(m − 2) 4 u = m(m − 2) 4 u m+2 m−2 on Sm \\ Σ, m ≥ 3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) in the distributional sense with u singular at every point of Σ ⊂ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Here we denote by gSm the standard Riemannian metric on Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) and its counterpart on a general manifold (M, g) are known as the singular Yamabe problem, and has been extensively studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Just as the classical Yamabe problem in the com- pact setting, the questions concerning metrics of constant positive scalar curvature are consid- erably more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remarkable breakthroughs and geometrically appealing examples were obtained by Schoen and Yau [37] and Schoen [36] when the ambient manifold is the m-sphere Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The former established that if Sm \\ Σ admits a complete metric with scalar curvature bounded below by a positive constant, then the Hausdorff dimension of Σ is at most (m − 2)/2, and the latter constructed several examples of domains Sm \\Σ that admit complete conformally flat metrics with constant positive scalar curvature, including the case where Σ is any finite set with at least two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Subsequently, Mazzeo and Smale [34] and Mazzeo and Pacard [32,33] generalized the existence results, allowing Σ to be a disjoint union of submanifolds with di- mensions between 1 and (m − 2)/2 when the ambient manifold (M, g) is a general compact manifold with constant nonnegative scalar curvature, and between 0 and (m − 2)/2 in the case (M, g) = (Sm, gSm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the past two decades, it has been realized that the conformal Laplacian, namely the op- erator appearing as the linear part of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1), falls into a particular family of operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' These operators are called conformally covariant elliptic operators of order k and of bidegree ((m − k)/2, (m + k)/2), acting on manifolds (M, g) of dimension m > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Many important geometric operators are in this class, for instance, the conformal Laplacian, the Paneitz operator, the Dirac operator, see also [10, 13, 20] for more examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' All such operators share several analytical properties, in particular, they are associated to the non-compact embedding of Sobolev space Hk/2 �→ L2m/(m−k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And often, they have a central role in conformal geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let (M, g, σ) be an m-dimensional spin manifold, m ≥ 2, with a fixed Riemannian metric g and a fixed spin structure σ : PSpin(M) → PSO(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The Dirac operator Dg is defined in terms of a representation ρ : Spin(m) → Aut(Sm) of the spin group which is compatible with Clifford multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let S(M) := PSpin(M) ×ρ Sm be the associated bundle, which we call the spinor bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then the Dirac operator Dg acts on smooth sections of S(M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Dg : C∞(M, S(M)) → C∞(M, S(M)), is a first order conformally covariant operator of bidegree ((m − 1)/2, (m + 1)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We point out here that the spinor bundle S(M) has complex dimension 2[ m 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Analogously to the conformal Laplacian, where the scalar curvature is involved, the Dirac operator on a spin manifold has close relations with the mean curvature function associated to 3 conformal immersions of the universal covering into Euclidean spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This theory is referred as the spinorial Weierstraß representation, and we refer to [2,3,17,25–27,31,41–43] and refer- ences therein for more details in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In a similar way as in the Yamabe problem, the spinorial analogue of the Yamabe equation (related with a normalized positive constant mean curvature) reads as Dgψ = |ψ| 2 m−1 g ψ on (M, g) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) where | · |g stands for the induced hermitian metric on fibers of the spinor bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' One may also consider the equation with an opposite sign Dgψ = −|ψ| 2 m−1 g ψ on (M, g) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) which corresponds to negative constant mean curvature surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, since the spectrum of Dg is unbounded on both sides of R and is symmetric about the origin on many manifolds (say, for instance dim M ̸≡ 3(mod 4)), the two problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) are of the same struc- ture from analytical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Although conformally covariant operators share many properties, only few statements can be proven simultaneously for all of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Particularly, the behavior of solutions of the conformally invariant equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' From the analytic perspective, some of the conformally covariant operators are bounded from below (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' the Yamabe and the Paneitz operator), whereas others are not (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' the Dirac operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Some of them act on functions, while others on sections of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For the Dirac operators, additional structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' spin structure) is used for defining it, and hence, more attention needs to be payed on such an exceptional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In this paper we initiate an investigation into the singular solutions of the nonlinear Dirac equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) when the ambient manifold is Sm, which is perhaps the most geometrically ap- pealing instance of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' As was described earlier, for a given closed subset Σ ⊂ Sm, it is to find metrics g = |ψ|4/(m−1) gSm gSm which are complete on Sm \\ Σ and such that ψ satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) with (M, g) = (Sm \\ Σ, gSm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This is the singular spinorial Yamabe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us mention that, up until now, no existence examples have been known for the singular solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Our first main result is follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let Σ ⊂ Sm be a pair of antipodal points, for m ≥ 2, or an equatorial circle for m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' There is a one-parameter family Sm of spinors ψ solving the problem DgSmψ = |ψ| 2 m−1 gSm ψ on Sm \\ Σ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) such that g = |ψ| 4 m−1 gSm gSm is a complete metric on Sm \\ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, (1) if Σ is a pair of antipodal points, the family Sm is parameterized by µ ∈ [− (m−1)m 2m+1m , +∞)\\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (2) if Σ is an equatorial circle, the family Sm is parameterized by O = ∪k∈NOk, where each Ok ⊂ (0, +∞) is a bounded open set, Ok ∩ Oj = ∅ for k ̸= j and O is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us remark that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4), or more generally Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2), is invariant under several Lie group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For instance, the canonical action of S1 = {eiθ ∈ C : θ ∈ [0, 2π]} on spinors 4 keeps the equation invariant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' if ψ is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) then eiθψ is also a solution, for every fixed θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, for the case m ≡ 2, 3, 4(mod 8), the spinor bundle has a quaternionic structure which commutes with Clifford multiplication, see for instance the construction in [18, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7] or [28, Page 33, Table III].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In these cases, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) is invariant under the action of the unit quaternions S3 = {q = H : |q| = 1} on spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, in general, it is crucial to distinguish solutions of Dirac equations under various group actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For instance, these symmetries were exploited in [29] to construct families of solutions on the sphere and the S1 symmetry was used in [30] to exhibit also non-trivial solutions for the sub-critical Dirac equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thanks to our constructions, the solutions in the family Sm obtained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1 are distinguished via their parameterizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And if G is a group that keeps Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) invariant, our construction shows a larger family G × Sm of singular solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' As we will see in Section 3, via a conformal change of the metric gSm, problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) can be transformed to DgRmψ = |ψ| 2 m−1 gRm ψ on Rm \\ {0} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) when Σ is a pair of antipodal points and DgRm−1ψ = f(x) 1 m−1|ψ| 2 m−1 gRm−1ψ on Rm−1 \\ {0} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) when Σ is an equatorial circle, where f(x) = 2 1+|x|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' To obtain the results for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) in consistence with similar results for the classical Yamabe equation, a fundamental idea is to express the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) on the cylinder R×Sl, l = m−1 or m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By introducing the cylindrical coordinates (t, θ) ∈ R × Sl: t = − ln |x|, θ = x |x| for x ∈ Rl+1, one may be expecting that the ansatz ϕ(t, θ) = |x| l 2ψ(x) could turn Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) into a more manageable problem via a separation of variables process leading to a ”radial” solution ψ(x) = ψ(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This is the very case for many elliptic problems (with a corresponding change of the exponent on |x|), including the Yamabe equation, fractional Yamabe equation [12] and the Q-curvature problem [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, we point out that in the scalar case, there is a symmetrization process that behaves well with elliptic operators, reducing the problem to the study of an ODE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' But when dealing with differential operators acting on vector bundles (spinor bundle in our case), one does not have a general symmetrization process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In particular, even on the Euclidean spaces Rm, one cannot use the radial ansatz ψ = ψ(r), r = |x| for x ∈ Rm, to reduce a Dirac equation to an ODE system in terms of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that the spinorial Yamabe equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6)) contains 2[ m 2 ] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 2[ m−1 2 ]) un- known complex-functions, which is a considerably large number as m grows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Instead of blindly “guessing” a particular ansatz, our starting point is the spin structure, or more precisely the spin representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, we use the matrix representation of Clifford multiplication to con- struct a “nice” function space E(Rm) for spinor fields which is invariant under the action of the Dirac operator DgRm, see in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3 for the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We find that the space E(Rm) is of 5 particular interest from two perspectives (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1 below): First of all, when the dimen- sion m = 2, 3, 4, E(Rm) encapsulates several important and special formulations of spinors which are of interest to particle physicists when they study quantum electrodynamic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Many important physical simulations have been obtained by using these special spinors, see for instance [11,14,40,45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The second perspective is that, spinors in E(Rm) reduce the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) significantly in the sense that, for any dimension m ≥ 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) can be reduced to the following ODE systems of only two unknown functions � � � − f ′ 2 − m − 1 r f2 = (f 2 1 + f 2 2) 1 m−1f1 f ′ 1 = (f 2 1 + f 2 2) 1 m−1f2 for r > 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7) and � � � � � � � − f ′ 2 − m − 2 r f2 = � 2 1 + r2 � 1 m−1(f 2 1 + f 2 2) 1 m−1f1 f ′ 1 = � 2 1 + r2 � 1 m−1(f 2 1 + f 2 2) 1 m−1f2 for r > 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) where f1, f2 ∈ C1(0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' After using the Emden-Fowler change of variable r = e−t and writing f1(r) = −u(t)e m−1 2 t, f2(r) = v(t)e m−1 2 t in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7), we get a nondissipative Hamiltonian system of (u, v) � � � � � u′ + m − 1 2 u = (u2 + v2) 1 m−1v, −v′ + m − 1 2 v = (u2 + v2) 1 m−1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) And, by writing f1(r) = −u(t)e m−2 2 t and f2(r) = v(t)e m−2 2 t, we can transform (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) into � � � � � u′ + m − 2 2 u = cosh(t)− 1 m−1(u2 + v2) 1 m−1v −v′ + m − 2 2 v = cosh(t)− 1 m−1(u2 + v2) 1 m−1u (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10) which is a dissipative Hamiltonian system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us denote by H(u, v) = −m − 1 2 uv + m − 1 2m (u2 + v2) m m−1 the corresponding Hamiltonian energy for the systems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that H is constant along trajectories of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, the equilibrium points of H are (0, 0) and ± �(m − 1)(m−1)/2 2m/2 , (m − 1)(m−1)/2 2m/2 � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) where (0, 0) is a saddle point and the other two are center points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' then it follows easily that for µ ∈ [− (m−1)m 2m+1m , +∞) \\ {0} there is a periodic solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) at the level {H = µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We set D1 m for these periodic solutions, parameterized by their Hamiltonian energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We distinguish 6 a dichotomy within the set D1 m based on the sign of the Hamiltonian energy µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, D1 m = D1,+ m ∪ D1,− m , where D1,+ m := {(u, v) ∈ D1 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' H(u, v) > 0} and D1,− m := {(u, v) ∈ D1 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' H(u, v) < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We will call elements of D1,− m , positive Delaunay-type solutions and elements of D1,+ m , sign- changing Delaunay-type solutions for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This terminology is based on the similarities between D1,− m and the classical Delaunay solutions for the Yamabe problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We will clarify more these similarities along the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since any (u, v) ∈ D1 m will not reach the rest point (0, 0), we have u(t)2 + v(t)2 is bounded away from 0 for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Besides the above existence results, we have the following bifurcation phenomenon for the solutions (u, v) ∈ D1,− m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let m ≥ 2, the following facts hold for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9): (1) For every T > 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) has the constant 2T-periodic solutions ± �(m − 1)(m−1)/2 2m/2 , (m − 1)(m−1)/2 2m/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, for T ≤ √m−1 2 π, these are the only solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (2) Let T > √m−1 2 π and d ∈ N such that d √m−1 2 π < T ≤ (d+1) √m−1 2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) has d+1 inequivalent solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Particularly, these solutions are given by the constant solution and k periods of a solution (uT,k, vT,k) with fundamental period 2T/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3) The Hamiltonian energy H(uT,1, vT,1) ↗ 0 as T → +∞ and (uT,1, vT,1) is (locally) com- pact in the sense that (uT,1, vT,1) converges in C1 loc(R, R2) to the nontrivial homoclinic solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' That is, there exists t0 ∈ R such that (uT,1, vT,1) converges in C1 loc to (u0(· − t0), v0(· − t0)), where u0(t) = m(m−1)/2et/2 2m/2 cosh(t)m/2 and v0(t) = m(m−1)/2e−t/2 2m/2 cosh(t)m/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By translating the above results to system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7) (hence Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5)), we have Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let m ≥ 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) has a one-parameter family S1 m of singular solutions on Rm\\{0}, parameterized by [− (m−1)m 2m+1m , +∞)\\{0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, the following asymptotic estimates hold |ψ(x)| ̸= 0, |ψ(x)| = O(|x|− m−1 2 ) as |x| → +∞, |ψ(x)| = O(|x|− m−1 2 ) as |x| → 0, for each ψ ∈ S1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, if ψµ is the solution corresponding to µ ∈ [− (m−1)m 2m+1m , 0), then there exists λ > 0 such that ψµ converges in C1 loc(Rm) to ψ∞ = � 2λ λ2+|x|2 � m 2 � 1 − x λ � γ0 as µ → 0, where γ0 is a constant spinor with |γ0| = 1 √ 2 � m 2 � m−1 2 and “·” stands for the Clifford multiplication on spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 7 It is important here to notice the difference between the decay rate of singular solutions that we found in the previous Corollary and the one of regular solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5), studied in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, the decay rate of a regular solution is O(|x|−m+1) but the one of a singular solution is O(|x|− m−1 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10) we have Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let m ≥ 3, the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10) with initial datum u(0) = v(0) = µ > 0 has a solution (uµ, vµ) globally defined on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, there are exactly two types of initial data, which can be characterized by: Ak = � µ > 0 : vµ changes sign k times on (0, +∞) and lim |t|→+∞ Hµ(t) < 0 � , and Ik = � µ > 0 : vµ changes sign k times on (0, +∞) and Hµ(t) > 0 for all t ∈ R � for k ∈ N ∪ {0}, where Hµ(t) := −m − 2 2 uµvµ + m − 1 2m cosh(t)− 1 m−1(u2 µ + v2 µ) m m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In particular, (1) Ak ̸= ∅ is a bounded open set for all k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (2) if we set µk = sup Ak, then µk ∈ Ik and µ0 < µ1 < · · · < µj < µj+1 < · · · → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3) if we set νk = sup Ik, then νk < +∞ and (νk, νk + ε) ⊂ Ak+1 for some small ε > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4) if µ ∈ Ik, then (uµ(t), vµ(t)) → (0, 0) as |t| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' To be more precise, we have uµ(t)2 + vµ(t)2 = O(e−(m−2)t) as |t| → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (5) if µ ∈ Ak, then uµ(t)2 + vµ(t)2 is bounded from below by a positive constant for all t ∈ R and is unbounded as |t| → +∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' furthermore, up to a multiplication by constant, uµ(t)2 + vµ(t)2 is upper bounded by cosh(t) for all |t| large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By setting D2 m = {(uµ, vµ) : µ ∈ ∪k≥0Ak}, we call these unbounded solution the Delaunay- type solution for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' As a direct consequence of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5, we have a characterization of singular solutions for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) on Rm−1 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let m ≥ 3, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) has a one-parameter family S2 m of singular solutions on Rm−1 \\ {0}, parameterized by ∪k≥0Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, the following asymptotic estimates hold |x|− m−2 2 < |ψ(x)| ≲ |x|− m−1 2 as |x| → 0 and |x|− m−2 2 < |ψ(x)| ≲ |x|− m−3 2 as |x| → +∞ for each ψ ∈ S2 m 8 This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' First, in Section 2, we lay down the necessary geometric preliminaries that we will need to formulate our problem, including the main ansatz that will be adopted to find our families of singular solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Next, in Section 3, we use the ansatz to formulate the problem as a Hamiltonian system in R2 (autonomous in the case of a point singularity and non-autonomous in the case of a one dimensional singularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In section 4, we study the properties of the solutions of the Hamiltonian system in the two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This allows us to prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 2 Geometric preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1 General preliminaries about spin geometry Let (M, g) be an m-dimensional Riemannian manifold (not necessarily compact) with a chosen orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let PSO(M) be the set of positively oriented orthonormal frames on (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This is a SO(m)-principal bundle over M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' A spin structure on M is a pair σ = (PSpin(M), ϑ) where PSpin(M) is a Spin(m)-principal bundle over M and ϑ : PSpin(M) → PSO(M) is a map such that the diagram PSpin(M) × Spin(m) � ϑ × Θ � PSpin(M) ϑ � � M PSO(M) × SO(m) � PSO(M) � commutes, where Θ : Spin(m) → SO(m) is the nontrivial double covering of SO(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' There is a topological condition for the existence of a spin structure, namely, the vanishing of the second Stiefel-Whitney class ω2(M) ∈ H2(M, Z2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Furthermore, if a spin structure exists, it need not be unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For these results we refer to [18,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In order to introduce the spinor bundle, we recall that the Clifford algebra Cl(Rm) is the associative R-algebra with unit, generated by Rm satisfying the relation x · y − y · x = −2(x, y) for x, y ∈ Rm (here (·, ·) is the Euclidean scalar product on Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' It turns out that Cl(Rm) has a smallest representation ρ : Spin(m) ⊂ Cl(Rm) → End(Sm) of dimension dimC(Sm) = 2[ m 2 ] such that Cl(Rm) := Cl(Rm)⊗C ∼= EndC(Sm) as C-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In case m is even, this irreducible representations is uniquely determined, but it splits into non-equivalent sub-representations S+ m and S− m as Spin(m)-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If m is odd, there are two irreducible Clm-representations S0 m and S1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Both of them coincide if considered as Spin(m)-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Define the chirality operator ωRm C = i[ m+1 2 ]e1 · e2 · · · em ∈ Clm with {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , em} being a positively oriented orthonormal frame on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In case m is even, we have ωRm C act as ±1 on S± m, and sections of S+ m (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' S− m) are called positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' negative) spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' While if m is odd, the chirality operator acts on Sj m as (−1)j, j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, for m odd, it will cause no confusion if we simply identify S0 m and S1 m as the same vector space, that is Sm = S0 m = S1 m, and equip them with Clifford multiplication of opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Associated to the above observations, the spinor bundle is then defined as S(M) := PSpin(M) ×ρ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Note that the spinor bundle carries a natural Clifford multiplication, a natural hermitian metric 9 and a metric connection induced from the Levi-Civita connection on TM (see [18, 28]), this bundle satisfies the axioms of Dirac bundle in the sense that (i) for any x ∈ M, X, Y ∈ TxM and ψ ∈ Sx(M) X · Y · ψ + Y · X · ψ + 2g(X, Y )ψ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (ii) for any X ∈ TxM and ψ1, ψ2 ∈ Sx(M), (X · ψ1, ψ2)g = −(ψ1, X · ψ2)g, where (·, ·)g is the hermitian metric on S(M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (iii) for any X, Y ∈ Γ(TM) and ψ ∈ Γ(S(M)), ∇S X(Y · ψ) = (∇XY ) · ψ + Y · ∇S Xψ, where ∇S is the metric connection on S(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The Dirac operator is then defined on the spinor bundle S(M) as the composition Dg : Γ(S(M)) ∇S −→ Γ(T ∗M ⊗ S(M)) −→ Γ(TM ⊗ S(M)) m −→ Γ(S(M)) where m denotes the Clifford multiplication m : X ⊗ ψ �→ X · ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us remark that there is an implicit g-dependence in the Clifford multiplication “m” or “·”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, considering a simple case where we replace g with a conformal metric ˜g = e2ug, the isometry X �→ e−uX from (TM, g) onto (TM, ˜g) defines a principal bundle isomorphism SO(TM, g) → SO(TM, ˜g) lifting to the spin level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then it induces a bundle isomorphism S(M, g) → S(M, ˜g), ψ �→ ˜ψ, fiberwisely preserving the Hermitian inner product and sending X · ψ to e−uX˜· ˜ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the sequel, when necessary, we shall write DM g and ·g, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=', to precise the underlying manifold M and the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2 Spinor bundle and the Dirac operator on product manifolds In this subsection our notation is close to [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let (N = M1 × M2, gN = gM1 ⊕ gM2) be a product of Riemannian spin mj-manifolds (Mj, gMj, σMj), j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We have PSpin(N) = (PSpin(M1) × PSpin(M2)) ×ζ Sm1+m2 where ζ : Spin(m1) × Spin(m2) → Spin(m1 + m2) is the Lie group homomorphism lifting the standard embedding SO(m1) × SO(m2) → SO(m1 + m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The spinor bundle over N can be identified with S(N) = � (S(M1) ⊕ S(M1)) ⊗ S(M2) both m1 and m2 are odd, S(M1) ⊗ S(M2) m1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 10 That is, we always put the even dimensional factor in the place of M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And the Clifford multi- plication on S(N) can be explicitly given in terms of the Clifford multiplications on its factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, for X ∈ TM1, Y ∈ TM2, ϕ ∈ Γ(S(M2)) and ψ = � ψ1 ⊕ ψ2 ∈ Γ(S(M1) ⊕ S(M1)) for both m1 and m2 odd ψ ∈ Γ(S(M1)) for m1 even we have (X ⊕ Y ) ·gN (ψ ⊗ ϕ) = (X ·gM1 ψ) ⊗ ϕ + (ωM1 C gM1 ψ) ⊗ (Y ·gM2 ϕ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) where in case m1 and m2 odd we set X ·gM1 ψ = (X ·gM1 ψ1) ⊕ (−X ·gM1 ψ2) and ωM1 C ·gM1 ψ = i(ψ2⊕−ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us remark that there are different ways to formulate the Clifford multiplication (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1), but such changes are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, due to the uniqueness of Cl(TM1 ⊕ TM2), any definition of the Clifford multiplication on S(N) can be identified with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) via a vector bundle isomorphism (see the examples in the next subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let ∇S(M1) and ∇S(M2) be the Levi-Civita connections on S(M1) and S(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By ∇S(M1)⊗S(M2) = ∇S(M1) ⊗ IdS(M2) + IdS(M1) ⊗ ∇S(M2) we mean the tensor product connection on S(M1) ⊗ S(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1), the Dirac operator on N is given by DN g = ˜DM1 gM1 ⊗ IdS(M2) + (ωM1 C gM1 IdS(M1)) ⊗ DM2 gM2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) where ˜DM1 gM1 = DM1 gM1 ⊕ −DM1 gM1 if both m1 and m2 are odd and ˜DM1 gM1 = DM1 gM1 if m1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For the case m1 + m2 even, we have the decomposition S(N) = S(N)+ ⊕ S(N)− and, moreover, when restrict DN g on those half-spinor spaces we get DN g : Γ(S(N)±) → Γ(S(N)∓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3 A particular ansatz in Euclidean spaces Let M = Rm be equipped with the Euclidean metric, then the spinor bundle is given by S(Rm) = Rm × Sm ∼= Rm × C2[ m 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Although, from the abstract setting, the Dirac operator can be given by DgRmψ = m � k=1 ek ·gRm ∇ekψ, ψ ∈ S(Rm) where {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , em} is a orthonormal base of Rm, we can have a more explicit representation of this operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact the Dirac operator can be formulated as a constant coefficient differential operator of the form DgRm = m � k=1 α(m) k ∂ ∂xk (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) where α(m) k is a linear map α(m) k : C2[ m 2 ] → C2[ m 2 ] satisfying the relation α(m) j α(m) k + α(m) k α(m) j = −2δij (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) 11 for all j, k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us give a possible construction of these {α(m) j } by using 2[ m 2 ] × 2[ m 2 ] complex matrices with a block structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We start with m = 1 and the 1-dimensional Dirac operator DgR = i d dx, that is we have α(1) 1 = i the pure imaginary unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For m is even, we define α(m) j = � 0 −iα(m−1) j iα(m−1) j 0 � for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , m − 1 and α(m) m = � 0 i Id i Id 0 � where “Id” is understood to be the identity on C2[ m−1 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And, if m is odd, we define α(m) j = α(m−1) j for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , m − 1 and α(m) m = i m+1 2 α(m−1) 1 · · α(m−1) m−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' It is illuminating to consider this construction in low dimensions: Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For m = 2, we have α(2) 1 = � 0 1 −1 0 � and α(2) 2 = �0 i i 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Writing a spinor field ψ : R2 → S(R2) in components as �ψ1 ψ2 � ∈ C2, we then have DgR2ψ = � 0 1 −1 0 � � ∂ψ1 ∂x1 ∂ψ2 ∂x1 � + �0 i i 0 � � ∂ψ1 ∂x2 ∂ψ2 ∂x2 � = � ∂ψ2 ∂x1 + i ∂ψ2 ∂x2 − ∂ψ1 ∂x1 + i ∂ψ1 ∂x2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) Thus, in this case, the Dirac operator is simply the Cauchy-Riemann operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Consider the product R2 = R × R and the identification S(R2) = (S(R) ⊕ S(R)) ⊗ S(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We see that the fiberwise isomorphism is given explicitly by (S(R) ⊕ S(R)) ⊗ S(R) ∋ �u1v u2v � ←→ 1 √ 2 �(u1 + u2)v (u1 − u2)v � ∈ S(R2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) for u1, u2, v ∈ Γ(S(R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In particular, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2), we see that �i d dx 0 0 −i d dx � �u1v u2v � − d dy � u2v −u1v � = � iu′ 1v − u2v′ −iu′ 2v + u1v′ � which coincides with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) (under the action of the isomorphism in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For m = 3, we have α(3) 1 = � 0 1 −1 0 � , α(3) 2 = �0 i i 0 � and α(3) 3 = �−i 0 0 i � which are exactly the classical Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And for the product R3 = R2 × R, it is easy to obtain from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) that DgR3 = DgR2 ⊗ IdS(R) + �−1 0 0 1 � ⊗ DgR fitting into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 12 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For m = 4, we have α(4) 1 = � � � � 0 −i i i −i 0 � � � � , α(4) 2 = � � � � 0 1 1 −1 −1 0 � � � � , α(4) 3 = � � � � 0 −1 0 0 1 1 0 0 −1 0 � � � � and α(4) 4 = � � � � 0 i 0 0 i i 0 0 i 0 � � � � And for the product R4 = R2 × R2, we have S(R4) = S(R2) ⊗ S(R2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By considering a bundle isomorphism S(R2) ⊗ S(R2) ∋ �u1 u2 � ⊗ �v1 v2 � ←→ � � � � −iu1v1 −iu2v2 iu1v2 iu2v1 � � � � ∈ S(R4) for u1, u2, v1, v2 ∈ Γ(S(R2)), one easily verifies the correspondence DgR4 = DgR2 ⊗ IdS(R2) + �−1 0 0 1 � ⊗ DgR2 which justifies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Meanwhile,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' for the product R4 = R3 ×R and the associated spinor bundle S(R4) = (S(R3) ⊕ S(R3)) ⊗ S(R),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' we have the fiberwise isomorphism (S(R3) ⊕ S(R3)) ⊗ S(R) ∋ � � � � ψ1ϕ ψ2ϕ ψ3ϕ ψ4ϕ � � � � ←→ 1 √ 2 � � � � (ψ4 − ψ2)ϕ (ψ3 − ψ1)ϕ (ψ2 + ψ4)ϕ −(ψ1 + ψ3)ϕ � � � � ∈ S(R4) for �ψ1 ψ2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' �ψ3 ψ4 � ∈ S(R3) and ϕ ∈ S(R) such that the action of �DgR3 0 0 −DgR3 � ⊗ IdS(R) + i � 0 IdS(R3) −IdS(R3) 0 � ⊗ DgR on (S(R3)⊕S(R3))⊗S(R) coincides with the action of DgR4 on S(R4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This verifies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Note the analogy with dimension two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We could continue this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For general m, one can compute the matrices {α(m) j }, the chirality operator ωRm C and, particularly when m is even, the corresponding bundle isomorphism to decompose the Dirac operator in a product structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, these explicit formulas are seldom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' It is always simpler to use the abstract setting of the Clifford module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 13 It is interesting to note that the aforementioned explicit formula for the Dirac operator mo- tivates a “nice” function space which is invariant under the actions of the Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' More precisely, let us set E(Rm) := � ψ(x) = f1(|x|)γ0 + f2(|x|) |x| x · γ0 : x ∈ Rm, f1, f2 ∈ C∞(0, ∞) and γ0 ∈ S2[ m 2 ] C � = � ψ(x) = f1(|x|)γ0 + f2(|x|) |x| m � k=1 xkα(m) k γ0 : f1, f2 ∈ C∞(0, ∞) and γ0 ∈ S2[ m 2 ] C � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' where S2[ m 2 ] C stands for the complex unit sphere in the spin-module Sm ∼= C2[ m 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, following the rule of the Clifford multiplication or the relation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4), it is easy to check that DgRmψ = − � f ′ 2(|x|) + (m − 1)f2(|x|) |x| � γ0 + f ′ 1(|x|) |x| x · γ0 ∈ E(Rm) ∀ψ ∈ E(Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, in order to make sure that ψ is continuous at the origin, one may consider a further restriction to the subspace E0(Rm) = � ψ(x) = f1(|x|)γ0+f2(|x|) |x| x·γ0 ∈ E : f ′ 1(t) = O(t) and f2(t) = O(t) as t ↘ 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1) It is interesting to see that the specific ansatz provided in E(Rm) contains some important formulations of spinors, which are of interest to many physicists when they are dealing with spinor fields in quantum electrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, to the best of our knowledge, it can be traced back to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Finkelstein, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' LeLevier and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Ruderman [14] in 1951 when they investigated a nonlinear Dirac equation in R3 × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By separating the time variable, the authors introduced a very special formulation of a spinor field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' ψ(r, θ1, θ2) = � � � � � f1(r) 0 if2(r) cos θ1 if2(r) sin θ1eiθ2 � � � � � or � � � � � if2(r) cos θ1 if2(r) sin θ1eiθ2 f1(r) 0 � � � � � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7) where (r, θ1, θ2) ∈ (0, +∞) × [0, π] × [0, 2π] is the spherical coordinates on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And subsequently, this ansatz has been commonly used in particle physics where spinors play a crucial role, see for instance [40, 45] and [11] for a 2-dimensional analogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Now, in our setting, we understand that the above spinor field belongs to the sub-bundle S(R3) ⊕ S(R3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Consider the standard spherical coordinates x1 = r cos θ1, x2 = r sin θ1 cos θ2, x3 = r sin θ1 sin θ2 cos θ3 and x4 = r sin θ1 sin θ2 sin θ3 for r > 0, θ1, θ2 ∈ [0, π] and θ3 ∈ [0, 2π], if we restrict to θ2 = π 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' the variable x2 is separated out, treated as the time variable) and take γ0 = � � � � 1 0 0 0 � � � � ∈ S4 C, 14 we soon derive that f1(|x|)γ0 + f2(|x|) |x| 4 � k=1 xkα(4) k γ0 = � � � � � if2(r) cos θ1 if2(r) sin θ1eiθ3 f1(r) 0 � � � � � which is exactly the latter one in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (2) Although the special ansatz (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7) for a spinor has been known for over half a century, it is still new and important to have the family E(Rm) for general dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Particularly, the ansatz in E(Rm) reduces the Dirac equation significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, for the semilinear equations of the form DgRmψ = h(|x|, |ψ|)ψ, ψ : Rm → Sm ∼= C2[ m 2 ] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) where h : [0, +∞) × [0, ∞) → R is a given function, the ansatz in E(Rm) transforms it equivalently to � � � � � − f ′ 2 − m − 1 r f2 = h � r, � f 2 1 + f 2 2 � f1, f ′ 1 = h � r, � f 2 1 + f 2 2 � f2, for r > 0 making the problem much easier to deal with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3) This ansatz was also used to study several mathematical physics models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We cite for instance [6–8] for the study of Dirac-type equation, [15,39] for the study of particle like solutions of coupled Dirac type equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4) The space E(Rm) is somehow natural within spinor fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, if one looks at the parallel spinors on Rm and the Dirac bubbles [9] (corresponding to Killing spinors on the sphere), then one notices that they all belong to E(Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, we can think about E(Rm) as a generalized special class of spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 3 Set up of the problems Let us consider the m-sphere Sm to be Rm ∪ {∞}, where the coordinates x ∈ Rm is given by the standard stereographic projection from the north pole αm : Sm \\ {P m+1 N } → Rm (here P m+1 N = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , 0, 1) ∈ Sm ⊂ Rm+1 stands for the north pole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For clarity, we use the sub- or superscripts to indicate the underlying dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By setting P m+1 S = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , 0, −1) for the south pole, we can see that the manifold R × Sm−1 is conformally equivalent to Sm \\ {P m+1 N , P m+1 S }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The conformal diffeomorphism can be explicitly formulated by Sm \\ {P m+1 N , P m+1 S } αm −→ Rm \\ {0} βm −→ R × Sm−1 ξ = (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , ξm+1) �−→ x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , xm) �−→ (ln |x|, x/|x|) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) 15 where we have (α−1 m )∗gSm = 4 (1+|x|2)2gRm and (βm)∗(gR ⊕ gSm−1) = 1 |x|2gRm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This observation leads to some further considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Typical examples arise from the (con- nected) domain Ω ⊂ Sn whose complement is an equatorial circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Without loss of generality, we may consider the domain Sm \\ S1 = � (ξ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , ξm+1) ∈ Rm+1 : � k ξ2 k = 1, ξ2 1 + ξ2 m+1 < 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we have the following conformal equivalence Ω = Sm \\ S1 αm −→ Rm \\ {(R, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , 0)} βm −→ R × (Sm−1 \\ {P m N , P m S }) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) We now consider the solutions of the spinorial Yamabe equation on the sphere (Sm, gSm), that are singular at a prescribed closed set Σ ⊂ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' More specifically, we will consider the problem DgSmφ = |φ| 2 m−1 gSm φ on Ω = Sm \\ Σ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) when Σ is given by a pair of antipodal points, say {P m+1 N , P m+1 S }, or an equatorial circle S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Before discussing the Delaunay family of solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3), let us recall the transforma- tion formula of the Dirac operator under conformal changes (see [21,23]): Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let g0 and g = f 2g0 be two conformal metrics on a Riemannian spin m- manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, there exists an isomorphism of vector bundles F : S(M, g0) → S(M, g) which is a fiberwise isometry such that Dg � F(ψ) � = F � f − m+1 2 Dg0 � f m−1 2 ψ �� , where Dg0 and Dg are the Dirac operators on M with respect to the metrics g0 and g, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In what follows, our discussions will be build upon this formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1 The singular set is a pair of antipodal points In this setting, without loss of generality, we assume Σ = {P m+1 N , P m+1 S } ⊂ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, as a direct consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1, we have that if ψ is a solution to the equation DgRmψ = |ψ| 2 m−1 gRm ψ on Rm \\ {0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) then φ = F(f − m−1 2 ψ) (f(x) = 2 1+|x|2) is a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that since Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) has the same structure as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8), we shall look at solutions of the form ψ(x) = f1(|x|)γ0 + f2(|x|) |x| x · γ0 ∈ E(Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) Then, applying the Emden-Fowler change of variable r = e−t and write f1(r) = −u(t)e m−1 2 t and f2(r) = v(t)e m−1 2 t, we are led to consider the following system � � � � � u′ + m − 1 2 u = (u2 + v2) 1 m−1v, −v′ + m − 1 2 v = (u2 + v2) 1 m−1u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) 16 This system is easily integrated and is nondissipative, in particular, the Hamiltonian energy H(u, v) = −m − 1 2 uv + m − 1 2m � u2 + v2� m m−1 is constant along solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The equilibrium points for system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) are (0, 0) and ± �(m − 1)(m−1)/2 2m/2 , (m − 1)(m−1)/2 2m/2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And there is a special homoclinic orbit u0(t) = m(m−1)/2et/2 2m/2 cosh(t)m/2, v0(t) = m(m−1)/2e−t/2 2m/2 cosh(t)m/2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7) corresponding to the level set H = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' it limits on the origin as t tends to ±∞, and encloses a bounded set Λ in the first quadrant of the (u, v)-plane, given by {H ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' It is easy to see that orbits not enclosed by this level set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' those orbits in {H > 0}, must pass across the u-axis and v-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' That is u and v must change sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Observe that the equilibrium point (0, 0) is contained exactly in two orbits: the homoclinic one and the stationary orbit (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, for orbits (u(t), v(t)) in {H ̸= 0}, we must have that u2 + v2 ̸= 0 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And thus, we have an unbounded one parameter family of periodic solutions D1 m = � (u, v) is a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) : u(0) = v(0) = µ > 0, µ ̸= m(m−1)/2 2m/2 � , which induces correspondingly a family of singular solutions S1 m to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remark that |ψ(x)| → +∞ as |x| → 0 and |ψ(x)| = O(|x|− m−1 2 ) as |x| → +∞ for each ψ ∈ S1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, these solutions give rise to distinguished singular solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If we take into account just the periodic solutions in D1 m, we will call them the Delaunay-type solutions of the spinorial Yamabe problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Although we do not know them explicitly, in Section 4, we will study the bifurcation phenomenon for solution in the first quadrant of (u, v)- plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2 The singular set is an equatorial circle First of all, we need to observe that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) can be interpreted as an equation on R × (Sm−1 \\ {P m N , P m S }) by a conformal change of the Riemannian metric gSm on Sm \\ S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Consider the product metric on R × Sm−1, given in (τ, ϑ)-coordinates by ¯g = dτ 2 + dϑ2, where ϑ = (ϑ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , ϑm−1) parameterizes the unit sphere Sm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then it follows from the conformal equiv- alence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) that (α−1 m ◦ β−1 m )∗gSm = 4e2τ (1 + e2τ)2¯g = 1 cosh(τ)2¯g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And as a direct consequence of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1, we have that if ϕ is a solution to the equation D¯gϕ = |ϕ| 2 m−1 ¯g ϕ on R × (Sm−1 \\ {P m N , P m S }) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) 17 then φ = F(cosh(τ) m−1 2 ϕ) is a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) with F being a bundle isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us remark that the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) on product manifolds indicates a way to construct singular solutions for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, if m is odd (hence m ≥ 3), then m − 1 is even and we can consider a special spinor of the form ϕ = 1 ⊗ ˜ψ so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) is reduced to DgSm−1 ˜ψ = | ˜ψ| 2 m−1 gSm−1 ˜ψ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) where ˜ψ = ˜ψ(ϑ) is a spinor on Sm−1 \\ {P m N , P m S }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And once again, by using the conformal formula in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) can be equivalently transformed to DgRm−1ψ = f(x) 1 m−1|ψ| 2 m−1 gRm−1ψ on Rm−1 \\ {0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10) where f(x) = 2 1+|x|2 for x ∈ Rm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And the solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10) are in one-to-one correspondence via the identification ˜ψ ↔ f − m−2 2 ψ for spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Now, by considering the ansatz ψ(x) = f1(|x|)γ0 + f2(|x|) |x| x · γ0 ∈ E(Rm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' and applying the change of variable r = e−t, we can reduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10) to the system � � � � � u′ + m − 2 2 u = cosh(t)− 1 m−1(u2 + v2) 1 m−1v −v′ + m − 2 2 v = cosh(t)− 1 m−1(u2 + v2) 1 m−1u (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) where f1(r) = −u(t)e m−2 2 t and f2(r) = v(t)e m−2 2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If m is even, then the spinor bundle on R × (Sm−1 \\ {P m N , P m S }) can be identified with S(R) ⊗ (S(Sm−1) ⊕ S(Sm−1)) and the Dirac operator can be formulated as D¯g = �DgSm−1 0 0 −DgSm−1 � ⊗ IdS(R) + i � 0 IdS(Sm−1) −IdS(Sm−1) 0 � ⊗ DgR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, considering a spinor of the form ϕ = 1 ⊗ ( ˜ψ1 ⊕ ˜ψ1) for ˜ψ1, ˜ψ1 ∈ Γ(S(Sm−1)), we may reduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) to the following Dirac system � DgSm−1 ˜ψ1 −DgSm−1 ˜ψ2 � = � | ˜ψ1|2 gSm−1 + | ˜ψ2|2 gSm−1 � 1 m−1 � ˜ψ1 ˜ψ2 � on Sm−1 \\ {P m N , P m S }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10), we can transform the above system to � DgRm−1ψ1 −DgRm−1ψ2 � = f(x) 1 m−1� |ψ1|2 gRm−1 + |ψ2|2 gRm−1 � 1 m−1 � ψ1 ψ2 � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12) on Rm−1 \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 18 Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' using the ansatz ψ1(x) = f1(|x|)γ0 + f2(|x|) |x| x · γ0 and ψ2(x) = f3(|x|)γ0 + f4(|x|) |x| x · γ0 in E(Rm−1) and applying the change of variable r = e−t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' we then get the following system � � � � � � � � � � � � � � � � � � � � � u′ 1 + m − 2 2 u1 = cosh(t)− 1 m−1� u2 1 + u2 2 + v2 1 + v2 2 � 1 m−1v1 −v′ 1 + m − 2 2 v1 = cosh(t)− 1 m−1� u2 1 + u2 2 + v2 1 + v2 2 � 1 m−1u1 u′ 2 + m − 2 2 u2 = cosh(t)− 1 m−1� u2 1 + u2 2 + v2 1 + v2 2 � 1 m−1v2 −v′ 2 + m − 2 2 v2 = cosh(t)− 1 m−1� u2 1 + u2 2 + v2 1 + v2 2 � 1 m−1u2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='13) where we have substituted f1(r) = −u1(t)e m−2 2 t, f2(r) = v1(t)e m−2 2 t, f3(r) = u2(t)e m−2 2 t and f4(r) = v2(t)e m−2 2 t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, we can consider the solutions for which u1 = u2 and v1 = v2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' these are the solutions having the simplest and clearest structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By writing u = √ 2u1 and v = √ 2v1, we can turn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='13) into � � � � � u′ + m − 2 2 u = cosh(t)− 1 m−1� u2 + v2� 1 m−1v −v′ + m − 2 2 v = cosh(t)− 1 m−1� u2 + v2� 1 m−1u which exactly coincides with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Clearly, the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) has an Hamiltonian structure, where the Hamiltonian energy is given by H(t, u, v) = −m − 2 2 uv + m − 1 2m cosh(t)− 1 m−1(u2 + v2) m m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' It is evident that this system is dissipative and there is no periodic solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, one may consider solutions that are not converging to (0, 0) as t → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' More precisely, we will characterize the following family of solutions D2 m = � (u, v) is a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) : u2(t) + v2(t) → +∞ as t → ±∞ � which induces a family of singular solutions S2 m to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence these solutions gives rise to singular solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In this setting, we shall call the family D2 m the Delaunay-type solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 4 Analysis of the ODE systems This section contains our main study of the dynamical systems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We point out that both systems have a variational structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, if we denote z = (u, v) ∈ R2, systems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) can be rewritten as ˙z = dz dt = J∇zH(t, z) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) 19 where J = � 0 1 −1 0 � and H stands for the corresponding Hamiltonian energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The functionals ΦT(z) = 1 2 � T −T (−J ˙z, z)dt − � T −T H(t, z)dt and Φ(z) = 1 2 � R (−J ˙z, z)dt − � R H(t, z)dt can be used to obtain periodic solutions and homoclinic solutions for (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In par- ticular, there is one-to-one correspondence between 2T-periodic solutions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1) and critical points of ΦT (as long as H(t, z) is periodic in the t-variable or independent of t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Similarly, critical points of Φ correspond to homoclinic solutions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=', z(t) → (0, 0) as t → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For the autonomous system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6), we point out that the existence of a 2T-periodic solu- tion for every T > T0, some T0 > 0, and the asymptotic behavior of these solutions as T ↗ +∞ have been already investigated in [1,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By summarizing their results, we have Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' There exists T0 > 0 such that for every T > T0 the Hamiltonian system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) has a non-constant 2T-periodic solution zT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The family {zT : T > T0} is compact in the following sense: for any sequence Tn ↗ +∞, up to a subsequence if necessary, zTn converges in C1 loc(R, R2) to a nontrivial solution z∞ of the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) on R satisfying lim |t|→+∞ z∞(t) = lim |t|→+∞ ˙z∞(t) = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=', z∞ is a homoclinic orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that the previous proposition does not provide a clear description of the behavior of the solutions zT as T ↘ T0 or a characterization of z∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For instance, from the arguments in [1, 44], we do not have an estimate of T0 and we do not know if there are non-constant so- lutions below T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, if H has a “good” structure around its equilibrium points, then one can use Lyapunov’s center theorem to exhibit a family of small amplitude periodic solutions bifurcating from the equilibrium solution and also have an estimate on T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Nevertheless, this does not provide uniqueness of the family of non-constant solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the sequel, we will perform different approaches to characterize the Delaunay-type fam- ilies D1 m and D2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We also want to point out that an alternative method can be used to find periodic solutions of family D1,− m using variational analysis and by tracking the least energy so- lution, we can characterize the homoclinic energy z∞, corresponding to the least energy solution for the functional Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This procedure was used in a more general setting of product manifolds in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1 The nondissipative case: Bifurcation of the positive periodic orbits In order to analyse the dynamical system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6), we recall that H(u, v) = −m − 1 2 uv + m − 1 2m � u2 + v2� m m−1 20 for u, v ∈ R and m ≥ 2, which is independent of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We will focus on the periodic solu- tions/orbits of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) in the first quadrant of the (u, v)-plane, that is u, v : R/2TZ → (0, +∞) for all T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Such solutions will be referred as positive solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' System (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) has an “obvious” constant solution u = v ≡ (m−1)(m−1)/2 2m/2 for all T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' From now on, we intend to look at non-constant solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By setting z = u2 + v2 and w = u2 − v2, we have uv = √ z2−w2 2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) becomes � � � z′ = −2λw zz′ − ww′ = 1 λzp−1z′√ z2 − w2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) where we denote λ = m−1 2 > 0 and p = m m−1 ∈ (1, 2] for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' After multiplication by (z2 − w2)−1/2 in the second equation, we obtain d dt �√ z2 − w2 � = d dt � 1 λpzp� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thus, for any solution z and w, there exists a constant K such that √ z2 − w2 = 1 λpzp + K, that is, w2 = z2 − � 1 λpzp + K �2 and 1 λpzp + K ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) For K ∈ R, let us denote FK(s) = s2 − � 1 λpsp + K �2 for s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remark that, if (z, w) is a non-constant 2T-periodic solution of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2), then z must achieve the maximum and minimum in one period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence z′ has at least two zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This, together with the first equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2), implies that FK should vanish at least twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, the conditions on K are particularly restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, for K = 0, we can combine the first equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) together to obtain (z′)2 = 4λ2z2 − 4 p2z2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, if there exist t0 and t1 such that z(t0) < z(t1) and z′(t0) = z′(t1) = 0, we have z(t0) = 0 and z(t1) = (m 2 )m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Clearly, this should corresponds to the homoclinic solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7) and can not be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For K < 0, by analyzing the algebraic equation FK(s) = 0, we can see that Fk has exactly two zeros 0 < s0 < s1 on (0, +∞) given by the relations � � � � � � � s0 = − 1 λpsp 0 − K, s1 = 1 λpsp 1 + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' But we find 1 λpsp 0+K < 0, which fails to satisfy the second inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' So the remaining range for K is (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, it is obvious that K can not be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If K > 0 is small, FK has exactly two zeros on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We only prove the case p = m m−1 ∈ (1, 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' m > 2, since p = 2 is much easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that F ′ K(s) = 2s − 2 λ � 1 λpsp + K � sp−1 for s ≥ 0 and p ∈ (1, 2], we have F ′ K(0) = 0 and F ′ K(s) < 0 in (0, δ1) for some δ1 > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Observe that the two maps s �→ λs2−p and s �→ 1 λpsp + K have exactly two intersections for K > 0 small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We denote the horizontal coordinates of these two intersections by 0 < s0,1 < s0,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we have F ′ K < 0 on (0, s0,1) ∪ (s0,2, +∞) and F ′ K > 0 on (s0,1, s0,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, FK(s0,1) < 0 is a strict local minimum, whereas FK(s0,2) is a strict local maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since F0(1) = 1 − 1 λ2p2 > 0 (we used the facts λ = m−1 2 , p = m m−1 and m > 2), we have FK(1) > 0 for all small K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence FK(s0,2) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This implies FK has exactly two zeros on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let K0 := sup � K > 0 : FK has two zeros � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We remark that, for K > 0, FK can not have a third zero in (0, +∞) since F ′ K changes sign at most twice and FK(0) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' K0 < +∞ and FK0 has only one zero, which is the global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Furthermore, FK(s) < 0 for all K > K0 and s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since K0 < +∞ is obvious, we only need to check the remaining statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' To begin with, we mention that ∂ ∂K FK(s) = −2 � 1 λpsp + K � < 0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4) provided that K > 0 and s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, if F ˆ K(s ˆ K) > 0 for some ˆK > 0 and s ˆ K > 0, we have FK(s ˆ K) > 0 for all K ∈ (0, ˆK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, due to the continuity of FK with respect to K, there exists ε > 0 such that FK(s ˆ K) > 0 for K ∈ ( ˆK, ˆK + ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, we can see that � K > 0 : FK has two zeros � = (0, K0) is an open interval and that max FK0 ≤ 0 (otherwise FK0 will have two zeros).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By choosing a sequence Kn ↗ K0 and sn > 0 such that FKn(sn) > 0, we have {sn} is bounded and FKn(sn) → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore FK0 has only one zero, which is the global maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The last assertion comes from the fact (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The value of K0 can be explicitly computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Precisely, we have K0 = � 1 − 1 p � λ 1 p−1 = 1 m �m − 1 2 �m−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, K = K0 is the largest positive number such that the equation s = 1 λpsp + K has a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the sequel, let K ∈ (0, K0), we set 0 < s0 < s1 the points such that FK vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' It is worth pointing out that s0 and s1 are functions of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then FK is positive on the interval (s0, s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) is now equivalent to dz 2λ � FK(z) = ±dt, 22 which can be solved by ηK(z) = ±t + C, where ηK(z) = � s s0 dz 2λ � FK(z) and C ∈ R is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Of course, ηK is defined on the interval (s0, s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By noting that s0 and s1 are simple roots of FK (that is F ′ K(sj) ̸= 0 for j = 0, 1), we have ηK(s1) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, we have η′ K(s) > 0 and η′ K(s) → +∞ as s → s0 or s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, ηK has an inverse η−1 K which increases from s0 to s1 on the interval [0, ηK(s1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Now, solutions to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='3) can be represented as z(t) = η−1 K (±t + C) for C ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Setting zK(t) = � η−1 K (t) t ∈ [0, ηK(s1)], η−1 K (−t) t ∈ [−ηK(s1), 0], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) it follows that zK is a 2ηK(s1)-periodic solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) and can not have smaller period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, this zK (jointly with the corresponding wK from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2)) gives rise to a positive solution (uK, vk) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) with H(uK, vK) = − λK 2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The mapping K �→ ηK(s1) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Particularly, lim K↘0 ηK(s1) = +∞ and lim K↗K0 ηK(s1) = √m − 1 2 π Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For starters, we shall write s0 = s0(K) and s1 = s1(K) to emphasize that s0 and s1 are functions of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that s0 and s1 are solutions to the equation s = 1 λpsp + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By the implicit function theorem, we have s0 and s1 are C1 functions, in particular, � � � � � � 1 − 1 λs0(K)p−1� s′ 0(K) = 1, � 1 − 1 λs1(K)p−1� s′ 1(K) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since we have assumed s0 < s1, we have � 1 − 1 λs0(K)p−1� > 0 and � 1 − 1 λs1(K)p−1� < 0 which implies that s′ 0(K) > 0 and s′ 1(K) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The continuity of ηK(s1) is obvious and, without digging out very much from the function ηK(s1), we can evaluate the asymptotic behavior of ηK(s1) as K goes to the end points 0 and K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, to see the limiting behavior of ηK(s1) as K ↘ 0, we first observe that FK(0) < 0 and FK(2K) > 0 for all small K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence we have 0 < s0(K) < 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover λ1/(p−1) < s1(K) since s1(K) is the larger solution to the equation s = 1 λpsp + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then ηK(s1) ≥ � λ1/(p−1) 2K dz 2λ � FK(z) ≥ 1 2λ � λ1/(p−1) 2K dz z = 1 2λ � ln λ1/(p−1) − ln 2K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 23 Thus, by taking K → 0, we have limK↘0 ηK(s1) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For K close to K0, we set GK(t) = FK(tm−1), that is GK(t) = t2(m−1) − � 2 mtm + K �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By writing t0 = s1/(m−1) 0 and t1 = s1/(m−1) 1 , we can write GK in its factorization GK(t) = 4 m2(t − t0)(t1 − t)PK(t) with PK(t) = � tm + m 2 tm−1 + m 2 K �� a0tm−2 + a1tm−3 + · · · + am−3t + am−2 � , where a0 = 1, a1 = t0 + t1 − m 2 and aj = −t0t1aj−2 + (t0 + t1)aj−1 for j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' From elementary computations, we can simply write aj = tj+1 1 − tj+1 0 t1 − t0 − m 2 tj 1 − tj 0 t1 − t0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , m − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we can reformulate ηK(s1) as ηK(s1) = � t1 t0 tm−2dt � GK(t) = m 2 � 1 0 (t0 + (t1 − t0)τ)m−1dτ � τ(1 − τ)PK(t0 + (t1 − t0)τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7) Notice that, as K approaches K0, we have t0, t1 → m−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By the continuity of ηK(s1), we have lim K→K0 ηK(s1) = cm � 1 0 dτ � τ(1 − τ) = cmπ where cm = m(m − 1)m−1 2m � PK0( m−1 2 ) = √m − 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Recall that we are looking at the 2ηK(s1)-periodic solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2), then Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5 implies: (1) For every T > 0, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2) has the constant solution z0 ≡ (m−1)m−1 2m−1 and w0 ≡ 0, which gives the nontrivial constant solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And, for T ≤ √m−1 2 π, this is the only possible solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (2) Let d ∈ N with d √m−1 2 π < T ≤ (d + 1) √m−1 2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then for any k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' , d, we have T k ≥ T d > √m−1 2 π and there exists K = K(T/k) ∈ (0, K0) such that ηK(s1) = T/k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 24 (3) The solutions given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5) corresponds to the solutions obtained in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1, since the Hamiltonian energy H(uK, vK) → 0 and the minimal period ηK(s1) → +∞ as K → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, we have T0 = √m−1 2 π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We end this section by comparing the classical Delaunay solutions that appear in the study of the singular Yamabe problem and the solutions that we have just studied above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us recall the classical Delaunay solutions for the singular Yamabe problem as in [32, 35], that are obtained by solving the ODE u′′ − (m − 2)2 4 u + m(m − 2) 4 u m+2 m−2 = 0, u > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) This equation is clearly nondissipative, and the corresponding Hamiltonian energy is �H(u, u′) = 1 2|u′|2 − (m − 2)2 8 u2 + (m − 2)2 8 u 2m m−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By examining the level sets of �H, we see that all bounded positive solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) lie in the region of the (u, u′)-plane where �H is non-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the figures below, we show a few orbits for both the Hamitonians for the systems (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8) when m = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Figure 1: The orbits for the spinorial Yam- abe equation Figure 2: The orbits for the classical Yam- abe equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2 The dissipative case: Shooting method In this subsection, we investigate the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In particular, since we are looking for singular solutions of the spinorial Yamabe equation, we are interested in solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) such that (u(t), v(t)) ̸→ (0, 0) as t → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='00 2 0 1 上 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='2 0 4 08 0 1 上 0 2 E D25 In order to avoid unnecessary complexity and to get non-trivial solutions, we choose as initial conditions u(0) = v(0) = µ ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, the symmetry of the system allows us to consider only the case µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Recall that the Hamiltonian energy associated to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) is given by H(t, u, v) = −m − 2 2 uv + m − 1 2m cosh(t)− 1 m−1(u2 + v2) m m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We begin with: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For any µ > 0, there is (uµ, vµ) ∈ C1(R, R2), unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) satisfying uµ(0) = vµ(0) = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Furthermore, (uµ, vµ) depends continuously on µ, uniformly on [−T, T], for any T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' To begin with, we may write the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) in integral form as � � � � � � � u(t) = µ + � t 0 � cosh(s)− 1 m−1� u(s)2 + v(s)2� 1 m−1v(s) − m − 2 2 u(s) � ds v(t) = µ − � t 0 � cosh(s)− 1 m−1� u(s)2 + v(s)2� 1 m−1u(s) − m − 2 2 v(s) � ds for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since the right-hand side of the above equation is a Lipschitz continuous function of (u, v), the classical contraction mapping argument gives us a local existence of (uµ, vµ) on [0, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let [0, Tµ) be the maximal interval of existence for (uµ, vµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Clearly, if we define uµ(t) := vµ(−t) and vµ(t) := uµ(−t) for t < 0, we have (uµ, vµ) is a solution on (−Tµ, Tµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose that Tµ < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we have |uµ(t)| + |vµ(t)| → +∞ as |t| → Tµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us denote Hµ(t) = H(t, uµ(t), vµ(t)), t ∈ (−Tµ, Tµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' A simple computation implies d dtHµ(t) = d dt � cosh(t)− 1 m−1 �m − 1 2m (u2 µ + v2 µ) m m−1 ≤ 0, ∀t ≥ 0 so that the energy Hµ is non-increasing along the solution (uµ, vµ), on [0, Tµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, since we have |uµ(t)| + |vµ(t)| → +∞ as t → Tµ, we find Hµ(t) ≥ −m − 2 2 uµ(t)vµ(t) + m − 1 2m cosh(Tµ)− 1 m−1(uµ(t)2 + vµ(t)2) m m−1 → +∞ as t → Tµ, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence we have uµ and vµ are globally defined on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In what follows, we state some basic properties for solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Given µ > 0, then the following holds: If, for some t0 ̸= 0, we have uµ(t0) = 0, then vµ(t0) ̸= 0 and u′ µ(t0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 26 If, for some t0 > 0, we have vµ(t0) = 0, then uµ(t0) ̸= 0 and v′ µ(t0) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, both uµ and vµ can not change sign infinitely many times in a bounded interval [−T, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Observe that the only rest point of system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) is (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Furthermore, for t0 ̸= 0, the Cauchy problem for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) is locally well-posed for any initial datum (u(t0), v(t0)) ∈ R2, for both t > t0 and t < t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thus, a rest point cannot be reached in a finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In order to see that both uµ and vµ can only change sign a finite number of times in a bounded interval [−T, T], we assume by contradiction that there exists {tu j } and {tv j} in [−T, T] such that tu j → Tu and tv j → Tv as j → ∞, uµ(tu j ) = vµ(tv j) = 0 for all j, and uµ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' vµ) changes sign a finite number of times on [−|Tu| + δ, |Tu| − δ] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' [−|Tv| + δ, |Tv| − δ]) for any δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If |Tu| < |Tv|, then vµ will not change sign in a left neighborhood of |Tu| and in a right neighborhood of −|Tu|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then the first equation in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) implies that u′ µ(tu j ) has the same sign as vµ, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence |Tu| ≥ |Tv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Similarly, one obtains |Tv| ≥ |Tu|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore |Tu| = |Tv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, it can not happen that Tu = −Tv while uµ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' vµ) keeps a definite sign around Tv (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Tu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, we must have Tu = Tv = T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In particular, we have uµ(T0) = vµ(T0) = 0, which is also impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Given µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If (uµ, vµ) is a bounded solution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=', |uµ(t)| + |vµ(t)| ≤ M for all t ∈ R and some M > 0, then (uµ, vµ) → (0, 0) as |t| → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By symmetry, we only need to prove the result for t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Multiplying by uµ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' vµ) the equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11), we have � � � � � uu′ = cosh(t)− 1 m−1(u2 µ + v2 µ) 1 m−1uµvµ − m − 2 2 u2 µ, −vv′ = cosh(t)− 1 m−1(u2 µ + v2 µ) 1 m−1uµvµ − m − 2 2 v2 µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thus we need to show that uµ(t)2 + vµ(t)2 → 0 as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose by contradiction that, for arbitrary small ε > 0, there exists t0 > 0 large such that cosh(t0)− 1 m−1M m m−1 ≤ 2ε and uµ(t0)2 + vµ(t0)2 ≥ 2δ0, for some δ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since 1 2(u2 µ)′ ≤ ε − m − 2 2 u2 µ, we find uµ(t)2 ≤ 2ε m − 2 − 2ε m − 2e(m−2)(t0−t) + uµ(t0)2e(m−2)(t0−t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, by enlarging t0, we can assume without loss of generality that vµ(t0)2 > δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And hence, we obtain −1 2(v2 µ)′ ≤ ε − m − 2 2 v2 µ, which implies vµ(t)2 ≥ 2ε m − 2 − 2ε m − 2e(m−2)(t−t0) + vµ(t0)2e(m−2)(t−t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By taking ε < m−2 2 δ0, we have vµ(t)2 → +∞ as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This contradicts the boundedness of vµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 27 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' From the above result, we can conclude that, if there exists t0 > 0 such that Hµ(t0) ≤ 0, the corresponding solution (uµ, vµ) must be unbounded as t → ±∞ (since the energy Hµ(t) = H(t, uµ(t), vµ(t)) is decreasing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If (uµ, vµ) is a solution such that lim|t|→+∞ Hµ(t) ∈ [−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then uµ(t)2 + vµ(t)2 = O(cosh(t)) as |t| → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since Hµ(t) is decreasing, we can take t0 > 0 such that Hµ(t0) ≤ 0 and 0 ≥ Hµ(t) ≥ m − 1 2m cosh(t)− 1 m−1(uµ(t)2 + vµ(t)2) m m−1 − m − 2 4 (uµ(t)2 + vµ(t)2) for all t ≥ t0 Notice that uµ(t)2 + vµ(t)2 can not reach 0 in a finite time, we soon have uµ(t)2 + vµ(t)2 ≤ cm cosh(t) for all t ≥ t0 and cm > 0 depends only on m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let (uµ, vµ) be a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) such that vµ changes sign a finite number of times on R, then there exists T > 0 such that uµ(t)vµ(t) > 0 for all |t| ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since vµ changes sign a finite number of times on R, we suppose without loss of gener- ality that vµ(t) > 0 for all t ≥ T1, some T1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Assume, by contradiction, that uµ(t) < 0 for all t > T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then the second equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) implies that v′ µ(t) > 0 for t > T1, that is, vµ(t) is increasing for t > T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence we have lim t→+∞ vµ(t) = v∞ ∈ (0, +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that, by the second equation again, we have v′ ≥ m − 2 2 v for t ≥ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We deduce that vµ(t) ≥ vµ(T1)e m−2 2 (t−T1) for t ≥ T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence v∞ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, since uµ and vµ have opposite sign, we find Hµ(t) ≥ m − 1 2m cosh(t)− 1 m−1(uµ(t)2 + vµ(t)2) m m−1 > m − 1 2m cosh(t)− 1 m−1vµ(t) 2m m−1 → +∞ as t → +∞, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let t0 ≥ T1 be such that uµ(t0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, it follow from the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) that u′ µ(t0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If there exists ˆt0 > t0 such that uµ(ˆt0) = 0 and uµ(t) > 0 on (t0, ˆt0), we soon derive that u′ µ(t) < 0 in a left neighborhood of ˆt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thus, by the the first equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) again, we get vµ(ˆt0) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This is impossible since we have assumed vµ(t) > 0 for all t > T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, by taking T > t0, we conclude uµ(t) > 0 for all t ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let (uµ, vµ) be a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) such that uµ changes sign a finite number of times on R, then there exists T > 0 such that uµ(t)vµ(t) > 0 for all |t| ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 28 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose that we have uµ(t) > 0 for all t ≥ T, some T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12, we can not have vµ(t) < 0 for all t > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose that there exists t0 > T1 such that vµ(t0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then v′ µ(t0) < 0 and vµ enters to negative values, and can not have further zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, if there is ˆt0 > t0 such that vµ(ˆt0) = 0 and vµ(t) < 0 on (t0, ˆt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We will have v′ µ(ˆt0) ≥ 0, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we obtain a contradiction with Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let (uµ, vµ) be a bounded solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) such that vµ (or uµ) changes sign a finite number of times on R, then uµ(t)2 + vµ(t)2 = O(e−(m−2)t) as |t| → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By virtue of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='13, we can take T > 1 large enough such that uµ(t)vµ(t) > 0 for all t ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, it can be derived from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) that −(u2 µ + v2 µ)′′ + (m − 2)2(u2 µ + v2 µ) = 4(m − 2) cosh(t)− 1 m−1(u2 µ + v2 µ) 1 m−1uµvµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, from the boundedness of uµ and vµ, we have � − (u2 µ + v2 µ)′′ + (m − 2)2(u2 µ + v2 µ) > 0 − (u2 µ + v2 µ)′′ + (m − 2)2(u2 µ + v2 µ) ≤ δe− 1 m−1 t(u2 µ + v2 µ) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='9) for t sufficiently large, where δ > 0 is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let Γ1(t) = e−(m−2)t and Γ2(t) = arctan(t)e−(m−2)t, for t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' One checks easily that −Γ′′ 1 + (m − 2)2Γ1 = 0 and − Γ′′ 2 + (m − 2)2Γ2 ≥ 2(m − 2) 1 + t2 e−(m−2)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By taking C1, C2 > 0 such that C1Γ1(T0) ≤ uµ(T0)2 + vµ(T0)2 ≤ C2Γ2(T0), for some T0 > T, we find � � � − (u2 µ + v2 µ − C1Γ1)′′ + (m − 2)2(u2 µ + v2 µ − C1Γ1) > 0, − (u2 µ + v2 µ − C2Γ2)′′ + � (m − 2)2 − 2(m − 2) (1 + t2) arctan(t) � (u2 µ + v2 µ − C2Γ2) < 0, for all t > T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, by the comparison principle, we have C1Γ1(t) ≤ uµ(t)2 + vµ(t)2 ≤ C2Γ2(t), for all t > T0, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let (uµ, vµ) be a solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) such that vµ changes sign a finite number of times on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If Hµ(t) = H(t, uµ(t), vµ(t)) > 0 for all t > 0, then Hµ(t) ≤ Ce−c|t| as t → ±∞, for some constants C, c > 0 possibly depending on µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 29 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We only prove the result for t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Note that d dtHµ(t) = d dt � cosh(t)− 1 m−1 �m − 1 2m (u2 µ + v2 µ) m m−1 = − 1 2m cosh(t)− 1 m−1 et − e−t et + e−t (u2 µ + v2 µ) m m−1 ≤ −1 − δ 2m cosh(t)− 1 m−1(u2 µ + v2 µ) m m−1 ≤ − 1 − δ m − 1Hµ(t), for t ≥ Tδ, where δ > 0 can be fixed arbitrarily small and the last inequality comes from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, we have Hµ(t) ≤ Hµ(Tδ)e− 1−δ m−1 t for all t ≥ Tδ, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Now, for µ > 0 and (uµ, vµ) the corresponding solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11), we introduce the sets Ak, Bk and Ik defined for k ∈ N ∪ {0} by Ak = � µ > 0 : vµ changes sign k times on (0, +∞) and lim |t|→+∞ Hµ(t) < 0 � , Bk = � µ > 0 : vµ changes sign k times on (0, +∞), Hµ(t) > 0 and (uµ, vµ) is unbounded � , Ik = � µ > 0 : vµ changes sign k times on (0, +∞), Hµ(t) > 0 and (uµ, vµ) is bounded � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that (0, 0) is a hyperbolic equilibrium point of the Hamiltonian energy H(t, ·, ·) for any t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' It is, then, immediate to see that A0 ̸= ∅ as it includes the interval (0, √ 2 2 ], since H(0, µ, µ) < 0 for all µ ∈ � 0, √ 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' As we will see later, tracking the sign changes of the solutions is crucial for the proof of Theo- rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The main idea is to study the stratified structure of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This will be done by checking their topology and boundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The boundedness, allows us to track the sup of Ak and Ik allowing us to prove that all the sets Ak are not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' As we will see below, the idea of tracking the signs coming from a limiting problem with explicit solutions and infinitely many sign changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This property will allow us to prove boundedness of the desired sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let us start first by discarding the sets Bk: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Bk = ∅ for all k ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose to the contrary that Bk ̸= ∅ for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let µ ∈ Bk and (uµ, vµ) be the corresponding solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, by substituting (uµ, vµ) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11), we obtain � � � � � u′ µvµ = cosh(t)− 1 m−1(u2 µ + v2 µ) 1 m−1v2 µ − m − 2 2 uµvµ, −uµv′ µ = cosh(t)− 1 m−1(u2 µ + v2 µ) 1 m−1u2 µ − m − 2 2 uµvµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10) 30 This gives u′ µvµ − uµv′ µ = cosh(t)− 1 m−1(u2 µ + v2 µ) m m−1 − (m − 2)uµvµ = 2m m − 1Hµ(t) + m − 2 m − 1uµvµ > m − 2 m − 1uµvµ, for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12, for t large enough, we can divide the above inequality by uµvµ to get (ln uµ − ln vµ)′ > m − 2 m − 1, where we have assumed without loss of generality that uµ(t) > 0 and vµ(t) > 0 for t large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence we have uµ(t) vµ(t) ≥ Ce m−2 m−1 t (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And therefore, there exists T > 0 such that uµ(t) > vµ(t) for all t > T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Now, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='10), we have u′ µvµ + uµv′ µ = cosh(t)− 1 m−1(u2 µ + v2 µ) 1 m−1(v2 µ − u2 µ) < 0 for t > T, that is, uµvµ is decreasing for all large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Assume that uµ(t)vµ(t) → a∞ ∈ [0, +∞) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='15, we have m − 1 2m cosh(t)− 1 m−1(uµ(t)2 + vµ(t)2) m m−1 → m − 2 2 a∞ as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, for arbitrary small ε > 0, there exists Tε > 0 such that � � � � � u′ µ ≤ ε − m − 2 2 uµ −v′ µ ≤ ε − m − 2 2 vµ for all t ≥ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This implies uµ(t) ≤ 2ε m − 2 − 2ε m − 2e m−2 2 (Tε−t) + uµ(Tε)e m−2 2 (Tε−t) and vµ(t) ≥ 2ε m − 2 − 2ε m − 2e m−2 2 (t−Tε) + vµ(Tε)e m−2 2 (t−Tε) for all t ≥ Tε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since µ ∈ Bk, we have |uµ(t)| + |vµ(t)| is unbounded as |t| → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, by fixing ε > 0 suitably small, we find vµ(t) ∼ e m−2 2 t and uµ(t) → 0 as t → +∞, this contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' There exists constants C0 > 0 such that, if for some T > 1, (1) Hµ(T) ≤ C0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 31 (2) uµ(T)vµ(T) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3) vµ changes sign k times on [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' then µ ∈ Ak ∪ Ik ∪ Ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose that µ ̸∈ Ak ∪ Ik, it remains to show that µ ∈ Ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Without loss of generality, let us assume that uµ(T) > 0 and vµ(T) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Set �T = inf � t > T : uµ(t) ≤ 0 � ∈ (T, +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If �T = +∞, we have vµ changes sign at most once in (T, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, as long as uµ > 0, the second equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) implies that v′ µ < 0 whenever vµ vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, vµ can not change sign more than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If vµ does not change sign on (T, +∞), we have µ ∈ Ak ∪ Ik, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, if vµ does change sign once in (T, +∞), we have uµ(t)vµ(t) < 0 for all large t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This contradicts Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, we have �T < +∞ and uµ( �T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' vµ changes sign exactly once in (T, �T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, by rewriting the second equation of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11), we have � vµ(t)e− m−2 2 t�′ = − cosh(t)− 1 m−1(uµ(t)2 + vµ(t)2) 1 m−1uµ(t)e− m−2 2 t < 0 for t ∈ (T, �T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If vµ stays positive on (T, �T), by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8, we have u′ µ ≥ 0 on a left neigh- borhood of �T, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' To proceed, let us set fµ = (uµ − vµ)/ √ 2 and gµ = (uµ + vµ)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then (fµ, gµ) satisfies the following system � � � � � f ′ = cosh(t)− 1 m−1(f 2 + g2) 1 m−1g − m − 2 2 g, −g′ = cosh(t)− 1 m−1(f 2 + g2) 1 m−1f + m − 2 2 f, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12) with Hamiltonian energy �H(t, f, g) = m − 2 4 f 2 − m − 2 4 g2 + m − 1 2m cosh(t)− 1 m−1(f 2 + g2) m m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Clearly, we have Hµ(t) = �H(t, fµ, gµ) for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And, by Claim 1, we can make T slightly larger so that uµ > vµ on [T, �T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' That is, we have fµ > 0 on [T, �T], gµ(T) > 0, gµ( �T) < 0 and gµ changes sign once in (T, �T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In what follows, we are going to prove that fµ stays positive on [T, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then the second equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12) shows that g′ µ < 0 for all t ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And hence µ ̸∈ Ij for any j ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In this case, we have fµ(t) > 0 and gµ(t) < 0 for all t ≥ �T, which implies vµ(t) < 0 for t ∈ [ �T, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' That is, vµ changes sign exactly once on (T, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore µ ∈ Ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose, by contradiction, that there exists �T > �T such that fµ( �T) = 0 and fµ > 0 on [T, �T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, the second equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12) implies that gµ is decreasing on [T, �T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And hence, 32 gµ( �T) < gµ( �T) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, we only need to consider the situation Hµ( �T) > 0, since the condition Hµ( �T) ≤ 0 will immediately trap the solution (uµ, vµ) in the third quadrant of (u, v)- plane for t > �T, and leads us to have µ ∈ Ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the case Hµ( �T) > 0, by fµ( �T) = 0 and gµ( �T) < 0, we have gµ( �T) < − �m(m − 2) 2(m − 1) � m−1 2 cosh( �T) 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let T < T1 < T2 < �T be such that m − 1 2m cosh( �T)− 1 m−1gµ(T1) 2m m−1 − m − 2 4 gµ(T1)2 = −C0 and m − 1 2m cosh( �T)− 1 m−1gµ(T2) 2m m−1 − m − 2 4 gµ(T2)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By assuming C0 suitably small, such T1 and T2 always exist, and we can have that gµ( �T) < gµ(T2) < gµ(T1) < gµ(T2)/2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In fact, by setting F(s) = m − 1 2m cosh( �T)− 1 m−1|s| 2m m−1 − m − 2 4 |s|2, s ∈ R we have gµ(T2) is nothing but the vanishing point of F in the negative line, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=', gµ(T2) = − �m(m − 2) 2(m − 1) � m−1 2 cosh( �T) 1 2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='13) and gµ(T1) is the smallest point such that F = −C0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, use the fact Hµ(t) ≤ C0 for all t > T, we have m − 2 4 fµ(t)2 ≤ C0 − F(gµ(t)) ≤ 2C0 for t ∈ [T1, T2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, we deduce 0 < fµ(t) ≤ δ0 := � 8C0 m − 2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='14) for t ∈ [T1, T2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that F ′(gµ(T2)) = − 1 m − 1 � m m − 1 � m−1 2 �m − 2 2 � m+1 2 cosh( �T) 1 2 < 0 and F ′′(gµ(T2)) = m − 2 2 �m(m + 1) (m − 1)2 − 1 � > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By using the second equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='14), we find C0 F ′(gµ(T2)) > gµ(T2) − gµ(T1) = � T2 T1 g′ µ(t)dt ≥ − � T2 T1 �� δ2 0 + gµ(T2)2� 1 m−1δ0 + m − 2 2 δ0 � dt ≥ −Cmgµ(T2) 2 m−1δ0(T2 − T1) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='15) 33 where Cm > 0 depends only on m (since we have assumed C0 is small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' On the other hand, we have d dtHµ(t) = − 1 2m cosh(t)− 1 m−1 et − e−t et + e−t (fµ(t)2 + gµ(t)2) m m−1 ≤ − 1 2m e − e−1 e + e−1 cosh( �T)− 1 m−1gµ(T1) 2m m−1 ≤ −cm cosh( �T)− 1 m−1gµ(T2) 2m m−1 for t ∈ [T1, T2], where in the last inequality we used |gµ(T1)| > 1 2|gµ(T2)| and cm = 1 2m �1 2 � 2m m−1 e − e−1 e + e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='15), we obtain Hµ(T2) − Hµ(T1) = � T2 T1 d dtHµ(t)dt ≤ −cm cosh( �T)− 1 m−1gµ(T2) 2m m−1(T2 − T1) ≤ cm cosh( �T)− 1 m−1gµ(T2) 2m m−1C0 CmF ′(gµ(T2))gµ(T2) 2 m−1δ0 = − �Cm cosh( �T) 1 2 − 1 m−1� C0 < −C0 provided that m ≥ 3 and C0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This implies Hµ(T2) ≤ 0 reaching a contradiction, and the proof is hereby completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The next lemma provides the main properties of the sets Ak and Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For all k ∈ N ∪ {0}, we have (1) Ak is an open set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (2) if µ ∈ Ik, then there exists ε > 0 such that (µ − ε, µ + ε) ⊂ Ak ∪ Ik ∪ Ak+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (3) if Ak ̸= ∅ and is bounded, then sup Ak ∈ Ik;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4) if both Ak and Ik are bounded, set µ = sup Ik, then there exists ε > 0 such that (µ, µ + ε) ⊂ Ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (1) is quite obvious, since it comes from the continuity of the solutions (uµ, vµ) with respect to the initial datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' To see (2), we fix µ ∈ Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we have Hµ(t) → 0 as |t| → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Given C0 as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17, there exists T > 1 such that Hµ(T) < C0, uµ(T)vµ(T) > 0 and vµ changes sign k times on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The continuity of the solution (uµ, vµ) with respect to µ implies that the same holds for an initial datum ˜µ ∈ (µ − ε, µ + ε) for ε > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then the conclusion follows by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' To check (3), let us set µ = sup Ak and take a sequence {µj} ⊂ Ak such that µj ↗ µ as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' If we suppose that µ ∈ Al for some l, then (1) suggests that µj ∈ Al for j large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence we have l = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This implies µ ∈ Ak which is absurd since Ak is an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that, by the continuity property of the solutions, the corresponding vµ can change sign only a finite 34 number of times on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore we must have that µ ∈ Is for some s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By (2), we have (µ − ε, µ + ε) ⊂ As ∪ Is ∪ As+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This implies s = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Finally, to see (4), we first observe that µ = sup Ik ∈ Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, let {µj} ∈ Ik be such that µj ↗ µ as j → +∞, we have µ ̸∈ Al for any l ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This is because Al is an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, arguing similarly as in (3), we get that µ ∈ Ik as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Now, by (2), we have (µ, µ + ε) ⊂ Ak ∪ Ak+1 for some ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since we have assumed the boundedness of Ak, we find sup Ak ≤ µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thus (µ, µ + ε) ⊂ Ak+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Our next result is the boundedness property of the sets Ak and Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Ak ∪ Ik is bounded for each k ∈ N ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Before prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='19, let us do some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Denoted by ε = µ−1 > 0, we consider the following rescaling � Uε(t) = εuµ � ε 2 m−1t � , Vε(t) = εvµ � ε 2 m−1t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We find the system for (Uε, Vε) is � � � � � U ′ ε = cosh � ε 2 m−1t �− 1 m−1(U 2 ε + V 2 ε ) 1 m−1Vε − ε 2 m−1 m − 2 2 Uε −V ′ ε = cosh � ε 2 m−1t �− 1 m−1(U 2 ε + V 2 ε ) 1 m−1Uε − ε 2 m−1 m − 2 2 Vε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='16) together with the initial datum Uε(0) = Vε(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The limiting problem associated to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='16) is � U ′ 0 = (U 2 0 + V 2 0 ) 1 m−1V0 −V ′ 0 = (U 2 0 + V 2 0 ) 1 m−1U0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17) with U0(0) = V0(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' There holds (Uε, Vε) → (U0, V0) as ε → 0 uniformly on [0, T], for all T > 0, where (U0, V0) is the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' First of all, we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='16) is equivalent to � � � � � � � Uε(t) = 1 + � t 0 � cosh � ε 2 m−1s �− 1 m−1(U 2 ε + V 2 ε ) 1 m−1Vε − ε 2 m−1 m − 2 2 Uε � ds Vε(t) = 1 − � t 0 � cosh � ε 2 m−1s �− 1 m−1(U 2 ε + V 2 ε ) 1 m−1Uε − ε 2 m−1 m − 2 2 Vε � ds (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='18) and, similarly, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17) is equivalent to � � � � � � � U0(t) = 1 + � t 0 (U 2 0 + V 2 0 ) 1 m−1V0 ds, V0(t) = 1 − � t 0 (U 2 0 + V 2 0 ) 1 m−1U0 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='19) 35 The Hamiltonian energy associated to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='16) is given by Hε(t, U, V ) = −ε 2 m−1 m − 2 2 UV + m − 1 2m cosh � ε 2 m−1t �− 1 m−1(U 2 + V 2) m m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And it is easy to see that Hε is decreasing along the flow, so that Hε(t, Uε(t), Vε(t)) ≤ Hε(0, 1, 1) < m − 2 2m 2 m m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This implies that Uε(t)2 + Vε(t)2 ≤ Cm cosh � ε 2 m−1t � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='20) for some constant Cm > 0 independent of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Fix T > 0 and consider t ∈ [0, T], we have |Uε(t) − U0(t)| + |Vε(t) − V0(t)| ≤ � t 0 cosh � ε 2 m−1t �− 1 m−1 ���(U 2 ε + V 2 ε ) 1 m−1Vε − (U 2 0 + V 2 0 ) 1 m−1V0 ���ds + � t 0 cosh � ε 2 m−1t �− 1 m−1 ���(U 2 ε + V 2 ε ) 1 m−1Uε − (U 2 0 + V 2 0 ) 1 m−1U0 ���ds + � t 0 � 1 − cosh � ε 2 m−1t �− 1 m−1� (U 2 0 + V 2 0 ) 1 m−1� |U0| + |V0| � ds + Cmε 2 m−1 cosh � ε 2 m−1T � 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='21) Since the first two integrands in the right-hand-side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='21) are locally Lipschitz, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='20) and the boundedness of U0 and V0, we have |Uε(t) − U0(t)| + |Vε(t) − V0(t)| ≲ � t 0 � |Uε − U0| + |Vε − V0| � ds + ε 2 m−1 cosh � ε 2 m−1T � 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Now, using the Gronwall inequality, we have |Uε(t) − U0(t)| + |Vε(t) − V0(t)| ≲ ε 2 m−1 for t ∈ [0, T], proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose the contrary, that Ak ∪ Ik is unbounded for some k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then we can find a sequence µj ∈ Ak ∪ Ik such that µj → +∞ as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By taking εj = µ−1 j , Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='20 implies that Vεj → V0 uniformly on [0, T] as j → ∞, for any fixed T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Notice that the solution (U0, V0) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17) can be explicitly formulated: U0(t) = √ 2 sin � 2 1 m−1t + π 4 � and V0(t) = √ 2 cos � 2 1 m−1t + π 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We can take T > 0 large enough so that V0 changes sign k +1 times on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Then, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='20, we have Vεj changes k + 1 times on [0, T] for all large j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' However, due to µj ∈ Ak ∪ Ik and Vεj(t) = εjvµj � ε2/(m−1) j t � , we have Vεj should change sign only k times on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And thus, we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 36 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let µ0 = sup A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='18, we have µ0 ∈ I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let now ν0 = sup I0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Applying Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='19 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='18, we have (ν0, ν0 + ε0) ⊂ A1 for some ε0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thus A1 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Let µ1 = sup A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We have µ1 > ν0 ≥ µ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' and so, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='18, µ1 ∈ I1, and then ν1 = sup I1 ∈ I1 and (ν1, ν1 + ε1) ⊂ A2, for some ε1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Iterating this argument, we construct two increasing sequences {µj} and {νj}, νj+1 ≥ µj+1 > νj ≥ µj, with µj ∈ Ij and (νj, νj + εj) ⊂ Aj+1, for some {εj} ⊂ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Next, we will show that µj → +∞ as j → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Suppose, by contradiction, that µj is bounded and µj → µ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' We can see that Hµ∞(t) > 0 for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Indeed, if Hµ∞(t0) ≤ 0 for some finite t0 > 0, it follows that (uµ∞(t), vµ∞(t)) will be trapped in one of the connected components of {(u, v) ∈ R2 : H(t, u, v < 0)}, for all t > t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='8 implies that vµ∞ changes sign a finite number of times in [0, t0], we have µ∞ ∈ Ak0 for some k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This contradicts the definition of µ∞ as Ak0 is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Moreover, vµ∞ must change sign infinite many times on (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Using the facts Hµ∞ is decreasing on (0, +∞) and bounded from below, we have H′ µ∞ ∈ L1(0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In particular, cosh(·)− 1 m−1(u2 µ∞ + v2 µ∞) m m−1 ∈ L1(0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='22) Multiplying by vµ∞ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' uµ∞) the equations in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11), we have � � � � � vµ∞u′ µ∞ = cosh(t)− 1 m−1(u2 µ∞ + v2 µ∞) 1 m−1v2 µ∞ − m − 2 2 uµ∞vµ∞, −uµ∞v′ µ∞ = cosh(t)− 1 m−1(u2 µ∞ + v2 µ∞) 1 m−1u2 µ∞ − m − 2 2 uµ∞vµ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This implies vµ∞u′ µ∞ + uµ∞v′ µ∞ = cosh(t)− 1 m−1(u2 µ∞ + v2 µ∞) 1 m−1(v2 µ∞ − u2 µ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Hence we have (uµ∞vµ∞)′ ∈ L1(0, +∞), which shows that uµ∞(t)vµ∞(t) → C∞ ∈ R as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Since vµ∞(t) changes sign infinitely many times as t → ∞, we have C∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='22), implies that Hµ∞(t) → 0 as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Therefore, one may take T > 0 sufficiently large such that Hµ∞(T) < C0 (where C0 > 0 is given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17), uµ∞(T)vµ∞(T) > 0 and vµ∞ changes sign kT times on [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='17, we have µ∞ ∈ AkT ∪ IkT ∪ AkT +1, reaching another contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Finally, in order to see that lim inft→+∞ |uµ(t)| + |vµ(t)| = +∞ for µ ∈ Ak, let us consider two possibilities: Hµ(t) → −∞ and Hµ(t) → H∞ ∈ (−∞, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the first case, we must have that uµ(t)vµ(t) → +∞ as t → +∞, which directly implies the assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In the latter case, we deduce that uµ(t)vµ(t) → C > 0 as t → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' And hence cosh(·)− 1 m−1(u2 µ + v2 µ) m m−1 converges to a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This shows that |uµ(t)| + |vµ(t)| grows as cosh(t)1/2m for t large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The upper bound of (uµ, vµ), µ ∈ Ak follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11, and the exponential decay of (uµ, vµ), µ ∈ Ik, follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Thus, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='5 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The numerical simulations performed on system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11) indicate the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' For each k ∈ N∪{0}, starting from µ larger than some µ∗ k ∈ Ak, the solution orbits will make a circle around a particular point (in either the first quadrant or the third quadrant) before going to 37 infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' As µ grows, the circle is becoming larger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' and once the circle touches the origin, we will have a homoclinic solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='11), which implies µ ∈ Ik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The set Ik seems to have only one point, and hence Ak are just open intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' In particular, we conjecture that ∪k≥0Ik is simply a countable set of discrete points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' This is illustrated in the following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' 3, where numerical experiments are performed on a 3-dimensional system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The first row shows the solution orbits (uµ, vµ) on R with three different initial datum in A0, and specifically µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' The second and third rows show the solutions with initial datum µ ∈ A1 and A2, respectively Figure 3: Unbounded trajectories with initial datum µ ∈ Ak, k = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Acknowledgements Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' is partly supported by NSF grant DMS 2154219, ” Regularity vs singularity formation in elliptic and parabolic equations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Abbondandolo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content=' Molina, Index estimates for strongly indefinite functionals, periodic orbits and homoclinic solutions of first order Hamiltonian systems, Cal.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='edu TIAN XU CENTER FOR APPLIED MATHEMATICS, TIANJIN UNIVERSITY, 300072, TIANJIN, CHINA xutian@amss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE2T4oBgHgl3EQfPQZw/content/2301.03757v1.pdf'} diff --git a/8dE4T4oBgHgl3EQfdQww/content/tmp_files/2301.05089v1.pdf.txt b/8dE4T4oBgHgl3EQfdQww/content/tmp_files/2301.05089v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9227f79e82f333b6811c64ed22676d4959f09afd --- /dev/null +++ b/8dE4T4oBgHgl3EQfdQww/content/tmp_files/2301.05089v1.pdf.txt @@ -0,0 +1,3067 @@ +1 +Approximate Information States for Worst-Case +Control and Learning in Uncertain Systems +Aditya Dave, Student Member, IEEE, Nishanth Venkatesh, Student Member, IEEE, and Andreas A. Malikopoulos, +Senior Member, IEEE +Abstract—In this paper, we investigate discrete-time decision- +making problems in uncertain systems with partially observed +states. We consider a non-stochastic model, where uncontrolled +disturbances acting on the system take values in bounded sets +with unknown distributions. We present a general framework +for decision-making in such problems by developing the notions +of information states and approximate information states. In +our definition of an information state, we introduce conditions +to identify for an uncertain variable sufficient to construct a +dynamic program (DP) that computes an optimal strategy. We +show that many information states from the literature on worst- +case control actions, e.g., the conditional range, are examples of +our more general definition. Next, we relax these conditions to +define approximate information states using only output variables, +which can be learned from output data without knowledge of +system dynamics. We use this notion to formulate an approximate +DP that yields a strategy with a bounded performance loss. +Finally, we illustrate the application of our results in control +and reinforcement learning using numerical examples. +Index Terms—Uncertain systems, worst-case control, approxi- +mate dynamic programming, offline reinforcement learning +I. INTRODUCTION +Decision-making under incomplete information is a funda- +mental problem in modern engineering applications involving +cyber-physical systems [1], e.g., connected and automated +vehicles [2], social media platforms [3], and robot swarms [4]. +In such applications, an agent is often required to sequentially +select control inputs to a dynamic system using only partial +observations at each instance of time, while simultaneously +accounting for uncontrolled disturbances that can interfere +with the system’s evolution. The most common modeling +paradigm for such decision-making problems is the stochastic +approach, where all disturbances to the system are considered +to be random variables with known distributions, and the +agent aims to select a decision-making strategy that minimizes +the expected incurred cost [5]. Stochastic models have been +utilized for problems in both control theory [6]–[13] and +reinforcement learning [14]–[18]. A decision-making strategy +derived using the stochastic approach performs optimally on +average across numerous operations of the system. However, +this performance degrades rapidly when there is a mismatch +between the distribution on disturbances considered in model- +ing and the realizations encountered during implementation +This research was supported by NSF under Grants CNS-2149520 and +CMMI-2219761. +The authors are with the Department of Mechanical Engineering, University +of Delaware, Newark, DE 19716 USA (email: adidave@udel.edu; +nish@udel.edu; andreas@udel.edu). +[19]. Furthermore, many safety-critical applications require +guarantees on the agent’s performance during each operation +[20]. Thus, in such applications it is inadequate to measure +performance using the expected cost. +The non-stochastic approach is an alternate modeling +paradigm for safety-critical systems, where all disturbances +are considered to belong to known sets with unknown distri- +butions. The agent aims to select a decision-making strategy +that minimizes the worst-case incurred cost across a finite +time horizon [21]. Because this approach focuses on robust- +ness against worst-case realizations of the disturbances, the +resulting strategy yields more conservative decisions than the +stochastic approach. At the expense of average performance, +this strategy provides concrete guarantees on the worst-case +performance during each operation of the system. Thus, this +approach has been widely applied to systems under attack +from an adversary, e.g., cyber-security [22] or cyber-physical +systems [23], and systems where a single failure can be +damaging, e.g., water reservoirs [24], or power systems [25]. +In this paper, we propose a framework for non-stochastic +decision-making using only partial observations in a dynamic +system. When the system’s dynamics are known to the agent, +this problem falls under the purview of control theory [26]. +However, many applications involve decision-making with an +incomplete knowledge of the dynamics as, e.g., automated +driving in mixed traffic [27] and human-robot coordination +[28], or decision-making without a reliable state-space model, +e.g., medical dead-end identification [29]. These restrictions +typically lead to formulating a reinforcement learning problem +[30], [31]. To account for both of these potential cases, we +formulate our problem using only output variables without +assuming a known state-space model. In our exposition, we +present rigorous definitions for the notions of information +states and approximate information states. Using these no- +tions, a surrogate state-space model can be constructed from +output variables. This surrogate model can be used to for- +mulate a control problem with full state observation, whose +solution yields either an optimal, or an approximate strategy +of the original problem. In reinforcement learning problems, +the surrogate model can be learned from output data. For +perfectly observed states, the agent can derive a decision- +making strategy using standard techniques [32]–[34]. +A. Related Work +1) Control theory: There have been numerous research +efforts in control theory to study dynamic decision-making +arXiv:2301.05089v1 [eess.SY] 12 Jan 2023 + +2 +problems given the system dynamics. For both stochastic +and non-stochastic models, an agent can derive an optimal +decision-making strategy offline using a dynamic program- +ming (DP) decomposition of the problem [35], [36]. For +systems with perfectly observed states, it is known that, at +each instance of time, the agent’s optimal action is simply a +function of the state. Using this property in a DP facilitates +the efficient computation of an optimal control strategy [37], +[38]. In contrast, for systems with partially observed states, +any optimal action is generally a function of the agent’s entire +memory of past observations and actions, which grows in +size with time [39]. Subsequently, the domain of the optimal +control strategy grows in size with time, and the corresponding +DP decomposition of the problem requires a large number +of computations for long time horizons [40]. This concern +is alleviated using an information state to construct a DP +decomposition instead of the memory [41]. +The most commonly used information state in stochastic +control is the belief state, i.e., a distribution on the state +space conditioned on the agent’s memory [42]–[44]. A general +notion of information states for stochastic control was recently +defined in [45]. For non-stochastic control problems, the DP +decomposition has been simplified using two well known +information states: (1) the conditional range, which is the +set of feasible states at any time consistent with the agent’s +memory [46] and can be used in both terminal cost [47]– +[51] and instantaneous cost problems [52]–[54]; and (2) the +maximum cost-to-come, which is the maximum accrued cost +at any time for each state in the conditional range [55] and +can be used in additive cost problems [56]–[58]. +A general notion of an information state for non-stochastic +terminal cost problems was presented in [59]. Information +states have also been derived for mixed problems considering +both stochastic and non-stochastic objectives in [60] and for +robust stochastic formulations in [61], [62]. The advantage +of using information states is that in many applications they +do not grow in size with time. Thus, they generally yield +a more computationally efficient DP decomposition than the +entire memory. However, in problems with large state spaces, +utilizing information states may not sufficiently simplify the +DP to be practical [63], [64]. +2) Reinforcement learning: The literature on reinforcement +learning is concerned with decision-making when the agent +may not have prior knowledge of the system’s dynamics [65]. +For systems with perfectly observed states, these problems +have been addressed using a variety of approaches [66]. In +the stochastic formulation, both model-based [67], [68] and +model-free approaches [69] have been utilized. In the non- +stochastic formulation, the worst-case reinforcement learning +problem was formulated and analyzed in [70]. Worst-case Q- +learning was proposed for reinforcement learning problems +in [71]–[74] and extended to problems with output-feedback +and partially known dynamics in [75]. Actor-critic methods +[76] and model-based off-policy learning approaches [77] have +also been developed for robust control. Alternate approaches +using online adaptive algorithms were proposed in [78], [79]. +However, in general, reinforcement learning is challenging +when the agent can only access partial observations, since +without knowledge of the system dynamics, the information +state must be learned from data [80]. +In the stochastic formulation, the notion of approximate +information states was presented in [81] to address the chal- +lenges of control and learning with partial observations. Ap- +proximate information states can improve the computational +tractability of control problems with large state spaces at the +cost of a bounded loss in performance [82]. The explicit +performance bounds of a finite-memory based approximate +information state were derived in [83]. In reinforcement learn- +ing, approximate information states can be learned from output +data and function as surrogate states to compute approximately +optimal strategies, whose performance has been empirically +validated in robotics [84] and medical care [85]. To the best of +our knowledge, no general theory of approximate information +states has yet been developed for non-stochastic formulations. +B. Contributions and Organization +In this paper, we develop a non-stochastic theory of approx- +imate information states for both instantaneous and terminal +cost problems, which can facilitate computationally efficient +control, and provide a principled approach to reinforcement +learning using partial observations. The contributions of this +paper are: (1) the introduction of a general notion of in- +formation states (Definition 3) which yields an optimal DP +decomposition for worst-case control (Theorem 1) and show +that many standard results in the literature are special cases +(Subsection III-C); (2) the introduction of the notion of ap- +proximate information states that can either be constructed +from output variables or learned from output data (Definition +4); (3) the formulation of an approximate DP (Theorem 3) +which computes a control strategy with a bounded loss of +optimality (Theorem 4); (4) the exposition of examples of +approximate information states (Subsection IV-D) along with +theoretical approximation bounds (Theorems 5 - 6); and (5) +the illustration of the approach in both control and learning +problems using numerical examples (Subsection V). +Note that while our theory shares conceptual similarities +to the theory of approximate information states for stochas- +tic problems in [81], our focus on non-stochastic problems +necessitates the use of a distinct mathematical framework +of uncertain variables [86] with set-valued uncertainties. We +bound the worst-case approximation loss rather than the ex- +pected loss. Note that we reported preliminary results for +terminal-cost control problems with finite feasible sets in +[59]. This paper extends the preliminary work as follows: (1) +we consider worst-case instantaneous cost problems which +subsume terminal cost problems; (2) we allow all variables +to take values in continuous spaces; and (3) we illustrate the +application of our results to a reinforcement learning problem. +The remainder of the paper proceeds as follows. In Section +II, we present our model and problem formulation. In Section +III, we define the notion of information states and prove the +optimality of the corresponding DP decomposition. In Section +IV, we present the notion of approximate information states, +a resulting approximate DP, and theoretical bounds on the +approximation loss. In Section V, we present a numerical + +3 +examples to illustrate the application of our results. In Section +VI, we draw concluding remarks and discuss ongoing work. +II. MODEL +A. Preliminaries +1) Uncertain Variables: In this paper, we utilize the math- +ematical framework for uncertain variables from [86]. An +uncertain variable is a non-stochastic analogue of a random +variable with set-valued uncertainty. For a sample space Ω +and a set X, an uncertain variable is a mapping X : Ω → X. +For any ω ∈ Ω, it has the realization X(ω) = x ∈ X. The +marginal range of X is the set [[X]] := {X(ω) | ω ∈ Ω}. For +two uncertain variables X ∈ X and Y ∈ Y, their joint range is +[[X, Y ]] := { +� +X(ω), Y (ω) +� +| ω ∈ Ω}. For a given realization y +of Y , the conditional range of X is [[X|y]] := {X(ω) | Y (ω) += y, ω ∈ Ω} and, generally, [[X|Y ]] := {[[X|y]] | y ∈ [[Y ]]}. +2) Hausdorff Distance: Consider that the feasible sets X, Y +are nonempty subsets of a metric space (S, η), where η(x, y) +is the distance between any x ∈ X and y ∈ Y. Then, we +define a distance between the two sets as follows. +Definition 1. The Hausdorff distance between X and Y is +H(X, Y) := max +� +sup +x∈X +inf +y∈Y +η(x, y), sup +y∈Y +inf +x∈X +η(x, y) +� +. +(1) +When the two sets X, Y are bounded, the Hausdorff distance +in (1) constitutes a pseudo-metric, i.e., H(X, Y) = 0 if and +only if closure(X) = closure(Y) [87, Appendix]. When both +X, Y are compact, the Hausdorff distance is a metric, i.e., +H(X, Y) = 0 if and only if X = Y [88, Chapter 1.12]. In +both cases, the distance H satisfies the triangle inequality. +3) L-invertible Functions: Consider a function f : X → Y. +For any y ∈ Y, the pre-image of the function is f −1(y) = +� +x ∈ X | f(x) = y +� +. Then, we use the Hausdorff distance to +define the notion of an L-invertible function as follows. +Definition 2. A function f : X → Y is called L-invertible if +there exists a constant Lf −1 ∈R≥0 such that for all y1, y2 ∈ Y: +H +� +f −1(y1), f −1(y2) +� +≤ Lf −1 · η(y1, y2). +(2) +For uncertain variables X ∈ X and Y ∈ Y such that Y = +f(X), the pre-image of f given a realization y ∈ [[Y ]] equals +the conditional range [[X|y]], i.e., f −1(y) = [[X|y]]. Thus, if f +is L-invertible, we equivalently state that for all y1, y2 ∈ [[Y ]]: +H +� +[[X|y1]], [[X|y2]] +� +≤ LX|Y · η(y1, y2), +(3) +where LX|Y = Lf −1. +B. Problem Formulation +We consider an agent which seeks to control the trajectory +of an uncertain system by selecting actions over T ∈ N +discrete time steps. At each time t = 0, . . . , T, the agent +receives an observation from the system, denoted by the +uncertain variable Yt ∈ Yt, and generates a control action +denoted by the uncertain variable Ut ∈ Ut. After generating +the action at each t, the agent incurs a cost denoted by the +uncertain variable Ct ∈ Ct ⊂ R≥0. To account for the case that +the agent may have no knowledge of a state-space model, we +describe the system dynamics using an input-output model, +as follows. At each t = 0, . . . , T, the system receives two +inputs: the control action Ut, and an uncontrolled disturbance +denoted by the uncertain variable Wt ∈ Wt. We consider that +the uncontrolled disturbances {Wt : t = 0, . . . , T} constitute a +sequence of independent uncertain variables. After receiving +the inputs at each t = 0, . . . , T, the system generates two +outputs: +Yt+1 = ht+1(W0:t, U0:t), +(4) +Ct = dt(W0:t, U0:t), +(5) +for some observation function ht+1 : �t +ℓ=0 Wℓ × �t +ℓ=0 Uℓ → +Yt+1 and cost function dt : �t +ℓ=0 Wℓ × �t +ℓ=0 Uℓ → Ct. The +initial observation is generated as Y0 = h0(W0). +The agent has perfect recall of the history of observations +and control actions. The memory of the agent at each t is +denoted by the uncertain variable Mt := (Y0:t, U0:t−1), which +takes values in the set Mt := �t +ℓ=0 Yℓ × �t−1 +ℓ=0 Uℓ. The agent +uses the memory Mt and a control law gt : Mt → Ut at +each t to generate the action Ut = gt(Mt). We denote the +control strategy by g := (g0, . . . , gT ) and the set of all feasible +control strategies by G. The performance of a strategy g ∈ G +is measured by the worst-case or maximum instantaneous cost +J (g) := +max +t=0,...,T +sup +w0:t∈[[W0:t]] +Ct. +(6) +Problem 1. The optimization problem of the agent is to derive +the control strategy g ∈ G such that infg∈G J (g), given the +marginal ranges {[[Ut]], [[Wt]], [[Ct]], [[Yt]] | t = 0, . . . , T} +and the functions {ht, dt | t = 0, . . . , T}. +If there exists a strategy g∗ ∈ G that achieves the optimal +performance in Problem 1, i.e., g∗ = arg ming∈G J (g), we +refer to it as an optimal control strategy for Problem 1. Our aim +is to tractably compute an optimal strategy if one exists. In our +modeling framework, we impose the following assumptions: +Assumption 1. We consider that the sets {Ut, Wt, Yt | t = +0, . . . , T} and {Ct | t = 0, . . . , T} are all bounded subsets of +a metric space (S, η) and R≥0, respectively. +Assumption 1 allows for both continuous and finite valued +feasible sets, while ensuring that the marginal range of each +uncertain variable in the problem formulation is also bounded. +Assumption 2. The observation functions {ht | t = 0, . . . , T} +of the system are both Lipschitz and L-invertible, whereas the +cost functions {dt | t = 0, . . . , T} are Lipschitz continuous. +Assumption 2 is satisfied by a large class of observation +functions, including: (1) all functions with compact domains +and finite co-domains; and (2) bi-Lipschitz functions, like +linear functions, with compact domains and compact co- +domains (see Appendix A). We will require both assumptions +in Section IV when deriving the main results. +Remark 1. In our exposition, we also consider a special case +of (6), called the maximum terminal cost criterion, given by +J tm(g) := +sup +w0:T ∈[[W0:T ]] +CT . +(7) + +4 +In addition to the general results for Problem 1, we often +present results specifically for systems which utilize (7) as +the performance measure. This serves two purposes: (1) the +results are often easier to interpret for a terminal cost problem; +and (2) these results can be extended to additive cost problems. +We explicitly present this extension in Subsection III-C. +Remark 2. We derive our results for Problem 1 with known +dynamics. However, our main results in Section IV can also +be used in learning problems with unknown dynamics. We +illustrate this application with an example in Subsection V-B. +III. DYNAMIC PROGRAMS AND INFORMATION STATES +In this section, we first present a memory-based DP decom- +position for Problem 1 which computes the optimal value of +the performance criterion 6. This will serve as a reference to +analyze subsequent DPs in the paper. Then, we highlight the +DP’s computational challenges and present information states +in Subsections III-A and III-B to alleviate them. In Subsection +III-C we present examples of information states. +To arrive at the memory-based DP, we construct a “new” +perfectly observed system whose state at each t = 0, . . . , T +is the memory Mt, which evolves as Mt+1 = (Mt, Ut, Yt+1). +Furthermore, for given realizations mt ∈ [[Mt]] and ut ∈ +[[Ut]], the maximum incurred cost at time t can be written as +sup +w0:t∈[[W0:t]] +Ct = +sup +ct∈[[Ct]]gct = +sup +mt,ut∈[[Mt,Ut]]g +sup +ct∈[[Ct|mt,ut]]gct, +for all t += +0, . . . , T, where [[Ct]]g, [[Mt, Ut]]g +and +[[Ct|mt, ut]]g are the respective marginal ranges and the con- +ditional range induced by strategy g. Recall that mt = (y0:t, +u0:t−1) and thus, we can expand the conditional range as +[[Ct|mt, ut]]g = +� +ct ∈ Ct +�� ∃ w0:t ∈ [[W0:t]] such that +ct = dt(w0:t, u0:t), yℓ = ht(w0:ℓ, u0:ℓ−1), ∀ℓ = 0, . . . , t +� += [[Ct|mt, ut]], +(8) +which shows that [[Ct|mt, ut]]g is independent of the choice of +strategy g, hence we can drop g. Next, we define et(mt, ut) := +supct∈[[Ct|mt,ut]] ct, independent of g, and state that +sup +mt,ut∈[[Mt,Ut]]g +sup +ct∈[[Ct|mt,ut]]gct = +sup +mt,ut∈[[Mt,Ut]]g et(mt, ut) += +sup +w0:t∈[[W0:t]] +et(Mt, Ut), +(9) +where, in the second equality, note that the marginal range of +external disturbances [[W0:t]] is independent of the strategy +g. Since et(Mt, Ut) is a function of the new state Mt and +control action Ut, it serves as an incurred cost at each +t = 0, . . . , T in our new perfectly observed system [53]. +The new instantaneous performance criterion is E(g) := +supt=0,...,T supw0:t∈[[W0:t]] et(Mt, Ut) and from (9), E(g) = +J (g) for any g. Subsequently, any strategy which achieves the +optimal performance in the new system is optimal for Problem +1. If such an optimal strategy exists, we can compute it using +a standard DP for perfectly observed systems, as follows. For +all t = 0, . . . , T, for each mt ∈ [[Mt]] and ut ∈ [[Ut]] we +recursively define the value functions +Qt(mt, ut) := max +� +sup +ct∈[[Ct|mt,ut]] +ct, +sup +mt+1∈[[Mt+1|mt,ut]] +Vt+1(mt+1) +� +, +(10) +Vt(mt) := +inf +ut∈[[Ut]] +Qt(mt, ut), +(11) +where VT +1(mT +1) := 0, identically. We define the ex- +tra value function VT +1 to ensure that the right hand side +(RHS) of (10) is well defined at time T. Then, we can +show using standard arguments [48], [53] that the optimal +value of Problem 1 is infg∈G J (g) = supm0∈[[M0]] V0(m0). +Furthermore, at any t = 0, . . . , T, if there exists an action +u∗ +t ∈ [[Ut]] which achieves the infimum in the RHS of (11), +then g∗ +t (mt) := arg minut∈[[Ut]] Qt(mt, ut) gives an optimal +control law at time t. If the infimum is achieved at each t, the +control strategy g∗ = (g∗ +0, . . . , g∗ +T ) is optimal for this system +and Problem 1. +Remark 3. The DP (10) - (11) can be specialized to the +terminal cost criterion (7) by defining for all t = 0, . . . , T −1, +Qtm +t (mt, ut) := +sup +mt+1∈[[Mt+1|mt,ut]] +V tm +t+1(mt+1), +(12) +V tm +t (mt) := +inf +ut∈[[Ut]] +Qtm +t (mt, ut), +(13) +where +Qtm +T (mT , uT ) +:= +supcT ∈[[CT |mT ,uT ]] cT +and +V tm +T (mT ) := infuT ∈[[Ut]] Qtm +T (mT , uT ). We will use this +terminal cost DP to simplify the exposition in Section IV. +Remark 4. A valid argument referring to the minimum of the +RHS of (11) at each t = 0, . . . , T is both a necessary and suf- +ficient condition to ensure the existence of an optimal control +strategy in Problem 1 [48], [89]. Consider that marginal ranges +of all uncertain variables are compact rather than just bounded. +From Assumption 2, the observation and cost functions at +each t are Lipschitz. Using these properties in (12) - (13), +we can show that the value functions are continuous and +the conditional ranges are compact for all t, which implies +that the minimum is achieved in the RHS of (11). Thus, +compactness of all marginal ranges and Assumptions 1 - 2 +constitute sufficient conditions for existence of an optimal +solution to Problem 1, which is consistent with the conditions +given in [90]. However, we continue using sup and inf in our +exposition since we use only Assumptions 1 - 2 to establish +our results without assuming compactness. +Remark 5. In the RHS of (11) at each t, we are required +to solve an optimization for each mt ∈ [[Mt]]. This is +computationally challenging for longer horizons as the size +of the set [[Mt]] increases with time t with addition of new +data. This concern motivates our search for an alternate DP +decomposition which can derive an optimal control strategy +while potentially achieving more favourable computational +properties. We present such a DP decomposition in Subsection +III-A by identifying an uncertain variable, called an informa- +tion state, which can be used to generate an optimal control +action at each time step instead of the memory. +A. Information States +In this subsection, we define information states for partially +observed uncertain systems, use them in a DP decomposition, +and prove it yields the optimal value for Problem 1. + +5 +Definition 3. An information state for Problem 1 at each t = +0, . . . , T is an uncertain variable Πt = σt(Mt) taking values in +a bounded set Pt and generated by a function σt : Mt → Pt. +Furthermore, for all t = 0, . . . , T, and for all mt ∈ [[Mt]] and +ut ∈ [[Ut]], it satisfies the following properties: +1) Sufficient to evaluate cost: +sup +ct∈[[Ct|mt,ut]] +ct = +sup +ct∈[[Ct|σt(mt),ut]] +ct. +(14) +2) Sufficient to predict itself: +[[Πt+1|mt, ut]] = [[Πt+1|σt(mt), ut]]. +(15) +We can use the information states from Definition 3 directly +in a DP, as follows. For all t = 0, . . . , T, for all πt ∈ [[Πt]] +and ut ∈ [[Ut]], we recursively define the value functions +¯Qt(πt, ut) := max +� +sup +ct∈[[Ct|πt,ut]] +ct, +sup +πt+1∈[[Πt+1|πt,ut]] +¯Vt+1(πt+1) +� +, +(16) +¯Vt(πt) := +inf +ut∈[[Ut]] +¯Qt(πt, ut), +(17) +where ¯VT +1(πT +1) := 0 identically. If the minimum in the +RHS of (17) exists at each t = 0, . . . , T, then this DP yields a +control law at time t as ¯g∗ +t (πt) := arg minut∈[[Ut]] ¯Qt(πt, ut). +Next, we prove that the DP (16) - (17) computes the same +value as the optimal DP (10) - (11). +Theorem 1. Let Πt = σt(Mt) be an information state at any +t. Then, for all t, and for all mt ∈ [[Mt]] and ut ∈ [[Ut]], +Qt(mt, ut)= ¯Qt +� +σt(mt), ut +� +and Vt(mt)= ¯Vt +� +σt(mt) +� +. (18) +Proof. Let mt ∈ [[Mt]] and ut ∈ [[Ut]] be given realizations +of Mt and Ut, respectively, for all t += +0, . . . , T. We +prove +the +result +by +mathematical +induction +starting +at +the last time step. At time T + 1, (18) holds trivially +because +VT +1(mT +1) += +¯VT +1(σT +1(mT +1)) += +0. +This forms the basis of our induction. Next, for any +t = 0, . . . , T, we consider the induction hypothesis that +Vt+1(mt+1) = +¯Vt+1(σt+1(mt+1)). Given the hypothesis, +we first prove that Qt(mt, ut) += +¯Qt(σt(mt), ut) by +comparing the RHS of (10) to the RHS of (16) term +by term. The first terms are equal by direct application +of (14) from Definition 3. Next, we use the induction +hypothesis +for +the +second +term +in +the +RHS +of +(10), +to +state +that +supmt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) += +supmt+1∈[[Mt+1|mt,ut]] ¯Vt+1(σt+1(mt+1)) += +supσt+1(mt+1)∈[[Πt+1|σt(mt),ut]] ¯Vt+1(σt+1(mt+1)), where, in +the second equality, we use the fact that [[Πt+1|mt, ut]] = +� +σt+1(mt+1) ∈ Pt+1 +��mt+1 ∈ [[Mt+1|mt, ut]] +� +and (15) +from Definition 3. This establishes that the second term in +the RHS of (10) equals the second term in the RHS of (16) +and subsequently, that given the induction hypothesis for +time t + 1, we have Qt(mt, ut) = +¯Qt(σt(mt), ut). Next, +we minimize both sides of the equality with respect to +ut ∈ [[Ut]], and use the definitions of the value functions in +(11) and (17) to write that Vt(mt) = infut∈Ut Qt(mt, ut) += infut∈Ut ¯Qt +� +σt(mt), ut +� += Vt +� +σt(mt) +� +, which proves the +induction hypothesis at time t. Thus, starting at time T + 1, +the result follows for all t using mathematical induction. +Theorem 1 implies that (16) - (17) is an optimal DP decom- +position for Problem 1, i.e., if an optimal strategy exists for +this DP, it yields an optimal solution to Problem 1 as follows. +Consider a control strategy ¯g∗ = (¯g∗ +0, . . . , ¯g∗ +T ) computed using +(16) - (17). We can construct a corresponding memory-based +strategy g = (g0, . . . , gT ) by defining gt(mt) := ¯g∗ +t (σt(mt)) +for all mt ∈ [[Mt]] and t = 0, . . . , T. Then, using Theorem 1, +we conclude that g achieves the infimum value at each t and +thus, constitutes an optimal solution to Problem 1. +Remark 6. In practice, using an information state to construct +the DP decomposition is useful computationally only if, for +most time steps in t = 0, . . . , T, either the value functions in +(16) - (17) have useful properties like concavity, or the set Pt is +smaller than Mt for some measure of size. Potentially useful +measures of sizes for sets include the number of elements, set +diameter, and set dimension. We present some examples of +information states for different systems in Subsection III-C. +B. Alternate Characterization of Information States +When exploring whether an uncertain variable is a valid +candidate to be considered an information state, it may be +difficult to verify the second property (15) in Definition 3. In +this subsection, we present two stronger conditions to replace +(15). Specifically, at each t = 0, . . . , T, to establish that Πt = +σt(Mt) is a valid information state, it is sufficient to satisfy +the following conditions instead of (15): +1) State-like evolution: There exists a function ¯ft : Pt × +Ut × Yt+1 → Pt+1, independent of the strategy g, such that +Πt+1 = ¯ft(Πt, Ut, Yt+1). +(19) +2) Sufficient to predict observations: For all mt ∈ Mt and +ut ∈ Ut, +[[Yt+1|mt, ut]] = [[Yt+1|σt(mt), ut]]. +(20) +Next, we prove that these two conditions, in addition to (14) +from Definition 3 are sufficient to identify an information state. +Lemma 1. For all t = 0, . . . , T, if an uncertain variable +Πt = σt(Mt) satisfies (19) - (20), it also satisfies (15). +Proof. For all t = 0, . . . , T and mt ∈ Mt, suppose that πt = +σt(mt) satisfy (19) - (20). Then, we substitute (19) into the +left hand side (LHS) of (15) to state that +[[Πt+1|mt, ut]] = [[ ¯ft(σt(mt), ut, Yt+1) | mt, ut]] += +� ¯ft(σt(mt), ut, yt+1) +�� yt+1 ∈ [[Yt+1|mt, ut]] +� +, +(21) +where, in the second equality, we write the conditional +range as a set. Next, using (20) on the range of ob- +servations +in +the +conditioning +of +(21), +we +can +state +that +� ¯ft(σt(mt), ut, yt+1) +�� yt+1 +∈ +[[Yt+1|mt, ut]] +� += +� ¯ft(σt(mt), ut, yt+1) +�� yt+1 +∈ +[[Yt+1|σt(mt), ut]] +� += +[[ ¯ft(σt(mt), ut, Yt+1) | σt(mt), ut]] = [[Πt+1|σt(mt), ut]], +which is equal to the RHS of (15). + +6 +C. Examples of Information States +In this subsection, we present examples of information +states which satisfy the conditions in Definition 3 for systems +with a given state-space model to describe their evolution. +At each t = 0, . . . , T, let Xt be a known set of feasible +states and let the system’s state be denoted by an uncertain +variable Xt +∈ Xt. The agent’s observation is given by +Yt = ht(Xt, Nt), where Nt ∈ Nt is a noise in observation, +and the agent incurs a cost Ct = dt(Xt, Ut) when they +implement an action Ut ∈ Ut. Starting at X0 ∈ X0, the +state evolution is given by Xt+1 = ft(Xt, Ut, Wt) for all t. +Each uncertain variable in {X0, Wt, Nt | t = 0, . . . , T} is +independent of all other uncertain variables in that set. Next, +we present information states for different cases which may +offer computational advantages over using the entire memory: +1) Systems with perfectly observed states: Consider that +Yt = Xt for all t = 0, . . . , T. Then, an information state +at each t is Πt = Xt, i.e., the state itself [48]. It takes values +in the set Xt and satisfies (14) - (15) for all t. Note that +it is always computationally advantageous to construct a DP +decomposition using the state at each time step instead of the +entire memory of the agent. +2) Systems with partially observed states: Generally in +a partially observed system with a known state space, +an information state at each t += +0, . . . , T is the con- +ditional range Πt += +[[Xt|Mt]], which is a set-valued +uncertain variable [53]. Explicitly, for a given realization +of the memory mt +∈ +Mt at time t, the conditional +range takes the realization Pt +:= +� +xt +∈ Xt +�� ∃x0 +∈ +X0, w0:t−1 ∈ �t−1 +ℓ=0 Wℓ, n0:t ∈ �t +ℓ=0 Nℓ such that yt = +ht(xt, nt), xℓ+1 = fℓ(xℓ, uℓ, wℓ), yℓ = hℓ(xℓ, nℓ) for all ℓ = +0, . . . , t−1 +� +. We denote the realization by Pt instead of πt to +highlight that it is a set. To establish that the conditional range +is a valid information state, it is easier to verify the alternate +conditions (19) and (20) instead of property (15) in Definition +3. Generally, it is computationally advantageous to construct a +DP decomposition using the conditional range instead of the +memory for systems with longer time horizons. +3) Systems with additive costs: Consider a system with +partially observed states with an additive performance cri- +terion J ad(g) := supx0,w0:T ,n0:T +�T +t=0 dt(Xt, Ut). We can +construct a DP and an information state for an additive cost +problem by recasting it as a terminal cost problem [48]. +At t = 0, we define A0 := 0 and for all t = 1, . . . , T, +we recursively define an uncertain variable At ∈ At as +At := At−1 + dt−1(Xt−1, Ut−1). Note that At tracks the +cost incurred by the system up to time t, i.e., before the +action Ut has been implemented. Then, at each t, we consider +an augmented state for the system, St = (Xt, At) and note +that it evolves as St+1 = +� +ft(Xt, Ut, Wt), At + dt(Xt, Ut) +� +. +Furthermore, this augmentation yields a terminal cost problem +with the cost AT + cT (XT , UT ). Thus, we can derive an +optimal control strategy using the terminal cost DP and, as +in case 2, an information state at each t is the conditional +range Πt = [[Xt, At|Mt]]. Generally, this information state is +useful for systems with longer time horizons. +Remark 7. The conditions in Definition 3 can help us identify +information states for systems with known dynamics and +simplify the DP decomposition. However, many applications +with large state spaces may require a further improvement +in computational tractability, even at the cost of optimality. +Moreover, in certain applications, we need to learn a represen- +tation of the information state using limited observations with +incomplete knowledge of the dynamics. Information states are +insufficient to account for these cases. Next, in Section IV, we +introduce approximate information states that can address the +above concerns. +IV. APPROXIMATE INFORMATION STATES +In this section, we define approximate information states by +relaxing the conditions given in Definition 3, and utilize them +to develop an approximate DP decomposition which computes +a sub-optimal control strategy for Problem 1. In Subsection +IV-A, we derive the preliminary results required to establish +useful properties of approximate information states. Then, +in Subsection IV-B, we prove these properties, namely, the +Lipischitz continuity of approximate value functions, and the +following error bounds: (1) an upper bound on the error when +the optimal value functions are estimated using approximate +value functions, and (2) an upper bound on the loss in +performance when control actions are generated using a sub- +optimal control strategy instead of an optimal strategy. +Definition 4. An approximate information state for Problem +1 at each t = 0, . . . , T is an uncertain variable ˆΠt = ˆσt(Mt) +taking values in a bounded set ˆPt and generated by an L- +invertible function ˆσt : Mt → ˆPt. Furthermore, for all t = +0, . . . , T, there exist parameters ϵt, δt, λt ∈ R≥0 such that for +all mt ∈ [[Mt]] and ut ∈ [[Ut]], it satisfies the properties: +1) Sufficient to approximate cost: +��� +sup +ct∈[[Ct|mt,ut]] +ct − +sup +ct∈[[Ct|ˆσt(mt),ut]] +ct +��� ≤ ϵt. +(22) +2) Sufficient to approximate evolution: We define the sets +Kt+1 := [[ˆΠt+1 | mt, ut]] and ˆKt+1 := [[ˆΠt+1 | ˆσt(mt), ut]]. +Then, it holds that +H(Kt+1, ˆKt+1) ≤ δt, +(23) +where recall that H is the Hausdorff distance in (1). +3) Lipschitz-like evolution: For all ˆπ1 +t , ˆπ2 +t ∈ [[ˆΠt]], +H +� +[[ˆΠt+1|ˆπ1 +t , ut]], [[ˆΠt+1|ˆπ2 +t , ut]] +� +≤ λt · η(ˆπ1 +t , ˆπ2 +t ), +(24) +where η is an appropriate metric on ˆPt. +Using the approximate information state in Definition 4, we +can construct a DP as follows. For all t, for all ˆπt ∈ [[ˆΠt]] +and ut ∈ [[Ut]], we recursively define the value functions +ˆQt(ˆπt, ut) := max +� +sup +ct∈[[Ct|ˆπt,ut]] +ct, +sup +ˆπt+1∈[[ˆΠt+1|ˆπt,ut]] +ˆVt+1(ˆπt+1) +� +, +(25) +ˆVt(ˆπt) := +inf +ut∈[[Ut]] +ˆQt(ˆπt, ut), +(26) + +7 +where ˆVT +1(ˆπT +1) := 0 identically. If there exists a minimiz- +ing argument in the RHS of (26) at each t = 0, . . . , T, then +ˆg∗ +t (ˆπt) := arg minut∈Ut ˆQt(ˆπt, ut) constitutes an approximate +control law at time t. Furthermore, we call ˆg∗ = (ˆg∗ +0, . . . , ˆg∗ +T ) +an approximately optimal strategy for Problem 1. In Subsec- +tion IV-B, we derive performance guarantees on the approxi- +mate DP and control strategy. +Remark 8. As we showed in Section III, we can specialize +this DP for terminal cost problems, with the value functions +for all t = 0, . . . , T − 1 given by +ˆQtm +t (ˆπt, ut) := +sup +ˆπt+1∈[[ˆΠt+1|ˆπt,ut]] +ˆV tm +t+1(ˆπt+1), +(27) +ˆV tm +t (ˆπt) := inf +ut∈Ut +ˆQtm +t (ˆπt, ut), +(28) +and ˆQtm +T (ˆπT , uT ) := supcT ∈[[CT |ˆπT ,uT ]] cT and ˆV tm +T (ˆπT ) := +infuT ∈UT ˆQtm +T (ˆπT , uT ) at time T. +Remark 9. The conditions in Definition 4 can be investigated +using only output variables. Thus, an approximate information +state can be learned from output data without knowledge of +dynamics, as illustrated in Subsection V-B. +A. Preliminary Results +In this subsection, we derive results necessary to prove the +properties of the approximate DP in Subsection IV-B. +Lemma 2. Consider three bounded subsets X, Y and Z of +a metric space (S, η). Let X ∈ X, Y ∈ Y and Z ∈ Z be +uncertain variables satisfying Y = g(X), where g : X → Y +is L-invertible, and Z = h(X), where h : X → Z is Lipschitz. +Then, there exists an LZ|Y ∈ R≥0 such that: +H([[Z|y1]],[[Z|y2]]) ≤ LZ|Y ·η(y1, y2), ∀y1, y2 ∈ [[Y ]]. (29) +Proof. We prove the result by constructing a feasible constant +LZ|Y +∈ R≥0 which ensures that (29) is satisfied for all +y1, y2 ∈ [[Y ]]. We begin by using the definition of the +Hausdorff distance in (1) to expand the LHS of (29) as +H +� +[[Z|y1]], [[Z|y2]] +� += max +� +sup +x1∈g−1(y1) +inf +x2∈g−1(y2) +η +� +h(x1), +h(x2) +� +, +sup +x2∈g−1(y2) +inf +x1∈g−1(y1) +η +� +h(x1), h(x2) +�� +, +(30) +where, note that [[Z|y]] = +� +z ∈ Z | z = h(x), ∀x ∈ g−1(y) +� +for any realization y +∈ +[[Y ]]. Next, recall that h is +Lipschitz +continuous +with +a +constant +Lh +∈ +R≥0. +Substituting +this +property +into +the +RHS +of +(30), +we +write that H +� +[[Z|y1]], [[Z|y2]] +� +≤ Lh · max +� +supx1∈g−1(y1) +infx2∈g−1(y2) η(x1, x2), supx2∈g−1(y2) infx1∈g−1(y1) η(x1, x2) +� += Lh · H +� +g−1(y1), g−1(y2) +� += Lh · Lg−1 · η(y1, y2), where, +in the second equality, we use the L-invertibile property of g. +Then, the result follows by selecting LZ|Y := Lh · Lg−1. +Lemma 3. Consider a bounded set X and two functions f : +X → R and g : X → R. Then, +| sup +x∈X +f(x) − sup +x∈X +g(x)| ≤ sup +x∈X +|f(x) − g(x)|, +(31) +| inf +x∈X +f(x) − inf +x∈X +g(x)| ≤ sup +x∈X +|f(x) − g(x)|. +(32) +Proof. First, we prove (31) by considering two mutually ex- +clusive cases which cover all possibilities. Case 1: We consider +supx∈X f(x) ≥ supx∈X g(x), which implies | supx∈X f(x)− +supx∈X g(x)| = supx∈X f(x) − supx∈X g(x). For any in- +finitesimally small β > 0, we define x(β) ∈ X as an +element which satisfies f(x(β)) + β ≥ supx∈X f(x). Then, +supx∈X f(x) − supx∈X g(x) +≤ +f(x(β)) + β − supx∈X +g(x) ≤ f(x(β)) + β − g(x(β)) ≤ supx∈X |f(x) − g(x)| + β +for all β > 0. Therefore, supx∈X f(x) − supx∈X g(x) ≤ +supx∈X |f(x) − g(x)|. Case 2: supx∈X f(x) < supx∈X g(x). +The proof can be completed using similar arguments as in Case +1. Then, (32) follows from similar arguments as (31). +Lemma 4. For any four scalars a, b, c, d ∈ R, +| max{a, b} − max{c, d}| ≤ max{|a − c|, |b − d|}. +(33) +Proof. We prove this result by considering four cases which +are mutually exclusive but cover all possibilities. Case 1: For +a ≥ b and c ≥ d: The result holds trivially. Case 2: For +a < b and c ≥ d: The LHS can be expanded as | max{a, b} − +max{c, d}| = |b−c|. Next, if b ≥ c, we use c ≥ d to conclude +that |b − c| < |b − d|, else if c > b, we use b > a to conclude +that |c − b| < |c − a|. Thus, | max{a, b} − max{c, d}| ≤ +max{|a − c|, |b − d|}. Case 3: For a < b and c < d: The +result holds trivially. Case 4: For a ≥ b and c < d: The proof +follows from the same sequence of arguments as Case 2. +Lemma 5. Consider two bounded subsets A, B of a metric +space (X, η). Let f : X → R be a bounded continuous +function with a Lipschitz constant Lf ∈ R≥0 on X. Then, +�� sup +a∈A +f(a) − sup +b∈B +f(b) +�� ≤ Lf · H(A, B). +(34) +Proof. We prove this result by considering two cases which +are mutually exclusive but cover all the possibilities. Case 1: +supa∈A f(a) ≥ supb∈B f(b), which implies | supa∈A f(a) − +supb∈B f(b)| = supa∈A f(a) − supb∈B f(b). We define the +non-empty set A1(β) := {a ∈ A | f(a) + β ≥ supb∈B f(b)} +for +any +infinitesimal +β +> +0. +Then, +supa∈A f(a) − +supb∈B f(b) +≤ +supa∈A1(β) f(a) + β − supb∈B f(b) +≤ +supa∈A1(β) infb∈B(f(a)−f(b))+β ≤ supa∈A infb∈B |f(a)− +f(b)| + β ≤ Lf · supa∈A infb∈B η(a, b) + β for all β > +0. This implies that | supa∈A f(a) − supb∈B f(b)| ≤ Lf · +supa∈A infb∈B η(a, b) ≤ Lf · H(A, B), where, in the second +inequality, we invoke the definition of the Hausdorff dis- +tance in (1) to complete the proof. Case 2: supa∈A f(a) < +supb∈B f(b) and we can prove the result using the same +sequence of arguments as case 1. +As a direct consequence of Lemma 5, we can also establish +the following property. Consider two bounded subsets Y, Z of +Rn, n ∈ N. For two uncertain variables Y ∈ Y and Z ∈ Z, +let the conditional range [[Z|y]] satisfy H +� +[[Z|y1]], [[Z|y2]] +� +≤ +LZ|Y · η(y1, y2) for all realizations y1, y2 ∈ Y of Y . Then, +for a continuous function f : Z → R≥0 with a Lipschitz + +8 +constant Lf, we can use (34) from Lemma 5 to state that for +all y1, y2 ∈ [[Y ]]: +��� +sup +z1∈[[Z|y1]] +f(z1) − +sup +z2∈[[Z|y2]] +f(z2) +��� ≤ LZ|Y ·Lf ·η(y1, y2). (35) +B. Properties of Approximate Information States +In this subsection, we present several properties of the +approximate DP (25) - (26). To begin, we prove in Theorem 2 +that each approximate value function is Lipschitz continuous. +This property subsequently allows us to establish error bounds. +Theorem 2. In the approximate DP (25) - (26), the value +functions ˆQt(ˆπt, ut) and ˆVt(ˆπt) are Lipschitz continuous with +respect to ˆπt ∈ [[ˆΠt]] for all ut ∈ [[Ut]] and t = 0, . . . , T. +Proof. We prove the Lipschitz continuity of the value func- +tions by constructing a valid candidate for the Lipschitz con- +stant L ˆVt at each t = 0, . . . , T, using mathematical induction. +At time T + 1, recall that ˆVT +1(ˆπT +1) = 0 identically and +thus, ˆVT +1(ˆπT +1) is trivially Lipschitz continuous with a con- +stant L ˆVT +1 = 0. This forms the basis of our induction. Then, +at each t = 0, . . . , T, we consider the induction hypothesis that +ˆQt+1(ˆπt+1, ut+1) and ˆVt+1(ˆπt+1) are Lipschitz continuous +with respect to ˆπt+1 ∈ [[ˆΠt+1]] for all ut+1 ∈ [[Ut+1]], and +denote the constant by L ˆVt+1 ∈ R≥0. +At time t, we first prove the result for the value function +ˆQt(ˆπt, ut). Let ˆπ1 +t , ˆπ2 +t ∈ [[ˆΠt]] be two possible realizations +of ˆΠt. Then, using the definition (25) of ˆQt(ˆπt, ut) and (33) +from Lemma 4, we state that +| ˆQt(ˆπ1 +t , ut) − ˆQt(ˆπ2 +t , ut)| ≤ max +���� +sup +c1 +t ∈[[Ct|ˆπ1 +t ,ut]] +c1 +t +− +sup +c2 +t ∈[[Ct|ˆπ2 +t ,ut]] +c2 +t +���, +��� +sup +ˆπ1 +t+1∈[[ˆΠt+1|ˆπ1 +t ,ut]] +ˆVt+1(ˆπ1 +t+1) +− +sup +ˆπ2 +t+1∈[[ˆΠt+1|ˆπ2 +t ,ut]] +ˆVt+1(ˆπ2 +t+1) +��� +� +. +(36) +We consider the RHS of (36) term by term. In the first term, +we note that for all ˆπt ∈ [[ˆΠt]], +sup +ct∈[[Ct|ˆπt,ut]] +ct = +sup +mt∈[[Mt|ˆπt]] +� +sup +ct∈[[Ct|mt,ut]] +ct +� +. +(37) +In the RHS of (37), recall from Assumption 2 that the +uncertain variable Ct is a Lipschitz function of (W0:t, U0:t), +and (Mt, Ut) is an L-invertible function of (W0:t, U0:t). Thus, +using (29) from Lemma 2, there exists a constant LC|M,U such +that H([[Ct|m1 +t, ut]], [[Ct|m2 +t, ut]]) ≤ LM|C,U ·η(m1 +t, m2 +t) for +all m1, m2 ∈ [[Mt]]. Furthermore, we use (35) to state that +��� +sup +c1 +t ∈[[Ct|m1 +t ,ut]] +c1 +t − +sup +c2 +t ∈[[Ct|m2 +t ,ut]] +c2 +t +��� +≤ LM|C,U · Lct · η(m1 +t, m2 +t). +(38) +Then, consider a function et : Mt × Ut → R≥0 defined as +et(mt, ut) := supct∈[[Ct|mt,ut]] ct. As a direct consequence +of (38), et is Lipschitz continuous with respect to mt with a +constant Let := LM|C,U · Lct. Using (37) and the definition +of et in the first term in the RHS of (36), +��� +sup +c1 +t ∈[[Ct|ˆπ1 +t ,ut]] +c1 +t − +sup +c2 +t ∈[[Ct|ˆπ2 +t ,ut]] +c2 +t +��� += +��� +sup +m1 +t ∈[[Mt|ˆπ1 +t ]] +et(m1 +t, ut) − +sup +m2 +t ∈[[Mt|ˆπ2 +t ]] +et(m2 +t, ut) +���. +(39) +In (39), recall that the uncertain variable ˆΠt is an L-invertible +function of Mt and thus, the conditional range [[Mt|ˆπt]] +satisfies (3). Then, we use (35) once more to state that +��� +sup +c1 +t ∈[[Ct|ˆπ1 +t ,ut]] +c1 +t − +sup +c2 +t ∈[[Ct|ˆπ2 +t ,ut]] +c2 +t +���≤LMt|ˆΠt·Let·η(ˆπ1 +t , ˆπ2 +t ). (40) +In the second term in the RHS of (36), we use the induction +hypothesis and (34) from Lemma 5 to write that +��� +sup +ˆπ1 +t+1∈[[ˆΠt+1|ˆπ1 +t ,ut]] +ˆVt+1(ˆπ1 +t+1)− +sup +ˆπ2 +t+1∈[[ˆΠt+1|ˆπ2 +t ,ut]] +ˆVt+1(ˆπ2 +t+1) +��� +≤ L ˆVt+1 · H +� +[[ˆΠt+1|ˆπ1 +t , ut]], [[ˆΠt+1|ˆπ2 +t , ut]] +� +≤ L ˆVt+1 · λt · η(ˆπ1 +t , ˆπ2 +t ), +(41) +where, in the second inequality, we use the third prop- +erty (24) of approximate information states in Definition +4. Then, the proof for +ˆQt(ˆπt, ut) is complete by substi- +tuting (40) and (41) into the RHS of (36) and defining +L ˆ +Qt := max +� +LMt|ˆΠt · Let, L ˆVt+1 · λt +� +. To prove the result +for ˆVt(ˆπt), we use (32) from Lemma 3 to state that +�� ˆVt(ˆπ1 +t )− +ˆVt(ˆπ2 +t ) +�� = +�� infut∈[[Ut]] ˆQt(ˆπ1 +t , ut) − infut∈[[Ut]] ˆQt(ˆπ2 +t , ut) +�� +≤ suput∈[[Ut]] +�� ˆQt(ˆπ1 +t , ut) − ˆQt(ˆπ2 +t , ut) +�� ≤ L ˆ +Qt · η(ˆπ1 +t , ˆπ2 +t ), +which proves the induction hypothesis at time t. Thus, the +result holds using mathematical induction. +Next, we establish an upper bound on the approximation +error when the value functions of the optimal DP (10) - (11) +are estimated using the approximate DP (25) - (26) at each t. +Theorem 3. Let L ˆVt+1 be the Lipschitz constant of ˆVt+1 for +all t = 0, . . . , T. Then, for all mt ∈ [[Mt]] and ut ∈ [[Ut]], +|Qt(mt, ut) − ˆQt(ˆσt(mt), ut)| ≤ αt, +(42) +|Vt(mt) − ˆVt(ˆσt(mt))| ≤ αt, +(43) +where αt = max(ϵt, αt+1 + L ˆVt+1 · δt) for all t = 0, . . . , T +and αT +1 = 0. +Proof. For all t = 0, . . . , T, let mt ∈ [[Mt]] and ut ∈ [[Ut]] +be realizations of Mt and Ut, respectively. We prove both +results by mathematical induction, starting with time step +T + 1. At T + 1, by definition, VT +1(mT +1, uT +1) = +VT +1(ˆσT +1(mT +1)) = 0. This forms the basis of our math- +ematical induction. Then, at each t = 0, . . . , T, we consider +the induction hypothesis |Vt+1(mt+1)− ˆVt+1(ˆσt+1(mt+1))| ≤ +αt+1. At time t, we first prove (42). Using (33) from Lemma +4 in the LHS of (42) to state that +|Qt(mt, ut) − ˆQt(ˆσt(mt), ut)| ≤ max +���� +sup +ct∈[[Ct|mt,ut]] +ct +− +sup +ct∈[[Ct|ˆσt(mt),ut]] +ct +���, +��� +sup +mt+1∈[[Mt+1|mt,ut]] +Vt+1(mt+1) +− +sup +ˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] +ˆVt+1(ˆπt+1) +��� +� +. +(44) + +9 +We consider the RHS of (44) term-by-term. By direct applica- +tion of (22) in Definition 4, the first term in the RHS satisfies +��� +sup +ct∈[[Ct|mt,ut]] +ct − +sup +ct∈[[Ct|ˆσt(mt),ut]] +ct +��� ≤ ϵt. +(45) +For the second term in the RHS of (44), we use the triangle +inequality to write that +��� +sup +mt+1∈[[Mt+1|mt,ut]] +Vt+1(mt+1) − +sup +ˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] +ˆVt+1(ˆπt+1) +��� ≤ +��� +sup +mt+1∈[[Mt+1|mt,ut]] +Vt+1(mt+1) +− +sup +ˆσt+1(mt+1)∈[[ˆΠt+1|mt,ut]] +ˆVt+1(ˆσt+1(mt+1)) +���+ +��� +sup +ˆπt+1∈[[ˆΠt+1|mt,ut]] +ˆVt+1(ˆπt+1) − +sup +ˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] +ˆVt+1(ˆπt+1) +���. +(46) +For the first term in the RHS of (46), we first note +that +supˆσt+1(mt+1)∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆσt+1(mt+1)) += +supmt+1∈[[Mt+1|mt,ut]] ˆVt+1(ˆσt+1(mt+1)) +because +[[ˆΠt+1 +| mt, ut]] = {ˆσt+1(mt+1) ∈ ˆPt | mt+1 ∈ [[Mt+1 | mt, ut]]}. +Then, +we +can +state +that +�� supmt+1∈[[Mt+1|mt,ut]] +Vt+1(mt+1)−supˆσt+1(mt+1)∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆσt+1(mt+1)) +�� +≤ supmt+1∈[[Mt+1|mt,ut]] +��Vt+1(mt+1) − ˆVt+1(ˆσt+1(mt+1)) +�� +≤ αt+1, where, in the first inequality, we use (31) from +Lemma 3; and, in the second inequality, we use the +induction hypothesis for time t + 1. The second term in the +RHS of (46) satisfies +�� supˆπt+1∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆπt+1) − +supˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] ˆVt+1(ˆπt+1) +�� ≤ L ˆVt+1· δt using (34) +from Lemma 5 and (23) from Definition 4. Substituting +the +respective +inequalities +for +each +term +in +the +RHS +of +(46) +yields +�� supmt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) +− +supˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] ˆVt+1(ˆπt+1) +�� ≤ αt+1 + L ˆVt+1 · δt. +We complete the proof for (42) by substituting the inequalities +in the RHS of (45) and (46) into the RHS of (44). Next, +we prove (43) at time t. Using the definition of the value +functions in the LHS of (43), we write that +|Vt(mt) − ˆVt(ˆσt(mt))| = +��� +inf +ut∈[[Ut]] +Qt(mt, ut) − +inf +ut∈[[Ut]] +ˆQt(ˆσt(mt), ut) +��� ≤ +sup +ut∈[[Ut]] +|Qt(mt, ut) − ˆQt(ˆσt(mt), ut)| +≤ max{ϵt, αt+1 + L ˆVt+1 · δt}, +(47) +where in the first inequality, we use (32) from Lemma 3; and +in the second inequality, we use (42). Thus, the results hold +for all t = 0, . . . , T using mathematical induction. +After bounding the approximation error for value functions, +we also seek to bound the maximum performance loss in the +implementation of an approximately optimal strategy. Con- +sider an approximate strategy ˆg∗ := (ˆg∗ +0, . . . , ˆg∗ +T ) computed +using (25) - (26), where ˆg∗ +t (ˆπt) = arg minut∈[[Ut]] ˆQt(ˆπt, ut) +for all t = 0, . . . , T. We can construct an approximate +memory-based strategy gap = (gap +0 , . . . , gap +T ) by selecting the +control law gap +t (mt) := ˆg∗ +t (ˆσt(mt)) for all t = 0, . . . , T. +Note that gap is equivalent to ˆg∗ because they generate the +same actions at each t and subsequently, yield the same +performance. Thus, we evaluate the performance of gap to +determine the quality of approximation. To this end, for all +t = 0, . . . , T, for all mt ∈ [[Mt]] and ut ∈ [[Ut]], we define +Θt(mt, ut) := max +� +sup +ct∈[[Ct|mt,ut]] +ct, +sup +mt+1∈[[Mt+1|mt,ut]] +Λt+1(mt+1) +� +, +(48) +Λt(mt) :=Θt(mt, gap +t (mt)), +(49) +where ΛT +1(mT +1) := 0, identically. Then, the performance +of the memory-based approximate strategy gap is Λ0(m0). In +contrast, recall that the performance of an optimal strategy g∗ +is the optimal value V0(m0) computed using (10) - (11). Next, +we bound the difference in performance between gap and g∗. +Theorem 4. Let L ˆVt+1 be the Lipschitz constant of ˆVt+1 for +all t = 0, . . . , T. Then, for all mt ∈ [[Mt]] and ut ∈ [[Ut]], +|Qt(mt, ut) − Θt(mt, ut)| ≤ 2αt, +(50) +|Vt(mt) − Λt(mt)| ≤ 2αt. +(51) +where αt = max(ϵt, αt+1 + L ˆVt+1 · δt) for all t = 0, . . . , T +and αT +1 = 0. +Proof. We begin by recursively defining the value functions +that compute the performance of the strategy ˆg. For all t = +0, . . . , T and for each ˆπt ∈ [[ˆΠt]] and ut ∈ [[Ut]], let +ˆΘt(ˆπt, ut) := max +� +sup +ct∈[[Ct|ˆπt,ut]] +ct, +sup +ˆπt+1∈[[ˆΠt+1|ˆπt,ut]] +ˆΛt+1(ˆπt+1) +� +, +(52) +ˆΛt(ˆπt) :=ˆΘt(ˆπt, ˆgt(ˆπt)), +(53) +where ˆΛT +1(ˆπT +1) := 0, identically. Note that +ˆΘt(ˆπt, ut) = ˆQt(ˆπt, ut) +and +ˆΛt(ˆπt) = ˆVt(ˆπt), +(54) +for all t = 0, . . . , T, since ˆgt(ˆπt) = arg minut∈Ut ˆQt(ˆπt, ut). +We first prove (50) for all t = 0, . . . , T. At time t, using +the triangle inequality and (54) in the LHS of (50): +|Qt(mt, ut)−Θt(mt, ut)| ≤ |Qt(mt, ut)− +ˆQt(ˆσt(mt), ut)| + |ˆΘt(ˆσt(mt), ut) − Θt(mt, ut)| +≤ αt+|ˆΘt(ˆσt(mt), ut) − Θt(mt, ut)|, +(55) +where, in the second inequality, we use (42) from Theorem 3. +Then, to prove (50), it suffices to show that +|ˆΘt(ˆσt(mt), ut) − Θt(mt, ut)| ≤ αt. +(56) +Next, we use mathematical induction starting at time T + 1 +to prove (56) in addition to |ˆΛt(ˆσt(mt)) − Λt(mt)| ≤ αt for +all t = 0, . . . , T. At time T + 1, using the definitions it holds +that ˆΛT +1(ˆσT +1(mT +1)) = ΛT +1(mT +1) = 0. This forms +the basis of our induction. Next, for all t = 0, . . . , T, we +consider the induction hypothesis that |ˆΛt+1(ˆσt+1(mt+1)) − +Λt+1(mt+1)| ≤ αt+1. Given the hypothesis, (56) holds at time +t using the same sequence of arguments as in the proof for + +10 +Theorem 3. Next, using the definitions of the value functions +from (49) and (53), we write that +|ˆΛt(ˆσt(mt)) − Λt(mt)| = |ˆΘt(ˆσt(mt), ˆgt(ˆσt(mt)) − Θt(mt, +gt(mt))| = |ˆΘt(ˆσt(mt), ˆut)−Θt(mt, ˆut)| ≤ αt, +(57) +where, in the second equality, we use the definition of the +control law to write that gt(mt) = ˆgt(ˆσt(mt)) =: ˆut; and +in the inequality, we use (56). This proves the induction +hypothesis for time t given the hypothesis for time t + 1. +Thus, using mathematical induction (56) holds for all t = +0, . . . , T. Subsequently, we complete the proof for (50) for +all t = 0, . . . , T by substituting (56) into the RHS of (55). +Furthermore, note that (51) follows directly from (50) using +the same sequence of arguments used to prove (57). +Remark 10. We can specialize the results of both Theorem 3 +and Theorem 4 to terminal cost problems, where the optimal +DP is given by (12) - (13) and the approximate DP is given by +(27) - (28). The approximation bounds in both theorems hold +for terminal cost problems with a recursively defined constant +αt := αt+1 +L ˆV tm +t+1 ·δt for all t = 0, . . . , T −1 and αT := ϵT . +C. Alternate Characterization +In this subsection, we provide stronger but simpler condi- +tions which can identify an approximate information state as +alternatives to (23) and (24). These conditions prescribe that +an approximate information state ˆΠt = ˆσt(Mt) must satisfy +for all t = 0, . . . , T: +1) State-like evolution: There exists a Lipschitz continuous +function ˆft : ˆPt × Ut × Yt+1 → Pt+1, independent of the +strategy g, such that +ˆΠt+1 = ˆft(ˆΠt, Ut, Yt+1). +(58) +2) Sufficient to approximate observations: For all mt ∈ +[[Mt]] and ut +∈ +[[Ut]], we define the sets Kob +t+1 +:= +[[Yt+1 | mt, ut]] and ˆKob +t+1 := [[Yt+1 | ˆσt(mt), ut]]. Then, +H(Kob +t+1, ˆKob +t+1) ≤ δob +t , +(59) +where δob +t ∈ R≥0 is a known constant. +3) Lipschitz-like observation prediction: There exists a +constant λob +t ∈ R≥0 such that for all ˆπ1 +t , ˆπ2 +t ∈ [[ˆΠt]], +H +� +[[Yt+1|ˆπ1 +t , ut]], [[Yt+1|ˆπ2 +t , ut]] +� +≤ λob +t · η(ˆπ1 +t , ˆπ2 +t ), +(60) +where η is an appropriate metric on ˆPt. +Next, we prove that in addition to (22) in Definition 4, +the conditions (58) - (60) are sufficient to characterize an +approximate information state instead of (23) and (24). +Lemma 6. For all t = 0, . . . , T, if an uncertain variable +ˆΠt = ˆσt(Mt) satisfies (58) - (59), it also satisfies (23). +Proof. Let mt ∈ [[Mt]] be a given realization of Mt and let +ˆπt = ˆσt(mt) satisfy (58) - (59), for all t = 0, . . . , T. Then, +using (58), we can write the LHS in (23) as H(Kt+1, ˆKt+1) = +H +� +[[ ˆft(ˆσt(mt), ut, Yt+1)|mt, ut]], [[ ˆft(ˆσt(mt), ut, Yt+1)| +ˆσt(mt), ut]] +� += +max +� +supyt+1∈Kob +t+1 inf ˆyt+1∈ ˆKob +t+1 +d +� ˆft(ˆσt(mt), ut, yt+1), ˆft(ˆσt(mt), ut, ˆyt+1) +� +, supˆyt+1∈ ˆKob +t+1 +infyt+1∈Kob +t+1 η( ˆft +� +ˆσt(mt), ut, yt+1), ˆft(ˆσt(mt), ut, ˆyt+1) +�� +, +where, +in +the +second +equality, +we +use +the +definition +of +the +Hausdorff +distance +from +(1). +Note +that +ˆft +is +globally +Lipschitz +because +the +approximate +information +state +takes +values +in +a +finite +set. +This +implies +that +d +� ˆft(ˆσt(mt), ut, yt+1), ˆft(ˆσt(mt), ut, ˆyt+1) +� +≤ L ˆ +ft · η(yt+1, +ˆyt+1), and thus H(Kt+1, ˆKt+1) ≤ L ˆ +ft max +� +supyt+1∈Kob +t+1 +inf ˆyt+1∈ ˆKob +t+1 η(yt+1, ˆyt+1), supˆyt+1∈ ˆKob +t+1 infyt+1∈Kob +t+1 +η(yt+1, ˆyt+1) +� += L ˆ +ft · H(Kob +t+1, ˆKob +t+1) ≤ L ˆ +ft · δob +t . +Lemma 7. For all t = 0, . . . , T, if an uncertain variable +ˆΠt = ˆσt(Mt) satisfies (58) - (60), it also satisfies (24). +Proof. Let ˆπ1 +t , ˆπ2 +t ∈ [[ˆΠt]] be two possible realizations of +an approximate information state ˆΠt, which satisfies (58) +- (60), for all t = 0, . . . , T. Then, using (58), we can +write the LHS in (24) as H +� +[[Πt+1|ˆπ1 +t , ut]], [[Πt+1|ˆπ2 +t , ut]] +� += H +� +[[ ˆft(ˆπ1 +t , ut, Yt+1)|ˆπ1 +t , ut]], [[ ˆft(ˆπ2 +t , ut, Yt+1)|ˆπ2 +t , ut]] ≤ +L ˆ +ft · +� +η(ˆπ1 +t , ˆπ2 +t ) + H +� +[[Yt+1|ˆπ1 +t , ut]], [[Yt+1|ˆπ2 +t , ut]] +�� +≤ L ˆ +ft · +(1 + λob +t ) · η(ˆπ1 +t , ˆπ2 +t ), where, in the first inequality, we use the +Lipschitz continuity of the function ˆft along with the triangle +inequality; and in the second inequality, we use (60). This +completes the proof by defining λt := L ˆ +ft · (1 + λob +t ). +D. Examples +In this subsection, we present two state-quantized [91] +approximate information states which satisfy Definition 4. +Consider a system as described in Subsection III-C with +compact feasible sets +� +Xt, Nt, Wt | t = 0, . . . , T +� +in a metric +space (S, η). Recall that Xt is the state space at any t. Then, +a finite subset +ˆ +Xt ⊂ Xt is a set of quantized states with +parameter γt ∈ R≥0 if maxxt∈Xt minˆxt∈ ˆ +Xt η(xt, ˆxt) ≤ γt. +The corresponding quantization function µt : Xt → +ˆ +Xt is +defined as µt(xt) := arg minˆxt∈ ˆ +Xt η(xt, ˆxt). Note that by +construction, η(xt, µt(xt)) ≤ γt for all xt ∈ Xt, for all t. +1) Perfectly Observed Systems: Consider a system where +Yt = Xt for all t = 0, . . . , T. Recall from Subsection III-C +that the Πt = Xt ∈ Xt for all t. Then, a feasible approximate +information state for such a system is the quantized state ˆΠt := +µt(Xt), which satisfies Definition 4 with ϵt = 2Ldt · γt and +δt = 2γt+1 + 2Lft · γt, where γT +1 = 0, and Ldt and Lft +are the Lipschitz constants for dt and ft, respectively (proof +in Appendix B). Note that because ˆΠt takes values in a finite +set, it trivially satisfies (24) in Definition 4. +2) Partially Observed Systems: For a partially observed sys- +tem, recall from Section III-C that an information state is given +by the conditional range Πt = [[Xt|mt]]. We construct an +approximate conditional range by quantizing each element in +Πt. Thus, the approximation is generated by the mapping νt : +B(Xt) → 2 ˆ +Xt, where B(Xt) is the set of all compact subsets of +Xt and 2 ˆ +Xt is the power set of ˆ +Xt. This transformation yields +the approximate range νt(Πt) := {µt(xt) ∈ ˆ +Xt | xt ∈ Πt}. +Then, the approximate range ˆΠt = νt(Πt) is an information +state for partially observed systems for all t = 0, . . . , T with +ϵt = 2Ldt · γt and δt = 2γt+1 + 2L ¯ +ft · Lht+1 · Lft · γt, where +γT +1 = 0, and L ¯ +ft, Lht+1 and Lft are Lipschitz constants of +¯ft, ht+1, and ft, respectively (proof in Appendix C). + +11 +V. NUMERICAL EXAMPLES +We present two numerical examples to illustrate our ap- +proach: (1) The Wall Defense Problem: a worst-case control +problem with partial observations, and (2) The Pursuit Evasion +Problem: a worst-case reinforcement learning problem with +partly unknown dynamics and partial observations. +A. The Wall Defense Problem +In the wall defense problem, we consider an agent who +defends a wall in a 5 × 5 grid world from an attacker over a +time horizon T. The wall is located across the central row of +the grid. We illustrate the wall defense problem for one initial +condition in Fig. 1(a). Here, the black colored cells constitute +the wall and the grey hatched cells are adjacent to the wall. The +solid blue triangle, solid red circle and red ring are the agent, +attacker and observation, respectively, at t = 0. The pink cells +are feasible positions of the attacker given the observation. +The attacker moves within the bottom two rows of the grid +and damages a wall cell when positioned in an adjacent cell. +At each t = 0, . . . , T, we denote the position of the attacker by +Xat +t ∈ X at = {(−2, −1), . . . , (2, −1), (−2, −2), . . . , (2, −2)}. +In contrast, the agent moves within the top two rows of the +grid and repairs a wall cell when positioned in an adjacent +cell. At each t, we denote the position of the agent by +Xag +t +∈ X ag = {(−2, 1), . . . , (2, 1), (−2, 2), . . . , (2, 2)}. The +state of the wall at each t is the accumulated damage denoted +by Dt = (D−2 +t , . . . , D2 +t ), where Di +t ∈ Di +t = {0, 1, 2, 3} +for all i = −2, . . . , 2 and Dt = ×2 +i=−2Di +t. The attacker +starts at the position Xat +0 +∈ X at, which evolves for all +t as Xat +t+1 += +I(Xat +t + Wt +∈ +X at) · (Xat +t + Wt) + (1 +−I(Xat +t +Wt ∈ X at))·Xat +t , where I is the indicator function and +Wt ∈ Wt is an uncontrolled disturbance with Wt = {(−1, 0), +(1, 0), (0, 0), (0, 1), (0, −1)}. At each t, the agent observes +their own position and the wall’s state. The agent also partially +observes the attacker’s position as Yt = I(Xat +t + Nt ∈ +X at) · (Xat +t + Nt) + (1 − I(Xat +t + Nt ∈ X at)) · Xat +t , where +Nt ∈ Nt = {(0, 0), (0, 1)} is the measurement noise. Given +the history of observations, the agent selects an action Ut ∈ +Ut = Wt at each t. Starting with Xag +0 ∈ X ag, the agent moves +as Xag +t+1 = I(Xag +t +Ut ∈ X ag)·(Xag +t +Ut)+(1−I(Xag +t +Ut ∈ +X ag)) · Xag +t . Starting with D0 = (0, 0, 0, 0, 0), the state of +the wall evolves as Di +t+1 = min +� +3, max +� +0, Di +t + I(Xat +t = +(i, −1)) − I(Xag +t += (i, 1)) +�� +for all t and i = −2, . . . , 2. At +each t, after selecting the action, the agent incurs a cost for +the damage to the wall, i.e., ct(Dt) = �2 +i=−2 Di +t. The agent’s +aim is to minimize the maximum instantaneous damage to the +wall, i.e., J (g) = maxt=0,...,T maxx0,w0:T ,n0:T ct(Dt). +Recall from Subsection III-C that an information state at +time t is Πt = +� +Xag +t , Dt, [[Xat +t |Mt]] +� +. We construct an approx- +imation of the conditional range [[Xat +t |Mt]] at time t using the +quantization approach from Subsection IV-D and define the +approximate range ˆAt = +� +µt(xt) ∈ +ˆ +X at|xt ∈ [[Xat +t |Mt]] +� +. +The set of quantized cells +ˆ +X at, with γt = 1 for all t, is +marked in Fig. 1(b) with dots. We consider the approximate +information state ˆΠt = +� +Xag +t , Dt, ˆAt, Y0 +� +for all t. The initial +observation Y0 in ˆΠt improves the prediction of ˆAt+1. For +five initial conditions, we compute the best control strategy +(a) The original grid +(b) The quantized grid +Fig. 1: The wall defense problem with the initial conditions +xag +0 = (0, 2) and y0 = (0, −2). +for T = 6 using both the information state (IS) and the +approximate information state (AIS). In Fig. 2, we present the +computational times (Run.) for both the DPs in seconds. Note +that the approximate DP has a faster run-time in all cases. We +also implement both strategies with random disturbances in the +system with T = 6. In Fig. 2, we also present the actual worst- +case costs across 5 × 103 implementations of both strategies +and note that the AIS has a bounded deviation from the IS. +Fig. 2: Costs and run-times for 5×103 simulations and T = 6. +B. Pursuit Evasion Problem +In the pursuit evasion problem, we consider an agent who +chases a moving target in a 9 × 9 grid world with static +obstacles. The agent aims to get close to the target over a time +horizon T. For each t = 0, . . . , T, we denote the position of +the agent by Xag +t +∈ X and that of the target by Xta +t ∈ X, +where X = +� +(−4, −4), . . . , (4, 4) +� +\ O is the set of feasible +grid cells and O ⊂ X is the set of obstacles. The target starts at +the position Xta +0 ∈ X, which is updated as Xta +t+1 = I(Xta +t +Wt +∈ X) · (Xta +t + Wt) + (1 − I(Xta +t + Wt ∈ X)) · Xta +t , where +Wt ∈ Wt = {(−1, 0), (1, 0), (0, 0), (0, 1), (0, −1)} is the +disturbance. At each t, the agent perfectly observes their +own position and nosily observes the target’s position as +Yt = I(Xta +t +Nt ∈ X)·(Xta +t +Nt)+(1−I(Xta +t +Nt ∈ X)) ·Xta +t , +where Nt ∈ Nt = Wt is the measurement noise. Next, starting +with Xag +0 ∈ X, the agent selects an action Ut ∈ Ut = Wt to +move as Xag +t+1 = I(Xag +t ++ Ut ∈ X) · (Xag +t ++ Ut) + (1 − +I(Xag +t ++ Ut ∈ X)) · Xag +t . At time T, the agent selects no +action and observes the target’s position Xta +T and incurs a +cost cT (Xta +T , Xag +T ) = η(Xta +T , Xag +T ) ∈ R≥0, where η is the +shortest distance between two cells, while avoiding obstacles. +The distance between two adjacent cells is 1 unit. The agent +seeks to minimize the worst-case terminal cost without prior +knowledge of either the observation function or the target’s +evolution dynamics. Note that this is a reinforcement learning +generalization of Problem 1. We illustrate the grid and one +initial set up in Fig. 3(a). Here, the black cells are obstacles. + +-2 +-1 +0 +1 +2 +2 +1 +0 +-1 +-2-2 +-1 +0 +1 +2 +2 +1 +0 +-1 +-2Initial Conditions +Strategy IS +Strategy AIS +xag +Yo +Run. (s) +Max. cost +Run. (s) +Max. cost +(1, 2) +(1,-2) +354.2 +2 +221.2 +2 +(0, 1) +(0,-1) +1209.7 +2 +607.3 +3 +(-2, 2) +(-1,-1) +686.1 +3 +446.4 +3 +(2, 2) +(-2, - 2) +19.22 +2 +12.81 +2 +(-2, 2) +(2, -1) +582.3 +3 +362.1 +3 +(-1, 1) +(1, -1) +1551.5 +2 +1075.1 +312 +The solid blue triangle, solid red circle and red ring are the +agent, target, and observation, respectively, at t = 0. The pink +cells are feasible positions of the target given the observation. +(a) The original problem (b) Actual +observation +prediction +(c) Learned observation +prediction +Fig. 3: The pursuit evasion problem with the initial conditions +xag +0 = (0, 2) and y0 = (3, −4). +We consider that the agent has access to 3×107 observation +trajectories from the target which are used to learn an approx- +imate information state representation offline, as characterized +in Subsection IV-C. First, we use the data on observation +trajectories to construct estimates of the conditional range +Kob +t+1 = [[Yt+1|Y0:t]] for all t = 0, . . . , T − 2 and Kob +T += +[[Xta +T |Y0:T −1]]. Then, taking inspiration from [45], we set-up +a deep neural network with an encoder-decoder structure for +each t = 0, . . . , T, as illustrated in Fig. 4. At each t, the +encoder ψt comprises of 3 layer neural network with sizes +(2, 14), (14, 12), (12+24, 24) and ReLU activation for the first +two layers, where the inputs are a 2-d vector of coordinates for +observation Yt and a 24-d vector for the previous approximate +information state ˆΠt−1. The encoder compresses these inputs +to a 24-d vector representing the approximate information state +ˆΠt. At each t, the decoder φt is a 4 layer neural network of +size (24, 48), (48, 56), (56, 64), (64, 74) with ReLU activation +for the first three layers and sigmoid activation for the last +layer. Its input is ˆΠt and its output is a 74-d vector with +each component taking values in [0, 1]. Each component of the +74-d output gives a set-inclusion value for a specific feasible +cell in the 9 × 9 grid, excluding obstacles. The output is thus +interpreted as the conditional range ˆKob +t+1 = [[Yt+1 | ˆΠt]] for +all t = 0, . . . , T − 2 and ˆKob +T = [[Xta +T | ˆΠT −1]]. We consider a +set-inclusion threshold of 0.5 for inclusion in ˆKob +t+1 at each t. +Fig. 4: The neural network architecture for approximate infor- +mation states at any t = 0, . . . , T − 1. +The learning objective of our neural network at each t +is to minimize H(Kob +t+1, ˆKob +t+1), which is consistent with the +characterization of approximate information states in Subsec- +tion IV-C. Note that at the terminal time step, this objective +also minimizes the difference in maximum costs. Since the +Hausdorff distance is not differentiable, we adapt the first +surrogate function proposed in [92] as a learning objective +to train the network weights. We train the network for 40 +epochs using 90% of the available data with a learning rate +of 0.0003 and test it against the other 10%. To illustrate the +training results, consider an out-of-sample initial observation +y0 = (3, −4). Then, the set Kob +1 +constructed using data is +shown by pink cells in 3(b) and the set ˆKob +1 generated by of +the trained network is shown by blue cells in 3(c). Note that +the trained network’s output matches the conditional range +constructed from data accurately except for one cell (0, −4). +We train a neural network for each t up to T = 4 to learn a +complete approximate information state representation for the +problem. Then, at each t, the agent uses the state (Xag +t , ˆΠt) in +the approximate DP (25) - (26) to compute an approximately +optimal control strategy. +We compare the performance of this approximate strategy +with a baseline strategy that uses the observation Yt at each +t instead of ˆΠt. Thus, for this baseline we train a network +to match the prediction [[Yt+1 | Yt]] to [[Yt+1 | Y0:t]] for all +t = 0, . . . , T −2 and [[Xta +T | YT −1]] to [[Xta +T | Y0:T −1]] at time +T − 1. The neural network structure is the same as before +except for a lack of ˆΠt−1 in the encoder input at each t and +we use the same training parameters as before. Subsequently, +the agent computes an approximately optimal strategy using +the approximate DP with the state (Xag +t , Yt) at each t. +For six initial conditions, we present in Fig. 5 the worst case +costs obtained when implementing both the approximately +optimal strategy (Maximum cost with AIS) and the baseline +strategy (Maximum cost without AIS) for T = 4. Across +104 simulations with randomly generated uncertainties, we +note that using the learned approximate information state +consistently improves worst-case performance when compared +to the baseline. Thus, learning an approximate information +state representation is a viable approach for worst-case re- +inforcement learning. In general, we expect our approach to +outperform the baseline more for longer time horizons. +Fig. 5: Worst-case costs for 104 simulations and T = 4. +VI. CONCLUSION +In this paper, we presented a principled approach to worst- +case control and learning in partially observed systems us- +ing non-stochastic approximate information states. We first +presented two sets of properties to characterize information +states and used them to construct a DP that yields an optimal +control strategy. Then, we proposed two sets of properties +to characterize approximate information states that can be +constructed from output variables with knowledge of the dy- +namics or learned from output data with incomplete knowledge +of the dynamics. We proved that approximate information +states can be used in a DP to compute an approximate +control strategy with a bounded loss in performance. We also + +-4-3 +¥-2 +-1 +0 +1 +2 +¥3 +4 +4 +3 +2 +1 +0 +-1 +-2 +-3 +-4-4-3 +¥-2 +-1 +0 +1 +2 +¥3 +4 +4 +3 +2 +1 +0 +-1 +Kpb +-2 +-3 +-4-4-3 +¥-2 +-1 +0 +1 +2 +¥3 +4 +4 +3 +2 +1 +0 +-1 +Rpb +-2 +-3 +-4dt ++t +Ilt-1 +24 +2 +14 +12 +24 +24 +48 +56 +64 +74 +Yt +tx0 +Yo +Maximum cost with AlS +Maximum cost without AlS +(3, 2) +(0,0) +2 +3 +(-4, 4) +(4,-4) +12 +12 +(4,4) +(-4,-2) +11 +12 +(-2, -3) +(- 4,3) +7 +8 +(-4, 2) +(1, 4) +7 +7 +(-3, 1) +(4, -1) +8 +913 +presented theoretical examples of this bound and numerical +examples to illustrate the performance of our approach in both +worst-case control and reinforcement learning. +Our ongoing work is to specialize the approach in this paper +to additive cost problems reported in [93]. Future work should +consider extending our results to problems with an infinite time +horizon, constructing tighter performance bounds for systems +with specific dynamics, and combining these results with other +reinforcement learning techniques, e.g., Q-learning [94]. +APPENDIX A – L-INVERTIBLE FUNCTIONS +In this appendix, we present two classes of functions which +are L-invertible: 1) all bi-Lipschitz functions which have +a compact domain and a compact co-domain, and 2) all +functions with a compact domain and a finite co-domain. +Lemma 8. Let X and Y be two compact subsets of a metric +space (S, η). Then, any bi-Lipischitz function f : X → Y is +L-invertible. +Proof. We begin by considering the pre-image set for any y ∈ +Y under the function f. Note that the function f is continuous +because it is bi-Lipschitz and the singleton {y} is a compact +subset of a metric space. Consequently, the pre-image f −1(y) +is a bounded subset of X. Next, let B(X) denote the set of all +bounded subsets of X. Given the first result, we can consider +a set-valued mapping f −1 : Y → B(X) which returns the +pre-image for each y ∈ Y. Then, for any y1, y2 ∈ Y, using +the definition of the Hausdorff distance in (1): +H +� +f −1(y1), f −1(y2) +� += max +� +sup +x1∈f −1(y1) +inf +x2∈f −1(y2) +η(x1, x2), +sup +x2∈f −1(y2) +inf +x1∈f −1(y1) +η(x1, x2) +� +. +(61) +In the RHS of (61), the bi-Lipschitz property of f implies that +there exist constants Lf, Lf ∈ R>0 such that Lfη(x1, x2) ≤ +|f(x1) − f(x2)| ≤ Lfη(x1, x2), for all x1, x2 ∈ X. Thus, for +all x1 ∈ g−1(y1) and x2 ∈ g−1(y2), we write that +η(x1, x2) ≤ L−1 +f +· η(y1, y2). +(62) +The proof is complete by substituting (62) into (61) and +defining the constant Lf −1 := L−1 +f . +Lemma 9. Let X be a compact subset and Y be a finite subset +of (S, η). Then, any function f : X → Y is L-invertible. +Proof. Let ||Y|| > 0 denote the minimum distance between +two distinct elements in the finite, non-empty set Y. Then, +for any y1, y2 ∈ Y such that y1 ̸= y2, H +� +f −1(y1), f −1(y2) +� +η(y1, y2) +≤ supy1,y2∈Y +H +� +f −1(y1), f −1(y2) +� +||Y|| +=: Lf −1, where Lf −1 ∈ +R≥0 is guaranteed to be finite because the set X is bounded +and thus, so is the numerator. Thus, the function f is L- +invertible as defined in (61). +APPENDIX B – APPROXIMATION BOUNDS FOR PERFECTLY +OBSERVED SYSTEMS +In this appendix, we derive the values of ϵt and δt for all t = +0, . . . , T when an approximate information state is constructed +using state quantization for a perfectly observed system, as +described in Subsection IV-D. We first state a property of the +Hausdorff distance which we will use in our derivation. +Lemma 10. Let X be a metric space with compact subsets +A, B, C, D ⊂ X. Then, it holds that +H +� +A ∪ B, C ∪ D +� +≤ max +� +H +� +A, C +� +, H +� +B, D +�� +. +(63) +Proof. The proof for this result is given in [88, Theorem +1.12.15]. +Next, we state and prove the main result of this appendix. +Theorem 5. Consider a perfectly observed system, i.e., Yt = +Xt, for all t = 0, . . . , T. Let µt : Xt → +ˆ +Xt such that +maxxt∈Xt η(xt, µt(xt)) ≤ γt at each t. Then, ˆΠt = µt(Xt) +is an approximate information state which satisfies (22) with +ϵt = 2Ldt · γt and (23) with δt = 2γt+1 + 2Lft · γt for all +t, where γT +1 = 0, and where Ldt, and Lft are Lipschitz +constants for dt and ft, respectively. +Proof. For all t = 0, . . . , T, let mt = (x0:t, u0:t−1) be +the realization of Mt and let the approximate information +state be ˆxt = µt(xt). We first derive the value of ϵt in +the RHS of (22). At time t, can expand the conditional +ranges to write that [[Xt|mt]] += +[[Xt|xt]] += +{xt} +and [[Xt|ˆxt]] += +{xt +∈ +X +| η(xt, ˆxt) +≤ +γt}. On +substituting +these +into +the +LHS +of +(22), +we +state +that +�� supct∈[[Ct|mt,ut]] ct +− +supct∈[[Ct|µt(xt),ut]] ct +�� += +��dt(xt, ut) +− +sup¯xt∈[[Xt|µt(xt)]] dt(¯xt, ut) +�� +≤ +sup¯xt∈[[Xt|µt(xt)]] |dt(xt, ut) +− +dt(¯xt, ut)| +≤ Ldt · sup¯xt∈[[Xt|µt(xt)]] η(xt, ¯xt) ≤ Ldt · +� +η(xt, µt(xt)) + +sup¯xt∈[[Xt|µt(xt)]] η(µt(xt), ¯xt) +� +, ≤ 2Ldt · γt =: ϵt, where, in +the third inequality, we use the triangle inequality. Next, to +derive the value of δt, we expand the LHS of (23) as +H +� +[[ ˆXt+1|xt, ut]], [[ ˆXt+1|µt(xt), ut]] +� +=H +�� +µt+1(ft(xt, ut, wt))|wt ∈ Wt +� +, +� +µt+1(ft(¯xt, ut, wt))|¯xt ∈ [[Xt|µt(xt)]], wt ∈ Wt +�� +≤ sup +wt∈Wt +H({µt+1(ft(xt, ut, wt))}, +{µt+1(ft(¯xt, ut, wt))|¯xt ∈ [[Xt|µt(xt)]]}), +(64) +where, +in +the +inequality, +we +use +(63) +from +Lemma +10 +and +the +fact +that +� +µt+1(ft(xt, ut, wt))|wt +∈ +Wt +� += +∪wt∈Wt +� +µt+1(ft(¯xt, ut, wt)) +� +. +Once +again +using +(63) +in +the +RHS +of +(64), +we +conclude +that +H +� +[[ ˆXt+1|xt, ut]], [[ ˆXt+1|µt(xt), ut]] +� +≤ +supwt∈Wt,¯xt∈[[Xt|µt(xt)]] η +� +µt+1 +� +ft(xt, ut, wt) +� +, µt+1 +� +ft(¯xt, +ut, wt) +�� +≤ supwt∈Wt,¯xt∈[[Xt|µt(xt)]] +� +η +� +µt+1(ft(xt, ut, wt)), +ft(xt, ut, wt) +� ++ +η +� +ft(xt, ut, wt), ft(¯xt, ut, wt) +� ++ +η +� +ft(¯xt, ut, wt), µt+1(ft(¯xt, ut, wt)) +�� +≤ +γt+1 + 2Lft · +γt + γt+1 =: δt, where, in the second inequality, we use the +triangle inequality. + +14 +APPENDIX C – APPROXIMATION BOUNDS FOR PARTIALLY +OBSERVED SYSTEMS +In this appendix, we derive the values of ϵt and δt for +all t = 0, . . . , T, when an approximate information state is +constructed using state quantization for a partially observed +system, as described in Subsection IV-D. +Theorem 6. Consider a partially observed system with Yt = +ht(Xt, Nt) for all t = 0, . . . , T. Let µt : Xt → ˆ +Xt such that +supxt∈Xt η(xt, µt(xt)) ≤ γt at each t. Then, ˆΠt = νt(Πt) +is an approximate information state with ϵt = 2Ldt · γt and +δt = 2γt+1 + 2L ¯ +ft · Lht+1 · Lft · γt for all t, where γT +1 = 0, +and where Ldt, L ¯ +ft, Lht+1, and Lft are Lipschitz constants +for the respective functions in the subscripts. +Proof. For all t = 0, . . . , T, let mt ∈ [[Mt]], Pt = [[Xt|mt]] ∈ +Pt, and ˆPt = νt(Pt) ∈ ˆPt be the realizations of the memory +Mt, the conditional range Πt and the approximate information +state ˆΠt, respectively. Note that the conditional range Pt +satisfies (14) and (15) from Definition 3. Next, to derive the +value of ϵt, we write the LHS of (22) using (14) as +�� +sup +ct∈[[Ct|mt,ut]] +ct − +sup +ct∈[[Ct|νt(Pt),ut]] +ct +�� += +�� sup +xt∈Pt +dt(xt, ut) − +sup +¯xt∈[[Xt|νt(Pt)]]) +dt(¯xt, ut) +�� +≤Ldt · H(Pt, [[Xt|νt(Pt)]]) +≤Ldt · +� +H(Pt, νt(Pt)) + H(νt(Pt), [[Xt|νt(Pt)]]) +� +, +(65) +where, in the equality, we use (14); in the first inequality, +we use (34) from Lemma 5; and in the second inequal- +ity we use the triangle inequality for the Hausdorff dis- +tance. We can expand the first term in the RHS of (65) +as H(Pt, νt(Pt)) = H(Pt, {µt(xt) ∈ +ˆ +Xt | xt ∈ Pt}) += H +� +∪xt∈Pt {xt}, ∪xt∈Pt{µt(xt) ∈ +ˆ +Xt} +� +≤ supxt∈Pt +η(xt, µt(xt)) ≤ γt, where we use (63) from Lemma 10 +in the first inequality. We can also expand the second +term in the RHS of (65) as H(νt(Pt), [[Xt|νt(Pt)]])) = +H +� +νt(Pt), {xt +∈ +Xt| inf ¯xt∈νt(Pt) η(xt, ¯xt) +≤ +γt} +� += +supxt∈[[Xt|νt(Pt)]] inf ¯xt∈νt(Pt) η(xt, ¯xt) ≤ γt, where the sec- +ond equality holds by expanding the Hausdorff distance and +noting that νt(Pt) ⊆ [[Xt|νt(Pt)]]. The proof is complete by +substituting the results for both terms in the RHS of (65). +Next, to derive the value of δt, we note that Pt = σt(mt). +Then, using the triangle inequality in the LHS of (23), +H +� +[[νt+1(Πt+1)|mt, ut]], [[νt+1(Πt+1)|νt(σt(mt)), ut]] +� +≤H +� +[[νt+1(Πt+1)|mt, ut]], [[Πt+1|mt, ut]] +� ++ +H +� +[[Πt+1|mt, ut]], [[Πt+1|νt(σt(mt)), ut]] +� ++ +H +� +[[Πt+1|νt(σt(mt)), ut]], [[νt+1(Πt+1)|νt(σt(mt)), ut]] +� +≤2γt+1 + H +� +[[Πt+1|mt, ut]], [[Πt+1|νt(σt(mt)), ut]] +� +, (66) +where, in the second inequality we use the fact that +H +� +Pt+1, νt+1(Pt+1) +� +≤ γt+1, which was proved above. We +can write the second term in the RHS of (66) using (15) from +Definition 3 as H +� +[[Πt+1|mt, ut]], [[Πt+1|νt(σt(mt)), ut]] +� += +H +� +[[Πt+1|Pt, ut]], [[Πt+1|νt(Pt), ut]] +� +. Furthermore, note that +[[Πt+1|νt(Pt), ut]] += +� ˜Pt+1 +∈ +[[Πt+1| ˜Pt, ut]] | ˜Pt +∈ +[[Πt|νt(Pt)]] +� += ∪ ˜ +Pt∈[[Πt|ν(Pt)]][[Πt+1| ˜Pt, ut]]. Next, we use +(63) from Lemma 10 to write that +H +� +[[Πt+1|Pt, ut]]), [[Πt+1|νt(Pt), ut]] +� +≤ +sup +˜ +Pt∈[[Πt|νt(Pt)]] +H +� +[[Πt+1|Pt, ut]]), [[Πt+1| ˜Pt, ut]] +� +≤L ¯ +ft · +sup +˜ +Pt∈[[Πt|νt(Pt)]] +H +� +[[Yt+1|Pt, ut]]), [[Yt+1| ˜Pt, ut]] +� +≤L ¯ +ft·Lht+1 · +sup +˜ +Pt∈[[Πt|νt(Pt)]] +H +� +[[Xt+1|Pt, ut]]), [[Xt+1| ˜Pt, ut]] +� +, +(67) +where, in the second inequality we use the same arguments +as in Lemma 6 and the third inequality can be proven +by +substituting +Yt+1 += +ht+1(Xt+1, Vt+1) +into +the +equation. 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Y¨uksel, “Convergence and near optimality of q- +learning with finite memory for partially observed models,” in 2021 +60th IEEE Conference on Decision and Control (CDC), pp. 1603–1608, +IEEE, 2021. +Aditya Dave (S’18) is a PhD candidate in the +Department of Mechanical Engineering at the Uni- +versity of Delaware, Newark, USA since 2017. He +received a B. Tech. in Mechanical Engineering from +the Indian Institute of Technology, Bombay, India, +in 2016. Prior to his PhD degree, he worked as +a Project Manager at Cairn Energy, Gurugram, In- +dia from 2016-2017. His current research interests +span several areas, including worst-case control, +reinforcement learning, decentralized systems, and +mechanism design. He is a student member of IEEE. +Nishanth Venkatesh S (S’21) is research engineer +at the Department of Mechanical Engineering at the +University of Delaware, Newark, USA. He received +a Master’s degree in Robotics from the University of +Delaware, Newark, USA, 2021. Prior to his Master’s +he received a B. Tech. in Mechanical Engineering +from the Indian Institute of Technology, Bombay, +India, in 2019. His current research interests span +several areas, including worst-case control, rein- +forcement learning, and decentralized systems. He +is a student member of IEEE. +Andreas A. Malikopoulos (S’06–M’09–SM’17) re- +ceived the Diploma in mechanical engineering from +the National Technical University of Athens, Greece, +in 2000. He received M.S. and Ph.D. degrees from +the department of mechanical engineering at the +University of Michigan, Ann Arbor, Michigan, USA, +in 2004 and 2008, respectively. He is the Terri +Connor Kelly and John Kelly Career Development +Associate Professor in the Department of Mechan- +ical Engineering at the University of Delaware, the +Director of the Information and Decision Science +(IDS) Laboratory, and the Director of the Sociotechnical Systems Center. +Prior to these appointments, he was the Deputy Director and the Lead of +the Sustainable Mobility Theme of the Urban Dynamics Institute at Oak +Ridge National Laboratory, and a Senior Researcher with General Motors +Global Research & Development. His research spans several fields, including +analysis, optimization, and control of cyber-physical systems; decentralized +systems; stochastic scheduling and resource allocation problems; and learning +in complex systems. The emphasis is on applications related to smart cities, +emerging mobility systems, and sociotechnical systems. He has been an +Associate Editor of the IEEE Transactions on Intelligent Vehicles and IEEE +Transactions on Intelligent Transportation Systems from 2017 through 2020. +He is currently an Associate Editor of Automatica and IEEE Transactions on +Automatic Control. He is a member of SIAM, AAAS, and a Fellow of the +ASME. + diff --git a/8dE4T4oBgHgl3EQfdQww/content/tmp_files/load_file.txt b/8dE4T4oBgHgl3EQfdQww/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6aa52a060f9e2d079241095e0758708dd2036e82 --- /dev/null +++ b/8dE4T4oBgHgl3EQfdQww/content/tmp_files/load_file.txt @@ -0,0 +1,1626 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf,len=1625 +page_content='1 Approximate Information States for Worst-Case Control and Learning in Uncertain Systems Aditya Dave, Student Member, IEEE, Nishanth Venkatesh, Student Member, IEEE, and Andreas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Malikopoulos, Senior Member, IEEE Abstract—In this paper, we investigate discrete-time decision- making problems in uncertain systems with partially observed states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We consider a non-stochastic model, where uncontrolled disturbances acting on the system take values in bounded sets with unknown distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We present a general framework for decision-making in such problems by developing the notions of information states and approximate information states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In our definition of an information state, we introduce conditions to identify for an uncertain variable sufficient to construct a dynamic program (DP) that computes an optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We show that many information states from the literature on worst- case control actions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', the conditional range, are examples of our more general definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we relax these conditions to define approximate information states using only output variables, which can be learned from output data without knowledge of system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We use this notion to formulate an approximate DP that yields a strategy with a bounded performance loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Finally, we illustrate the application of our results in control and reinforcement learning using numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Index Terms—Uncertain systems, worst-case control, approxi- mate dynamic programming, offline reinforcement learning I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' INTRODUCTION Decision-making under incomplete information is a funda- mental problem in modern engineering applications involving cyber-physical systems [1], e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', connected and automated vehicles [2], social media platforms [3], and robot swarms [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In such applications, an agent is often required to sequentially select control inputs to a dynamic system using only partial observations at each instance of time, while simultaneously accounting for uncontrolled disturbances that can interfere with the system’s evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The most common modeling paradigm for such decision-making problems is the stochastic approach, where all disturbances to the system are considered to be random variables with known distributions, and the agent aims to select a decision-making strategy that minimizes the expected incurred cost [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Stochastic models have been utilized for problems in both control theory [6]–[13] and reinforcement learning [14]–[18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A decision-making strategy derived using the stochastic approach performs optimally on average across numerous operations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' However, this performance degrades rapidly when there is a mismatch between the distribution on disturbances considered in model- ing and the realizations encountered during implementation This research was supported by NSF under Grants CNS-2149520 and CMMI-2219761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The authors are with the Department of Mechanical Engineering, University of Delaware, Newark, DE 19716 USA (email: adidave@udel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' nish@udel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' andreas@udel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, many safety-critical applications require guarantees on the agent’s performance during each operation [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, in such applications it is inadequate to measure performance using the expected cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The non-stochastic approach is an alternate modeling paradigm for safety-critical systems, where all disturbances are considered to belong to known sets with unknown distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent aims to select a decision-making strategy that minimizes the worst-case incurred cost across a finite time horizon [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Because this approach focuses on robust- ness against worst-case realizations of the disturbances, the resulting strategy yields more conservative decisions than the stochastic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At the expense of average performance, this strategy provides concrete guarantees on the worst-case performance during each operation of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, this approach has been widely applied to systems under attack from an adversary, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', cyber-security [22] or cyber-physical systems [23], and systems where a single failure can be damaging, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', water reservoirs [24], or power systems [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In this paper, we propose a framework for non-stochastic decision-making using only partial observations in a dynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' When the system’s dynamics are known to the agent, this problem falls under the purview of control theory [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' However, many applications involve decision-making with an incomplete knowledge of the dynamics as, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', automated driving in mixed traffic [27] and human-robot coordination [28], or decision-making without a reliable state-space model, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', medical dead-end identification [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' These restrictions typically lead to formulating a reinforcement learning problem [30], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To account for both of these potential cases, we formulate our problem using only output variables without assuming a known state-space model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In our exposition, we present rigorous definitions for the notions of information states and approximate information states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Using these no- tions, a surrogate state-space model can be constructed from output variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This surrogate model can be used to for- mulate a control problem with full state observation, whose solution yields either an optimal, or an approximate strategy of the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In reinforcement learning problems, the surrogate model can be learned from output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For perfectly observed states, the agent can derive a decision- making strategy using standard techniques [32]–[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Related Work 1) Control theory: There have been numerous research efforts in control theory to study dynamic decision-making arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='05089v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='SY] 12 Jan 2023 2 problems given the system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For both stochastic and non-stochastic models, an agent can derive an optimal decision-making strategy offline using a dynamic program- ming (DP) decomposition of the problem [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For systems with perfectly observed states, it is known that, at each instance of time, the agent’s optimal action is simply a function of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Using this property in a DP facilitates the efficient computation of an optimal control strategy [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In contrast, for systems with partially observed states, any optimal action is generally a function of the agent’s entire memory of past observations and actions, which grows in size with time [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Subsequently, the domain of the optimal control strategy grows in size with time, and the corresponding DP decomposition of the problem requires a large number of computations for long time horizons [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This concern is alleviated using an information state to construct a DP decomposition instead of the memory [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The most commonly used information state in stochastic control is the belief state, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', a distribution on the state space conditioned on the agent’s memory [42]–[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A general notion of information states for stochastic control was recently defined in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For non-stochastic control problems, the DP decomposition has been simplified using two well known information states: (1) the conditional range, which is the set of feasible states at any time consistent with the agent’s memory [46] and can be used in both terminal cost [47]– [51] and instantaneous cost problems [52]–[54];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and (2) the maximum cost-to-come, which is the maximum accrued cost at any time for each state in the conditional range [55] and can be used in additive cost problems [56]–[58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A general notion of an information state for non-stochastic terminal cost problems was presented in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Information states have also been derived for mixed problems considering both stochastic and non-stochastic objectives in [60] and for robust stochastic formulations in [61], [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The advantage of using information states is that in many applications they do not grow in size with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, they generally yield a more computationally efficient DP decomposition than the entire memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' However, in problems with large state spaces, utilizing information states may not sufficiently simplify the DP to be practical [63], [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2) Reinforcement learning: The literature on reinforcement learning is concerned with decision-making when the agent may not have prior knowledge of the system’s dynamics [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For systems with perfectly observed states, these problems have been addressed using a variety of approaches [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In the stochastic formulation, both model-based [67], [68] and model-free approaches [69] have been utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In the non- stochastic formulation, the worst-case reinforcement learning problem was formulated and analyzed in [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Worst-case Q- learning was proposed for reinforcement learning problems in [71]–[74] and extended to problems with output-feedback and partially known dynamics in [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Actor-critic methods [76] and model-based off-policy learning approaches [77] have also been developed for robust control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Alternate approaches using online adaptive algorithms were proposed in [78], [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' However, in general, reinforcement learning is challenging when the agent can only access partial observations, since without knowledge of the system dynamics, the information state must be learned from data [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In the stochastic formulation, the notion of approximate information states was presented in [81] to address the chal- lenges of control and learning with partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Ap- proximate information states can improve the computational tractability of control problems with large state spaces at the cost of a bounded loss in performance [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The explicit performance bounds of a finite-memory based approximate information state were derived in [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In reinforcement learn- ing, approximate information states can be learned from output data and function as surrogate states to compute approximately optimal strategies, whose performance has been empirically validated in robotics [84] and medical care [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To the best of our knowledge, no general theory of approximate information states has yet been developed for non-stochastic formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Contributions and Organization In this paper, we develop a non-stochastic theory of approx- imate information states for both instantaneous and terminal cost problems, which can facilitate computationally efficient control, and provide a principled approach to reinforcement learning using partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The contributions of this paper are: (1) the introduction of a general notion of in- formation states (Definition 3) which yields an optimal DP decomposition for worst-case control (Theorem 1) and show that many standard results in the literature are special cases (Subsection III-C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (2) the introduction of the notion of ap- proximate information states that can either be constructed from output variables or learned from output data (Definition 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (3) the formulation of an approximate DP (Theorem 3) which computes a control strategy with a bounded loss of optimality (Theorem 4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (4) the exposition of examples of approximate information states (Subsection IV-D) along with theoretical approximation bounds (Theorems 5 - 6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and (5) the illustration of the approach in both control and learning problems using numerical examples (Subsection V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that while our theory shares conceptual similarities to the theory of approximate information states for stochas- tic problems in [81], our focus on non-stochastic problems necessitates the use of a distinct mathematical framework of uncertain variables [86] with set-valued uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We bound the worst-case approximation loss rather than the ex- pected loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that we reported preliminary results for terminal-cost control problems with finite feasible sets in [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This paper extends the preliminary work as follows: (1) we consider worst-case instantaneous cost problems which subsume terminal cost problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (2) we allow all variables to take values in continuous spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and (3) we illustrate the application of our results to a reinforcement learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The remainder of the paper proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Section II, we present our model and problem formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Section III, we define the notion of information states and prove the optimality of the corresponding DP decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Section IV, we present the notion of approximate information states, a resulting approximate DP, and theoretical bounds on the approximation loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Section V, we present a numerical 3 examples to illustrate the application of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Section VI, we draw concluding remarks and discuss ongoing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' MODEL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Preliminaries 1) Uncertain Variables: In this paper, we utilize the math- ematical framework for uncertain variables from [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' An uncertain variable is a non-stochastic analogue of a random variable with set-valued uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For a sample space Ω and a set X, an uncertain variable is a mapping X : Ω → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For any ω ∈ Ω, it has the realization X(ω) = x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The marginal range of X is the set [[X]] := {X(ω) | ω ∈ Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For two uncertain variables X ∈ X and Y ∈ Y, their joint range is [[X, Y ]] := { � X(ω), Y (ω) � | ω ∈ Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For a given realization y of Y , the conditional range of X is [[X|y]] := {X(ω) | Y (ω) = y, ω ∈ Ω} and, generally, [[X|Y ]] := {[[X|y]] | y ∈ [[Y ]]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2) Hausdorff Distance: Consider that the feasible sets X, Y are nonempty subsets of a metric space (S, η), where η(x, y) is the distance between any x ∈ X and y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we define a distance between the two sets as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The Hausdorff distance between X and Y is H(X, Y) := max � sup x∈X inf y∈Y η(x, y), sup y∈Y inf x∈X η(x, y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (1) When the two sets X, Y are bounded, the Hausdorff distance in (1) constitutes a pseudo-metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', H(X, Y) = 0 if and only if closure(X) = closure(Y) [87, Appendix].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' When both X, Y are compact, the Hausdorff distance is a metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', H(X, Y) = 0 if and only if X = Y [88, Chapter 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In both cases, the distance H satisfies the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 3) L-invertible Functions: Consider a function f : X → Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For any y ∈ Y, the pre-image of the function is f −1(y) = � x ∈ X | f(x) = y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we use the Hausdorff distance to define the notion of an L-invertible function as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A function f : X → Y is called L-invertible if there exists a constant Lf −1 ∈R≥0 such that for all y1, y2 ∈ Y: H � f −1(y1), f −1(y2) � ≤ Lf −1 · η(y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (2) For uncertain variables X ∈ X and Y ∈ Y such that Y = f(X), the pre-image of f given a realization y ∈ [[Y ]] equals the conditional range [[X|y]], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', f −1(y) = [[X|y]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, if f is L-invertible, we equivalently state that for all y1, y2 ∈ [[Y ]]: H � [[X|y1]], [[X|y2]] � ≤ LX|Y · η(y1, y2), (3) where LX|Y = Lf −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Problem Formulation We consider an agent which seeks to control the trajectory of an uncertain system by selecting actions over T ∈ N discrete time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each time t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, the agent receives an observation from the system, denoted by the uncertain variable Yt ∈ Yt, and generates a control action denoted by the uncertain variable Ut ∈ Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' After generating the action at each t, the agent incurs a cost denoted by the uncertain variable Ct ∈ Ct ⊂ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To account for the case that the agent may have no knowledge of a state-space model, we describe the system dynamics using an input-output model, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, the system receives two inputs: the control action Ut, and an uncontrolled disturbance denoted by the uncertain variable Wt ∈ Wt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We consider that the uncontrolled disturbances {Wt : t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T} constitute a sequence of independent uncertain variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' After receiving the inputs at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, the system generates two outputs: Yt+1 = ht+1(W0:t, U0:t), (4) Ct = dt(W0:t, U0:t), (5) for some observation function ht+1 : �t ℓ=0 Wℓ × �t ℓ=0 Uℓ → Yt+1 and cost function dt : �t ℓ=0 Wℓ × �t ℓ=0 Uℓ → Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The initial observation is generated as Y0 = h0(W0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent has perfect recall of the history of observations and control actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The memory of the agent at each t is denoted by the uncertain variable Mt := (Y0:t, U0:t−1), which takes values in the set Mt := �t ℓ=0 Yℓ × �t−1 ℓ=0 Uℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent uses the memory Mt and a control law gt : Mt → Ut at each t to generate the action Ut = gt(Mt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We denote the control strategy by g := (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , gT ) and the set of all feasible control strategies by G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The performance of a strategy g ∈ G is measured by the worst-case or maximum instantaneous cost J (g) := max t=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=',T sup w0:t∈[[W0:t]] Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (6) Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The optimization problem of the agent is to derive the control strategy g ∈ G such that infg∈G J (g), given the marginal ranges {[[Ut]], [[Wt]], [[Ct]], [[Yt]] | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T} and the functions {ht, dt | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' If there exists a strategy g∗ ∈ G that achieves the optimal performance in Problem 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', g∗ = arg ming∈G J (g), we refer to it as an optimal control strategy for Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Our aim is to tractably compute an optimal strategy if one exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In our modeling framework, we impose the following assumptions: Assumption 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We consider that the sets {Ut, Wt, Yt | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T} and {Ct | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T} are all bounded subsets of a metric space (S, η) and R≥0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Assumption 1 allows for both continuous and finite valued feasible sets, while ensuring that the marginal range of each uncertain variable in the problem formulation is also bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Assumption 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The observation functions {ht | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T} of the system are both Lipschitz and L-invertible, whereas the cost functions {dt | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T} are Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Assumption 2 is satisfied by a large class of observation functions, including: (1) all functions with compact domains and finite co-domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and (2) bi-Lipschitz functions, like linear functions, with compact domains and compact co- domains (see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We will require both assumptions in Section IV when deriving the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In our exposition, we also consider a special case of (6), called the maximum terminal cost criterion, given by J tm(g) := sup w0:T ∈[[W0:T ]] CT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (7) 4 In addition to the general results for Problem 1, we often present results specifically for systems which utilize (7) as the performance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This serves two purposes: (1) the results are often easier to interpret for a terminal cost problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and (2) these results can be extended to additive cost problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We explicitly present this extension in Subsection III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We derive our results for Problem 1 with known dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' However, our main results in Section IV can also be used in learning problems with unknown dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We illustrate this application with an example in Subsection V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' DYNAMIC PROGRAMS AND INFORMATION STATES In this section, we first present a memory-based DP decom- position for Problem 1 which computes the optimal value of the performance criterion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This will serve as a reference to analyze subsequent DPs in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we highlight the DP’s computational challenges and present information states in Subsections III-A and III-B to alleviate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Subsection III-C we present examples of information states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To arrive at the memory-based DP, we construct a “new” perfectly observed system whose state at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T is the memory Mt, which evolves as Mt+1 = (Mt, Ut, Yt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, for given realizations mt ∈ [[Mt]] and ut ∈ [[Ut]], the maximum incurred cost at time t can be written as sup w0:t∈[[W0:t]] Ct = sup ct∈[[Ct]]gct = sup mt,ut∈[[Mt,Ut]]g sup ct∈[[Ct|mt,ut]]gct, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, where [[Ct]]g, [[Mt, Ut]]g and [[Ct|mt, ut]]g are the respective marginal ranges and the con- ditional range induced by strategy g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Recall that mt = (y0:t, u0:t−1) and thus, we can expand the conditional range as [[Ct|mt, ut]]g = � ct ∈ Ct �� ∃ w0:t ∈ [[W0:t]] such that ct = dt(w0:t, u0:t), yℓ = ht(w0:ℓ, u0:ℓ−1), ∀ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , t � = [[Ct|mt, ut]], (8) which shows that [[Ct|mt, ut]]g is independent of the choice of strategy g, hence we can drop g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we define et(mt, ut) := supct∈[[Ct|mt,ut]] ct, independent of g, and state that sup mt,ut∈[[Mt,Ut]]g sup ct∈[[Ct|mt,ut]]gct = sup mt,ut∈[[Mt,Ut]]g et(mt, ut) = sup w0:t∈[[W0:t]] et(Mt, Ut), (9) where, in the second equality, note that the marginal range of external disturbances [[W0:t]] is independent of the strategy g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Since et(Mt, Ut) is a function of the new state Mt and control action Ut, it serves as an incurred cost at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T in our new perfectly observed system [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The new instantaneous performance criterion is E(g) := supt=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=',T supw0:t∈[[W0:t]] et(Mt, Ut) and from (9), E(g) = J (g) for any g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Subsequently, any strategy which achieves the optimal performance in the new system is optimal for Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' If such an optimal strategy exists, we can compute it using a standard DP for perfectly observed systems, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, for each mt ∈ [[Mt]] and ut ∈ [[Ut]] we recursively define the value functions Qt(mt, ut) := max � sup ct∈[[Ct|mt,ut]] ct, sup mt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) � , (10) Vt(mt) := inf ut∈[[Ut]] Qt(mt, ut), (11) where VT +1(mT +1) := 0, identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We define the ex- tra value function VT +1 to ensure that the right hand side (RHS) of (10) is well defined at time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we can show using standard arguments [48], [53] that the optimal value of Problem 1 is infg∈G J (g) = supm0∈[[M0]] V0(m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, at any t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, if there exists an action u∗ t ∈ [[Ut]] which achieves the infimum in the RHS of (11), then g∗ t (mt) := arg minut∈[[Ut]] Qt(mt, ut) gives an optimal control law at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' If the infimum is achieved at each t, the control strategy g∗ = (g∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , g∗ T ) is optimal for this system and Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The DP (10) - (11) can be specialized to the terminal cost criterion (7) by defining for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T −1, Qtm t (mt, ut) := sup mt+1∈[[Mt+1|mt,ut]] V tm t+1(mt+1), (12) V tm t (mt) := inf ut∈[[Ut]] Qtm t (mt, ut), (13) where Qtm T (mT , uT ) := supcT ∈[[CT |mT ,uT ]] cT and V tm T (mT ) := infuT ∈[[Ut]] Qtm T (mT , uT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We will use this terminal cost DP to simplify the exposition in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A valid argument referring to the minimum of the RHS of (11) at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T is both a necessary and suf- ficient condition to ensure the existence of an optimal control strategy in Problem 1 [48], [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider that marginal ranges of all uncertain variables are compact rather than just bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' From Assumption 2, the observation and cost functions at each t are Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Using these properties in (12) - (13), we can show that the value functions are continuous and the conditional ranges are compact for all t, which implies that the minimum is achieved in the RHS of (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, compactness of all marginal ranges and Assumptions 1 - 2 constitute sufficient conditions for existence of an optimal solution to Problem 1, which is consistent with the conditions given in [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' However, we continue using sup and inf in our exposition since we use only Assumptions 1 - 2 to establish our results without assuming compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In the RHS of (11) at each t, we are required to solve an optimization for each mt ∈ [[Mt]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This is computationally challenging for longer horizons as the size of the set [[Mt]] increases with time t with addition of new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This concern motivates our search for an alternate DP decomposition which can derive an optimal control strategy while potentially achieving more favourable computational properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We present such a DP decomposition in Subsection III-A by identifying an uncertain variable, called an informa- tion state, which can be used to generate an optimal control action at each time step instead of the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Information States In this subsection, we define information states for partially observed uncertain systems, use them in a DP decomposition, and prove it yields the optimal value for Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 5 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' An information state for Problem 1 at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T is an uncertain variable Πt = σt(Mt) taking values in a bounded set Pt and generated by a function σt : Mt → Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, and for all mt ∈ [[Mt]] and ut ∈ [[Ut]], it satisfies the following properties: 1) Sufficient to evaluate cost: sup ct∈[[Ct|mt,ut]] ct = sup ct∈[[Ct|σt(mt),ut]] ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (14) 2) Sufficient to predict itself: [[Πt+1|mt, ut]] = [[Πt+1|σt(mt), ut]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (15) We can use the information states from Definition 3 directly in a DP, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, for all πt ∈ [[Πt]] and ut ∈ [[Ut]], we recursively define the value functions ¯Qt(πt, ut) := max � sup ct∈[[Ct|πt,ut]] ct, sup πt+1∈[[Πt+1|πt,ut]] ¯Vt+1(πt+1) � , (16) ¯Vt(πt) := inf ut∈[[Ut]] ¯Qt(πt, ut), (17) where ¯VT +1(πT +1) := 0 identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' If the minimum in the RHS of (17) exists at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, then this DP yields a control law at time t as ¯g∗ t (πt) := arg minut∈[[Ut]] ¯Qt(πt, ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we prove that the DP (16) - (17) computes the same value as the optimal DP (10) - (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let Πt = σt(Mt) be an information state at any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, for all t, and for all mt ∈ [[Mt]] and ut ∈ [[Ut]], Qt(mt, ut)= ¯Qt � σt(mt), ut � and Vt(mt)= ¯Vt � σt(mt) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let mt ∈ [[Mt]] and ut ∈ [[Ut]] be given realizations of Mt and Ut, respectively, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We prove the result by mathematical induction starting at the last time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time T + 1, (18) holds trivially because VT +1(mT +1) = ¯VT +1(σT +1(mT +1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This forms the basis of our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, for any t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, we consider the induction hypothesis that Vt+1(mt+1) = ¯Vt+1(σt+1(mt+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Given the hypothesis, we first prove that Qt(mt, ut) = ¯Qt(σt(mt), ut) by comparing the RHS of (10) to the RHS of (16) term by term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The first terms are equal by direct application of (14) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we use the induction hypothesis for the second term in the RHS of (10), to state that supmt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) = supmt+1∈[[Mt+1|mt,ut]] ¯Vt+1(σt+1(mt+1)) = supσt+1(mt+1)∈[[Πt+1|σt(mt),ut]] ¯Vt+1(σt+1(mt+1)), where, in the second equality, we use the fact that [[Πt+1|mt, ut]] = � σt+1(mt+1) ∈ Pt+1 ��mt+1 ∈ [[Mt+1|mt, ut]] � and (15) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This establishes that the second term in the RHS of (10) equals the second term in the RHS of (16) and subsequently, that given the induction hypothesis for time t + 1, we have Qt(mt, ut) = ¯Qt(σt(mt), ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we minimize both sides of the equality with respect to ut ∈ [[Ut]], and use the definitions of the value functions in (11) and (17) to write that Vt(mt) = infut∈Ut Qt(mt, ut) = infut∈Ut ¯Qt � σt(mt), ut � = Vt � σt(mt) � , which proves the induction hypothesis at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, starting at time T + 1, the result follows for all t using mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Theorem 1 implies that (16) - (17) is an optimal DP decom- position for Problem 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', if an optimal strategy exists for this DP, it yields an optimal solution to Problem 1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider a control strategy ¯g∗ = (¯g∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , ¯g∗ T ) computed using (16) - (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can construct a corresponding memory-based strategy g = (g0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , gT ) by defining gt(mt) := ¯g∗ t (σt(mt)) for all mt ∈ [[Mt]] and t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, using Theorem 1, we conclude that g achieves the infimum value at each t and thus, constitutes an optimal solution to Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In practice, using an information state to construct the DP decomposition is useful computationally only if, for most time steps in t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, either the value functions in (16) - (17) have useful properties like concavity, or the set Pt is smaller than Mt for some measure of size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Potentially useful measures of sizes for sets include the number of elements, set diameter, and set dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We present some examples of information states for different systems in Subsection III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Alternate Characterization of Information States When exploring whether an uncertain variable is a valid candidate to be considered an information state, it may be difficult to verify the second property (15) in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In this subsection, we present two stronger conditions to replace (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Specifically, at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, to establish that Πt = σt(Mt) is a valid information state, it is sufficient to satisfy the following conditions instead of (15): 1) State-like evolution: There exists a function ¯ft : Pt × Ut × Yt+1 → Pt+1, independent of the strategy g, such that Πt+1 = ¯ft(Πt, Ut, Yt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (19) 2) Sufficient to predict observations: For all mt ∈ Mt and ut ∈ Ut, [[Yt+1|mt, ut]] = [[Yt+1|σt(mt), ut]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (20) Next, we prove that these two conditions, in addition to (14) from Definition 3 are sufficient to identify an information state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, if an uncertain variable Πt = σt(Mt) satisfies (19) - (20), it also satisfies (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T and mt ∈ Mt, suppose that πt = σt(mt) satisfy (19) - (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we substitute (19) into the left hand side (LHS) of (15) to state that [[Πt+1|mt, ut]] = [[ ¯ft(σt(mt), ut, Yt+1) | mt, ut]] = � ¯ft(σt(mt), ut, yt+1) �� yt+1 ∈ [[Yt+1|mt, ut]] � , (21) where, in the second equality, we write the conditional range as a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, using (20) on the range of ob- servations in the conditioning of (21), we can state that � ¯ft(σt(mt), ut, yt+1) �� yt+1 ∈ [[Yt+1|mt, ut]] � = � ¯ft(σt(mt), ut, yt+1) �� yt+1 ∈ [[Yt+1|σt(mt), ut]] � = [[ ¯ft(σt(mt), ut, Yt+1) | σt(mt), ut]] = [[Πt+1|σt(mt), ut]], which is equal to the RHS of (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Examples of Information States In this subsection, we present examples of information states which satisfy the conditions in Definition 3 for systems with a given state-space model to describe their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, let Xt be a known set of feasible states and let the system’s state be denoted by an uncertain variable Xt ∈ Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent’s observation is given by Yt = ht(Xt, Nt), where Nt ∈ Nt is a noise in observation, and the agent incurs a cost Ct = dt(Xt, Ut) when they implement an action Ut ∈ Ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Starting at X0 ∈ X0, the state evolution is given by Xt+1 = ft(Xt, Ut, Wt) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Each uncertain variable in {X0, Wt, Nt | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T} is independent of all other uncertain variables in that set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we present information states for different cases which may offer computational advantages over using the entire memory: 1) Systems with perfectly observed states: Consider that Yt = Xt for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, an information state at each t is Πt = Xt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', the state itself [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' It takes values in the set Xt and satisfies (14) - (15) for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that it is always computationally advantageous to construct a DP decomposition using the state at each time step instead of the entire memory of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2) Systems with partially observed states: Generally in a partially observed system with a known state space, an information state at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T is the con- ditional range Πt = [[Xt|Mt]], which is a set-valued uncertain variable [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Explicitly, for a given realization of the memory mt ∈ Mt at time t, the conditional range takes the realization Pt := � xt ∈ Xt �� ∃x0 ∈ X0, w0:t−1 ∈ �t−1 ℓ=0 Wℓ, n0:t ∈ �t ℓ=0 Nℓ such that yt = ht(xt, nt), xℓ+1 = fℓ(xℓ, uℓ, wℓ), yℓ = hℓ(xℓ, nℓ) for all ℓ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , t−1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We denote the realization by Pt instead of πt to highlight that it is a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To establish that the conditional range is a valid information state, it is easier to verify the alternate conditions (19) and (20) instead of property (15) in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Generally, it is computationally advantageous to construct a DP decomposition using the conditional range instead of the memory for systems with longer time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 3) Systems with additive costs: Consider a system with partially observed states with an additive performance cri- terion J ad(g) := supx0,w0:T ,n0:T �T t=0 dt(Xt, Ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can construct a DP and an information state for an additive cost problem by recasting it as a terminal cost problem [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At t = 0, we define A0 := 0 and for all t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, we recursively define an uncertain variable At ∈ At as At := At−1 + dt−1(Xt−1, Ut−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that At tracks the cost incurred by the system up to time t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', before the action Ut has been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, at each t, we consider an augmented state for the system, St = (Xt, At) and note that it evolves as St+1 = � ft(Xt, Ut, Wt), At + dt(Xt, Ut) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, this augmentation yields a terminal cost problem with the cost AT + cT (XT , UT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, we can derive an optimal control strategy using the terminal cost DP and, as in case 2, an information state at each t is the conditional range Πt = [[Xt, At|Mt]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Generally, this information state is useful for systems with longer time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The conditions in Definition 3 can help us identify information states for systems with known dynamics and simplify the DP decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' However, many applications with large state spaces may require a further improvement in computational tractability, even at the cost of optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Moreover, in certain applications, we need to learn a represen- tation of the information state using limited observations with incomplete knowledge of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Information states are insufficient to account for these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, in Section IV, we introduce approximate information states that can address the above concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' APPROXIMATE INFORMATION STATES In this section, we define approximate information states by relaxing the conditions given in Definition 3, and utilize them to develop an approximate DP decomposition which computes a sub-optimal control strategy for Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Subsection IV-A, we derive the preliminary results required to establish useful properties of approximate information states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, in Subsection IV-B, we prove these properties, namely, the Lipischitz continuity of approximate value functions, and the following error bounds: (1) an upper bound on the error when the optimal value functions are estimated using approximate value functions, and (2) an upper bound on the loss in performance when control actions are generated using a sub- optimal control strategy instead of an optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' An approximate information state for Problem 1 at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T is an uncertain variable ˆΠt = ˆσt(Mt) taking values in a bounded set ˆPt and generated by an L- invertible function ˆσt : Mt → ˆPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, there exist parameters ϵt, δt, λt ∈ R≥0 such that for all mt ∈ [[Mt]] and ut ∈ [[Ut]], it satisfies the properties: 1) Sufficient to approximate cost: ��� sup ct∈[[Ct|mt,ut]] ct − sup ct∈[[Ct|ˆσt(mt),ut]] ct ��� ≤ ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (22) 2) Sufficient to approximate evolution: We define the sets Kt+1 := [[ˆΠt+1 | mt, ut]] and ˆKt+1 := [[ˆΠt+1 | ˆσt(mt), ut]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, it holds that H(Kt+1, ˆKt+1) ≤ δt, (23) where recall that H is the Hausdorff distance in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 3) Lipschitz-like evolution: For all ˆπ1 t , ˆπ2 t ∈ [[ˆΠt]], H � [[ˆΠt+1|ˆπ1 t , ut]], [[ˆΠt+1|ˆπ2 t , ut]] � ≤ λt · η(ˆπ1 t , ˆπ2 t ), (24) where η is an appropriate metric on ˆPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Using the approximate information state in Definition 4, we can construct a DP as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t, for all ˆπt ∈ [[ˆΠt]] and ut ∈ [[Ut]], we recursively define the value functions ˆQt(ˆπt, ut) := max � sup ct∈[[Ct|ˆπt,ut]] ct, sup ˆπt+1∈[[ˆΠt+1|ˆπt,ut]] ˆVt+1(ˆπt+1) � , (25) ˆVt(ˆπt) := inf ut∈[[Ut]] ˆQt(ˆπt, ut), (26) 7 where ˆVT +1(ˆπT +1) := 0 identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' If there exists a minimiz- ing argument in the RHS of (26) at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, then ˆg∗ t (ˆπt) := arg minut∈Ut ˆQt(ˆπt, ut) constitutes an approximate control law at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, we call ˆg∗ = (ˆg∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , ˆg∗ T ) an approximately optimal strategy for Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Subsec- tion IV-B, we derive performance guarantees on the approxi- mate DP and control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' As we showed in Section III, we can specialize this DP for terminal cost problems, with the value functions for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T − 1 given by ˆQtm t (ˆπt, ut) := sup ˆπt+1∈[[ˆΠt+1|ˆπt,ut]] ˆV tm t+1(ˆπt+1), (27) ˆV tm t (ˆπt) := inf ut∈Ut ˆQtm t (ˆπt, ut), (28) and ˆQtm T (ˆπT , uT ) := supcT ∈[[CT |ˆπT ,uT ]] cT and ˆV tm T (ˆπT ) := infuT ∈UT ˆQtm T (ˆπT , uT ) at time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The conditions in Definition 4 can be investigated using only output variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, an approximate information state can be learned from output data without knowledge of dynamics, as illustrated in Subsection V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Preliminary Results In this subsection, we derive results necessary to prove the properties of the approximate DP in Subsection IV-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider three bounded subsets X, Y and Z of a metric space (S, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let X ∈ X, Y ∈ Y and Z ∈ Z be uncertain variables satisfying Y = g(X), where g : X → Y is L-invertible, and Z = h(X), where h : X → Z is Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, there exists an LZ|Y ∈ R≥0 such that: H([[Z|y1]],[[Z|y2]]) ≤ LZ|Y ·η(y1, y2), ∀y1, y2 ∈ [[Y ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We prove the result by constructing a feasible constant LZ|Y ∈ R≥0 which ensures that (29) is satisfied for all y1, y2 ∈ [[Y ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We begin by using the definition of the Hausdorff distance in (1) to expand the LHS of (29) as H � [[Z|y1]], [[Z|y2]] � = max � sup x1∈g−1(y1) inf x2∈g−1(y2) η � h(x1), h(x2) � , sup x2∈g−1(y2) inf x1∈g−1(y1) η � h(x1), h(x2) �� , (30) where, note that [[Z|y]] = � z ∈ Z | z = h(x), ∀x ∈ g−1(y) � for any realization y ∈ [[Y ]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, recall that h is Lipschitz continuous with a constant Lh ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Substituting this property into the RHS of (30), we write that H � [[Z|y1]], [[Z|y2]] � ≤ Lh · max � supx1∈g−1(y1) infx2∈g−1(y2) η(x1, x2), supx2∈g−1(y2) infx1∈g−1(y1) η(x1, x2) � = Lh · H � g−1(y1), g−1(y2) � = Lh · Lg−1 · η(y1, y2), where, in the second equality, we use the L-invertibile property of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, the result follows by selecting LZ|Y := Lh · Lg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider a bounded set X and two functions f : X → R and g : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, | sup x∈X f(x) − sup x∈X g(x)| ≤ sup x∈X |f(x) − g(x)|, (31) | inf x∈X f(x) − inf x∈X g(x)| ≤ sup x∈X |f(x) − g(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (32) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' First, we prove (31) by considering two mutually ex- clusive cases which cover all possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 1: We consider supx∈X f(x) ≥ supx∈X g(x), which implies | supx∈X f(x)− supx∈X g(x)| = supx∈X f(x) − supx∈X g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For any in- finitesimally small β > 0, we define x(β) ∈ X as an element which satisfies f(x(β)) + β ≥ supx∈X f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, supx∈X f(x) − supx∈X g(x) ≤ f(x(β)) + β − supx∈X g(x) ≤ f(x(β)) + β − g(x(β)) ≤ supx∈X |f(x) − g(x)| + β for all β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Therefore, supx∈X f(x) − supx∈X g(x) ≤ supx∈X |f(x) − g(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 2: supx∈X f(x) < supx∈X g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The proof can be completed using similar arguments as in Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, (32) follows from similar arguments as (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For any four scalars a, b, c, d ∈ R, | max{a, b} − max{c, d}| ≤ max{|a − c|, |b − d|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (33) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We prove this result by considering four cases which are mutually exclusive but cover all possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 1: For a ≥ b and c ≥ d: The result holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 2: For a < b and c ≥ d: The LHS can be expanded as | max{a, b} − max{c, d}| = |b−c|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, if b ≥ c, we use c ≥ d to conclude that |b − c| < |b − d|, else if c > b, we use b > a to conclude that |c − b| < |c − a|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, | max{a, b} − max{c, d}| ≤ max{|a − c|, |b − d|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 3: For a < b and c < d: The result holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 4: For a ≥ b and c < d: The proof follows from the same sequence of arguments as Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider two bounded subsets A, B of a metric space (X, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let f : X → R be a bounded continuous function with a Lipschitz constant Lf ∈ R≥0 on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, �� sup a∈A f(a) − sup b∈B f(b) �� ≤ Lf · H(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (34) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We prove this result by considering two cases which are mutually exclusive but cover all the possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 1: supa∈A f(a) ≥ supb∈B f(b), which implies | supa∈A f(a) − supb∈B f(b)| = supa∈A f(a) − supb∈B f(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We define the non-empty set A1(β) := {a ∈ A | f(a) + β ≥ supb∈B f(b)} for any infinitesimal β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, supa∈A f(a) − supb∈B f(b) ≤ supa∈A1(β) f(a) + β − supb∈B f(b) ≤ supa∈A1(β) infb∈B(f(a)−f(b))+β ≤ supa∈A infb∈B |f(a)− f(b)| + β ≤ Lf · supa∈A infb∈B η(a, b) + β for all β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This implies that | supa∈A f(a) − supb∈B f(b)| ≤ Lf · supa∈A infb∈B η(a, b) ≤ Lf · H(A, B), where, in the second inequality, we invoke the definition of the Hausdorff dis- tance in (1) to complete the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Case 2: supa∈A f(a) < supb∈B f(b) and we can prove the result using the same sequence of arguments as case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' As a direct consequence of Lemma 5, we can also establish the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider two bounded subsets Y, Z of Rn, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For two uncertain variables Y ∈ Y and Z ∈ Z, let the conditional range [[Z|y]] satisfy H � [[Z|y1]], [[Z|y2]] � ≤ LZ|Y · η(y1, y2) for all realizations y1, y2 ∈ Y of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, for a continuous function f : Z → R≥0 with a Lipschitz 8 constant Lf, we can use (34) from Lemma 5 to state that for all y1, y2 ∈ [[Y ]]: ��� sup z1∈[[Z|y1]] f(z1) − sup z2∈[[Z|y2]] f(z2) ��� ≤ LZ|Y ·Lf ·η(y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (35) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Properties of Approximate Information States In this subsection, we present several properties of the approximate DP (25) - (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To begin, we prove in Theorem 2 that each approximate value function is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This property subsequently allows us to establish error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In the approximate DP (25) - (26), the value functions ˆQt(ˆπt, ut) and ˆVt(ˆπt) are Lipschitz continuous with respect to ˆπt ∈ [[ˆΠt]] for all ut ∈ [[Ut]] and t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We prove the Lipschitz continuity of the value func- tions by constructing a valid candidate for the Lipschitz con- stant L ˆVt at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, using mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time T + 1, recall that ˆVT +1(ˆπT +1) = 0 identically and thus, ˆVT +1(ˆπT +1) is trivially Lipschitz continuous with a con- stant L ˆVT +1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This forms the basis of our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, we consider the induction hypothesis that ˆQt+1(ˆπt+1, ut+1) and ˆVt+1(ˆπt+1) are Lipschitz continuous with respect to ˆπt+1 ∈ [[ˆΠt+1]] for all ut+1 ∈ [[Ut+1]], and denote the constant by L ˆVt+1 ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time t, we first prove the result for the value function ˆQt(ˆπt, ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let ˆπ1 t , ˆπ2 t ∈ [[ˆΠt]] be two possible realizations of ˆΠt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, using the definition (25) of ˆQt(ˆπt, ut) and (33) from Lemma 4, we state that | ˆQt(ˆπ1 t , ut) − ˆQt(ˆπ2 t , ut)| ≤ max ���� sup c1 t ∈[[Ct|ˆπ1 t ,ut]] c1 t − sup c2 t ∈[[Ct|ˆπ2 t ,ut]] c2 t ���, ��� sup ˆπ1 t+1∈[[ˆΠt+1|ˆπ1 t ,ut]] ˆVt+1(ˆπ1 t+1) − sup ˆπ2 t+1∈[[ˆΠt+1|ˆπ2 t ,ut]] ˆVt+1(ˆπ2 t+1) ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (36) We consider the RHS of (36) term by term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In the first term, we note that for all ˆπt ∈ [[ˆΠt]], sup ct∈[[Ct|ˆπt,ut]] ct = sup mt∈[[Mt|ˆπt]] � sup ct∈[[Ct|mt,ut]] ct � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (37) In the RHS of (37), recall from Assumption 2 that the uncertain variable Ct is a Lipschitz function of (W0:t, U0:t), and (Mt, Ut) is an L-invertible function of (W0:t, U0:t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, using (29) from Lemma 2, there exists a constant LC|M,U such that H([[Ct|m1 t, ut]], [[Ct|m2 t, ut]]) ≤ LM|C,U ·η(m1 t, m2 t) for all m1, m2 ∈ [[Mt]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, we use (35) to state that ��� sup c1 t ∈[[Ct|m1 t ,ut]] c1 t − sup c2 t ∈[[Ct|m2 t ,ut]] c2 t ��� ≤ LM|C,U · Lct · η(m1 t, m2 t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (38) Then, consider a function et : Mt × Ut → R≥0 defined as et(mt, ut) := supct∈[[Ct|mt,ut]] ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' As a direct consequence of (38), et is Lipschitz continuous with respect to mt with a constant Let := LM|C,U · Lct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Using (37) and the definition of et in the first term in the RHS of (36), ��� sup c1 t ∈[[Ct|ˆπ1 t ,ut]] c1 t − sup c2 t ∈[[Ct|ˆπ2 t ,ut]] c2 t ��� = ��� sup m1 t ∈[[Mt|ˆπ1 t ]] et(m1 t, ut) − sup m2 t ∈[[Mt|ˆπ2 t ]] et(m2 t, ut) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (39) In (39), recall that the uncertain variable ˆΠt is an L-invertible function of Mt and thus, the conditional range [[Mt|ˆπt]] satisfies (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we use (35) once more to state that ��� sup c1 t ∈[[Ct|ˆπ1 t ,ut]] c1 t − sup c2 t ∈[[Ct|ˆπ2 t ,ut]] c2 t ���≤LMt|ˆΠt·Let·η(ˆπ1 t , ˆπ2 t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (40) In the second term in the RHS of (36), we use the induction hypothesis and (34) from Lemma 5 to write that ��� sup ˆπ1 t+1∈[[ˆΠt+1|ˆπ1 t ,ut]] ˆVt+1(ˆπ1 t+1)− sup ˆπ2 t+1∈[[ˆΠt+1|ˆπ2 t ,ut]] ˆVt+1(ˆπ2 t+1) ��� ≤ L ˆVt+1 · H � [[ˆΠt+1|ˆπ1 t , ut]], [[ˆΠt+1|ˆπ2 t , ut]] � ≤ L ˆVt+1 · λt · η(ˆπ1 t , ˆπ2 t ), (41) where, in the second inequality, we use the third prop- erty (24) of approximate information states in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, the proof for ˆQt(ˆπt, ut) is complete by substi- tuting (40) and (41) into the RHS of (36) and defining L ˆ Qt := max � LMt|ˆΠt · Let, L ˆVt+1 · λt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To prove the result for ˆVt(ˆπt), we use (32) from Lemma 3 to state that �� ˆVt(ˆπ1 t )− ˆVt(ˆπ2 t ) �� = �� infut∈[[Ut]] ˆQt(ˆπ1 t , ut) − infut∈[[Ut]] ˆQt(ˆπ2 t , ut) �� ≤ suput∈[[Ut]] �� ˆQt(ˆπ1 t , ut) − ˆQt(ˆπ2 t , ut) �� ≤ L ˆ Qt · η(ˆπ1 t , ˆπ2 t ), which proves the induction hypothesis at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, the result holds using mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we establish an upper bound on the approximation error when the value functions of the optimal DP (10) - (11) are estimated using the approximate DP (25) - (26) at each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let L ˆVt+1 be the Lipschitz constant of ˆVt+1 for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, for all mt ∈ [[Mt]] and ut ∈ [[Ut]], |Qt(mt, ut) − ˆQt(ˆσt(mt), ut)| ≤ αt, (42) |Vt(mt) − ˆVt(ˆσt(mt))| ≤ αt, (43) where αt = max(ϵt, αt+1 + L ˆVt+1 · δt) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T and αT +1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, let mt ∈ [[Mt]] and ut ∈ [[Ut]] be realizations of Mt and Ut, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We prove both results by mathematical induction, starting with time step T + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At T + 1, by definition, VT +1(mT +1, uT +1) = VT +1(ˆσT +1(mT +1)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This forms the basis of our math- ematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, at each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, we consider the induction hypothesis |Vt+1(mt+1)− ˆVt+1(ˆσt+1(mt+1))| ≤ αt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time t, we first prove (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Using (33) from Lemma 4 in the LHS of (42) to state that |Qt(mt, ut) − ˆQt(ˆσt(mt), ut)| ≤ max ���� sup ct∈[[Ct|mt,ut]] ct − sup ct∈[[Ct|ˆσt(mt),ut]] ct ���, ��� sup mt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) − sup ˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] ˆVt+1(ˆπt+1) ��� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (44) 9 We consider the RHS of (44) term-by-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' By direct applica- tion of (22) in Definition 4, the first term in the RHS satisfies ��� sup ct∈[[Ct|mt,ut]] ct − sup ct∈[[Ct|ˆσt(mt),ut]] ct ��� ≤ ϵt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (45) For the second term in the RHS of (44), we use the triangle inequality to write that ��� sup mt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) − sup ˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] ˆVt+1(ˆπt+1) ��� ≤ ��� sup mt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) − sup ˆσt+1(mt+1)∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆσt+1(mt+1)) ���+ ��� sup ˆπt+1∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆπt+1) − sup ˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] ˆVt+1(ˆπt+1) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (46) For the first term in the RHS of (46), we first note that supˆσt+1(mt+1)∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆσt+1(mt+1)) = supmt+1∈[[Mt+1|mt,ut]] ˆVt+1(ˆσt+1(mt+1)) because [[ˆΠt+1 | mt, ut]] = {ˆσt+1(mt+1) ∈ ˆPt | mt+1 ∈ [[Mt+1 | mt, ut]]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we can state that �� supmt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1)−supˆσt+1(mt+1)∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆσt+1(mt+1)) �� ≤ supmt+1∈[[Mt+1|mt,ut]] ��Vt+1(mt+1) − ˆVt+1(ˆσt+1(mt+1)) �� ≤ αt+1, where, in the first inequality, we use (31) from Lemma 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and, in the second inequality, we use the induction hypothesis for time t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The second term in the RHS of (46) satisfies �� supˆπt+1∈[[ˆΠt+1|mt,ut]] ˆVt+1(ˆπt+1) − supˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] ˆVt+1(ˆπt+1) �� ≤ L ˆVt+1· δt using (34) from Lemma 5 and (23) from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Substituting the respective inequalities for each term in the RHS of (46) yields �� supmt+1∈[[Mt+1|mt,ut]] Vt+1(mt+1) − supˆπt+1∈[[ˆΠt+1|ˆσt(mt),ut]] ˆVt+1(ˆπt+1) �� ≤ αt+1 + L ˆVt+1 · δt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We complete the proof for (42) by substituting the inequalities in the RHS of (45) and (46) into the RHS of (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we prove (43) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Using the definition of the value functions in the LHS of (43), we write that |Vt(mt) − ˆVt(ˆσt(mt))| = ��� inf ut∈[[Ut]] Qt(mt, ut) − inf ut∈[[Ut]] ˆQt(ˆσt(mt), ut) ��� ≤ sup ut∈[[Ut]] |Qt(mt, ut) − ˆQt(ˆσt(mt), ut)| ≤ max{ϵt, αt+1 + L ˆVt+1 · δt}, (47) where in the first inequality, we use (32) from Lemma 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and in the second inequality, we use (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, the results hold for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T using mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' After bounding the approximation error for value functions, we also seek to bound the maximum performance loss in the implementation of an approximately optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Con- sider an approximate strategy ˆg∗ := (ˆg∗ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , ˆg∗ T ) computed using (25) - (26), where ˆg∗ t (ˆπt) = arg minut∈[[Ut]] ˆQt(ˆπt, ut) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can construct an approximate memory-based strategy gap = (gap 0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , gap T ) by selecting the control law gap t (mt) := ˆg∗ t (ˆσt(mt)) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that gap is equivalent to ˆg∗ because they generate the same actions at each t and subsequently, yield the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, we evaluate the performance of gap to determine the quality of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To this end, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, for all mt ∈ [[Mt]] and ut ∈ [[Ut]], we define Θt(mt, ut) := max � sup ct∈[[Ct|mt,ut]] ct, sup mt+1∈[[Mt+1|mt,ut]] Λt+1(mt+1) � , (48) Λt(mt) :=Θt(mt, gap t (mt)), (49) where ΛT +1(mT +1) := 0, identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, the performance of the memory-based approximate strategy gap is Λ0(m0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In contrast, recall that the performance of an optimal strategy g∗ is the optimal value V0(m0) computed using (10) - (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we bound the difference in performance between gap and g∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let L ˆVt+1 be the Lipschitz constant of ˆVt+1 for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, for all mt ∈ [[Mt]] and ut ∈ [[Ut]], |Qt(mt, ut) − Θt(mt, ut)| ≤ 2αt, (50) |Vt(mt) − Λt(mt)| ≤ 2αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (51) where αt = max(ϵt, αt+1 + L ˆVt+1 · δt) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T and αT +1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We begin by recursively defining the value functions that compute the performance of the strategy ˆg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T and for each ˆπt ∈ [[ˆΠt]] and ut ∈ [[Ut]], let ˆΘt(ˆπt, ut) := max � sup ct∈[[Ct|ˆπt,ut]] ct, sup ˆπt+1∈[[ˆΠt+1|ˆπt,ut]] ˆΛt+1(ˆπt+1) � , (52) ˆΛt(ˆπt) :=ˆΘt(ˆπt, ˆgt(ˆπt)), (53) where ˆΛT +1(ˆπT +1) := 0, identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that ˆΘt(ˆπt, ut) = ˆQt(ˆπt, ut) and ˆΛt(ˆπt) = ˆVt(ˆπt), (54) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, since ˆgt(ˆπt) = arg minut∈Ut ˆQt(ˆπt, ut).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We first prove (50) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time t, using the triangle inequality and (54) in the LHS of (50): |Qt(mt, ut)−Θt(mt, ut)| ≤ |Qt(mt, ut)− ˆQt(ˆσt(mt), ut)| + |ˆΘt(ˆσt(mt), ut) − Θt(mt, ut)| ≤ αt+|ˆΘt(ˆσt(mt), ut) − Θt(mt, ut)|, (55) where, in the second inequality, we use (42) from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, to prove (50), it suffices to show that |ˆΘt(ˆσt(mt), ut) − Θt(mt, ut)| ≤ αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (56) Next, we use mathematical induction starting at time T + 1 to prove (56) in addition to |ˆΛt(ˆσt(mt)) − Λt(mt)| ≤ αt for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time T + 1, using the definitions it holds that ˆΛT +1(ˆσT +1(mT +1)) = ΛT +1(mT +1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This forms the basis of our induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, we consider the induction hypothesis that |ˆΛt+1(ˆσt+1(mt+1)) − Λt+1(mt+1)| ≤ αt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Given the hypothesis, (56) holds at time t using the same sequence of arguments as in the proof for 10 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, using the definitions of the value functions from (49) and (53), we write that |ˆΛt(ˆσt(mt)) − Λt(mt)| = |ˆΘt(ˆσt(mt), ˆgt(ˆσt(mt)) − Θt(mt, gt(mt))| = |ˆΘt(ˆσt(mt), ˆut)−Θt(mt, ˆut)| ≤ αt, (57) where, in the second equality, we use the definition of the control law to write that gt(mt) = ˆgt(ˆσt(mt)) =: ˆut;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and in the inequality, we use (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This proves the induction hypothesis for time t given the hypothesis for time t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, using mathematical induction (56) holds for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Subsequently, we complete the proof for (50) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T by substituting (56) into the RHS of (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, note that (51) follows directly from (50) using the same sequence of arguments used to prove (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can specialize the results of both Theorem 3 and Theorem 4 to terminal cost problems, where the optimal DP is given by (12) - (13) and the approximate DP is given by (27) - (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The approximation bounds in both theorems hold for terminal cost problems with a recursively defined constant αt := αt+1 +L ˆV tm t+1 ·δt for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T −1 and αT := ϵT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Alternate Characterization In this subsection, we provide stronger but simpler condi- tions which can identify an approximate information state as alternatives to (23) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' These conditions prescribe that an approximate information state ˆΠt = ˆσt(Mt) must satisfy for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T: 1) State-like evolution: There exists a Lipschitz continuous function ˆft : ˆPt × Ut × Yt+1 → Pt+1, independent of the strategy g, such that ˆΠt+1 = ˆft(ˆΠt, Ut, Yt+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (58) 2) Sufficient to approximate observations: For all mt ∈ [[Mt]] and ut ∈ [[Ut]], we define the sets Kob t+1 := [[Yt+1 | mt, ut]] and ˆKob t+1 := [[Yt+1 | ˆσt(mt), ut]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, H(Kob t+1, ˆKob t+1) ≤ δob t , (59) where δob t ∈ R≥0 is a known constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 3) Lipschitz-like observation prediction: There exists a constant λob t ∈ R≥0 such that for all ˆπ1 t , ˆπ2 t ∈ [[ˆΠt]], H � [[Yt+1|ˆπ1 t , ut]], [[Yt+1|ˆπ2 t , ut]] � ≤ λob t · η(ˆπ1 t , ˆπ2 t ), (60) where η is an appropriate metric on ˆPt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we prove that in addition to (22) in Definition 4, the conditions (58) - (60) are sufficient to characterize an approximate information state instead of (23) and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, if an uncertain variable ˆΠt = ˆσt(Mt) satisfies (58) - (59), it also satisfies (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let mt ∈ [[Mt]] be a given realization of Mt and let ˆπt = ˆσt(mt) satisfy (58) - (59), for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, using (58), we can write the LHS in (23) as H(Kt+1, ˆKt+1) = H � [[ ˆft(ˆσt(mt), ut, Yt+1)|mt, ut]], [[ ˆft(ˆσt(mt), ut, Yt+1)| ˆσt(mt), ut]] � = max � supyt+1∈Kob t+1 inf ˆyt+1∈ ˆKob t+1 d � ˆft(ˆσt(mt), ut, yt+1), ˆft(ˆσt(mt), ut, ˆyt+1) � , supˆyt+1∈ ˆKob t+1 infyt+1∈Kob t+1 η( ˆft � ˆσt(mt), ut, yt+1), ˆft(ˆσt(mt), ut, ˆyt+1) �� , where, in the second equality, we use the definition of the Hausdorff distance from (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that ˆft is globally Lipschitz because the approximate information state takes values in a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This implies that d � ˆft(ˆσt(mt), ut, yt+1), ˆft(ˆσt(mt), ut, ˆyt+1) � ≤ L ˆ ft · η(yt+1, ˆyt+1), and thus H(Kt+1, ˆKt+1) ≤ L ˆ ft max � supyt+1∈Kob t+1 inf ˆyt+1∈ ˆKob t+1 η(yt+1, ˆyt+1), supˆyt+1∈ ˆKob t+1 infyt+1∈Kob t+1 η(yt+1, ˆyt+1) � = L ˆ ft · H(Kob t+1, ˆKob t+1) ≤ L ˆ ft · δob t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, if an uncertain variable ˆΠt = ˆσt(Mt) satisfies (58) - (60), it also satisfies (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let ˆπ1 t , ˆπ2 t ∈ [[ˆΠt]] be two possible realizations of an approximate information state ˆΠt, which satisfies (58) (60), for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, using (58), we can write the LHS in (24) as H � [[Πt+1|ˆπ1 t , ut]], [[Πt+1|ˆπ2 t , ut]] � = H � [[ ˆft(ˆπ1 t , ut, Yt+1)|ˆπ1 t , ut]], [[ ˆft(ˆπ2 t , ut, Yt+1)|ˆπ2 t , ut]] ≤ L ˆ ft · � η(ˆπ1 t , ˆπ2 t ) + H � [[Yt+1|ˆπ1 t , ut]], [[Yt+1|ˆπ2 t , ut]] �� ≤ L ˆ ft · (1 + λob t ) · η(ˆπ1 t , ˆπ2 t ), where, in the first inequality, we use the Lipschitz continuity of the function ˆft along with the triangle inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and in the second inequality, we use (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This completes the proof by defining λt := L ˆ ft · (1 + λob t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Examples In this subsection, we present two state-quantized [91] approximate information states which satisfy Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider a system as described in Subsection III-C with compact feasible sets � Xt, Nt, Wt | t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T � in a metric space (S, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Recall that Xt is the state space at any t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, a finite subset ˆ Xt ⊂ Xt is a set of quantized states with parameter γt ∈ R≥0 if maxxt∈Xt minˆxt∈ ˆ Xt η(xt, ˆxt) ≤ γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The corresponding quantization function µt : Xt → ˆ Xt is defined as µt(xt) := arg minˆxt∈ ˆ Xt η(xt, ˆxt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that by construction, η(xt, µt(xt)) ≤ γt for all xt ∈ Xt, for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 1) Perfectly Observed Systems: Consider a system where Yt = Xt for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Recall from Subsection III-C that the Πt = Xt ∈ Xt for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, a feasible approximate information state for such a system is the quantized state ˆΠt := µt(Xt), which satisfies Definition 4 with ϵt = 2Ldt · γt and δt = 2γt+1 + 2Lft · γt, where γT +1 = 0, and Ldt and Lft are the Lipschitz constants for dt and ft, respectively (proof in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that because ˆΠt takes values in a finite set, it trivially satisfies (24) in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2) Partially Observed Systems: For a partially observed sys- tem, recall from Section III-C that an information state is given by the conditional range Πt = [[Xt|mt]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We construct an approximate conditional range by quantizing each element in Πt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, the approximation is generated by the mapping νt : B(Xt) → 2 ˆ Xt, where B(Xt) is the set of all compact subsets of Xt and 2 ˆ Xt is the power set of ˆ Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' This transformation yields the approximate range νt(Πt) := {µt(xt) ∈ ˆ Xt | xt ∈ Πt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, the approximate range ˆΠt = νt(Πt) is an information state for partially observed systems for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T with ϵt = 2Ldt · γt and δt = 2γt+1 + 2L ¯ ft · Lht+1 · Lft · γt, where γT +1 = 0, and L ¯ ft, Lht+1 and Lft are Lipschitz constants of ¯ft, ht+1, and ft, respectively (proof in Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 11 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' NUMERICAL EXAMPLES We present two numerical examples to illustrate our ap- proach: (1) The Wall Defense Problem: a worst-case control problem with partial observations, and (2) The Pursuit Evasion Problem: a worst-case reinforcement learning problem with partly unknown dynamics and partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The Wall Defense Problem In the wall defense problem, we consider an agent who defends a wall in a 5 × 5 grid world from an attacker over a time horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The wall is located across the central row of the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We illustrate the wall defense problem for one initial condition in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Here, the black colored cells constitute the wall and the grey hatched cells are adjacent to the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The solid blue triangle, solid red circle and red ring are the agent, attacker and observation, respectively, at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The pink cells are feasible positions of the attacker given the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The attacker moves within the bottom two rows of the grid and damages a wall cell when positioned in an adjacent cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, we denote the position of the attacker by Xat t ∈ X at = {(−2, −1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , (2, −1), (−2, −2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , (2, −2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In contrast, the agent moves within the top two rows of the grid and repairs a wall cell when positioned in an adjacent cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t, we denote the position of the agent by Xag t ∈ X ag = {(−2, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , (2, 1), (−2, 2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , (2, 2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The state of the wall at each t is the accumulated damage denoted by Dt = (D−2 t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , D2 t ), where Di t ∈ Di t = {0, 1, 2, 3} for all i = −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , 2 and Dt = ×2 i=−2Di t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The attacker starts at the position Xat 0 ∈ X at, which evolves for all t as Xat t+1 = I(Xat t + Wt ∈ X at) · (Xat t + Wt) + (1 −I(Xat t +Wt ∈ X at))·Xat t , where I is the indicator function and Wt ∈ Wt is an uncontrolled disturbance with Wt = {(−1, 0), (1, 0), (0, 0), (0, 1), (0, −1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t, the agent observes their own position and the wall’s state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent also partially observes the attacker’s position as Yt = I(Xat t + Nt ∈ X at) · (Xat t + Nt) + (1 − I(Xat t + Nt ∈ X at)) · Xat t , where Nt ∈ Nt = {(0, 0), (0, 1)} is the measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Given the history of observations, the agent selects an action Ut ∈ Ut = Wt at each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Starting with Xag 0 ∈ X ag, the agent moves as Xag t+1 = I(Xag t +Ut ∈ X ag)·(Xag t +Ut)+(1−I(Xag t +Ut ∈ X ag)) · Xag t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Starting with D0 = (0, 0, 0, 0, 0), the state of the wall evolves as Di t+1 = min � 3, max � 0, Di t + I(Xat t = (i, −1)) − I(Xag t = (i, 1)) �� for all t and i = −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t, after selecting the action, the agent incurs a cost for the damage to the wall, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', ct(Dt) = �2 i=−2 Di t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent’s aim is to minimize the maximum instantaneous damage to the wall, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', J (g) = maxt=0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=',T maxx0,w0:T ,n0:T ct(Dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Recall from Subsection III-C that an information state at time t is Πt = � Xag t , Dt, [[Xat t |Mt]] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We construct an approx- imation of the conditional range [[Xat t |Mt]] at time t using the quantization approach from Subsection IV-D and define the approximate range ˆAt = � µt(xt) ∈ ˆ X at|xt ∈ [[Xat t |Mt]] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The set of quantized cells ˆ X at, with γt = 1 for all t, is marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 1(b) with dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We consider the approximate information state ˆΠt = � Xag t , Dt, ˆAt, Y0 � for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The initial observation Y0 in ˆΠt improves the prediction of ˆAt+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For five initial conditions, we compute the best control strategy (a) The original grid (b) The quantized grid Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 1: The wall defense problem with the initial conditions xag 0 = (0, 2) and y0 = (0, −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' for T = 6 using both the information state (IS) and the approximate information state (AIS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2, we present the computational times (Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=') for both the DPs in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that the approximate DP has a faster run-time in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We also implement both strategies with random disturbances in the system with T = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2, we also present the actual worst- case costs across 5 × 103 implementations of both strategies and note that the AIS has a bounded deviation from the IS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2: Costs and run-times for 5×103 simulations and T = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Pursuit Evasion Problem In the pursuit evasion problem, we consider an agent who chases a moving target in a 9 × 9 grid world with static obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent aims to get close to the target over a time horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, we denote the position of the agent by Xag t ∈ X and that of the target by Xta t ∈ X, where X = � (−4, −4), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , (4, 4) � \\ O is the set of feasible grid cells and O ⊂ X is the set of obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The target starts at the position Xta 0 ∈ X, which is updated as Xta t+1 = I(Xta t +Wt ∈ X) · (Xta t + Wt) + (1 − I(Xta t + Wt ∈ X)) · Xta t , where Wt ∈ Wt = {(−1, 0), (1, 0), (0, 0), (0, 1), (0, −1)} is the disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t, the agent perfectly observes their own position and nosily observes the target’s position as Yt = I(Xta t +Nt ∈ X)·(Xta t +Nt)+(1−I(Xta t +Nt ∈ X)) ·Xta t , where Nt ∈ Nt = Wt is the measurement noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, starting with Xag 0 ∈ X, the agent selects an action Ut ∈ Ut = Wt to move as Xag t+1 = I(Xag t + Ut ∈ X) · (Xag t + Ut) + (1 − I(Xag t + Ut ∈ X)) · Xag t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time T, the agent selects no action and observes the target’s position Xta T and incurs a cost cT (Xta T , Xag T ) = η(Xta T , Xag T ) ∈ R≥0, where η is the shortest distance between two cells, while avoiding obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The distance between two adjacent cells is 1 unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The agent seeks to minimize the worst-case terminal cost without prior knowledge of either the observation function or the target’s evolution dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that this is a reinforcement learning generalization of Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We illustrate the grid and one initial set up in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Here, the black cells are obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2 1 0 1 2 2 1 0 1 2-2 1 0 1 2 2 1 0 1 2Initial Conditions Strategy IS Strategy AIS xag Yo Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (s) Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' cost Run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (s) Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' cost (1, 2) (1,-2) 354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='2 2 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='2 2 (0, 1) (0,-1) 1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='7 2 607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='3 3 (-2, 2) (-1,-1) 686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='1 3 446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='4 3 (2, 2) (-2, - 2) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='22 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='81 2 (-2, 2) (2, -1) 582.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='3 3 362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='1 3 (-1, 1) (1, -1) 1551.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='5 2 1075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='1 312 The solid blue triangle, solid red circle and red ring are the agent, target, and observation, respectively, at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The pink cells are feasible positions of the target given the observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (a) The original problem (b) Actual observation prediction (c) Learned observation prediction Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 3: The pursuit evasion problem with the initial conditions xag 0 = (0, 2) and y0 = (3, −4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We consider that the agent has access to 3×107 observation trajectories from the target which are used to learn an approx- imate information state representation offline, as characterized in Subsection IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' First, we use the data on observation trajectories to construct estimates of the conditional range Kob t+1 = [[Yt+1|Y0:t]] for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T − 2 and Kob T = [[Xta T |Y0:T −1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, taking inspiration from [45], we set-up a deep neural network with an encoder-decoder structure for each t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t, the encoder ψt comprises of 3 layer neural network with sizes (2, 14), (14, 12), (12+24, 24) and ReLU activation for the first two layers, where the inputs are a 2-d vector of coordinates for observation Yt and a 24-d vector for the previous approximate information state ˆΠt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The encoder compresses these inputs to a 24-d vector representing the approximate information state ˆΠt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At each t, the decoder φt is a 4 layer neural network of size (24, 48), (48, 56), (56, 64), (64, 74) with ReLU activation for the first three layers and sigmoid activation for the last layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Its input is ˆΠt and its output is a 74-d vector with each component taking values in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Each component of the 74-d output gives a set-inclusion value for a specific feasible cell in the 9 × 9 grid, excluding obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The output is thus interpreted as the conditional range ˆKob t+1 = [[Yt+1 | ˆΠt]] for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T − 2 and ˆKob T = [[Xta T | ˆΠT −1]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We consider a set-inclusion threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='5 for inclusion in ˆKob t+1 at each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 4: The neural network architecture for approximate infor- mation states at any t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The learning objective of our neural network at each t is to minimize H(Kob t+1, ˆKob t+1), which is consistent with the characterization of approximate information states in Subsec- tion IV-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that at the terminal time step, this objective also minimizes the difference in maximum costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Since the Hausdorff distance is not differentiable, we adapt the first surrogate function proposed in [92] as a learning objective to train the network weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We train the network for 40 epochs using 90% of the available data with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='0003 and test it against the other 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' To illustrate the training results, consider an out-of-sample initial observation y0 = (3, −4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, the set Kob 1 constructed using data is shown by pink cells in 3(b) and the set ˆKob 1 generated by of the trained network is shown by blue cells in 3(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that the trained network’s output matches the conditional range constructed from data accurately except for one cell (0, −4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We train a neural network for each t up to T = 4 to learn a complete approximate information state representation for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, at each t, the agent uses the state (Xag t , ˆΠt) in the approximate DP (25) - (26) to compute an approximately optimal control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We compare the performance of this approximate strategy with a baseline strategy that uses the observation Yt at each t instead of ˆΠt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, for this baseline we train a network to match the prediction [[Yt+1 | Yt]] to [[Yt+1 | Y0:t]] for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T −2 and [[Xta T | YT −1]] to [[Xta T | Y0:T −1]] at time T − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The neural network structure is the same as before except for a lack of ˆΠt−1 in the encoder input at each t and we use the same training parameters as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Subsequently, the agent computes an approximately optimal strategy using the approximate DP with the state (Xag t , Yt) at each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For six initial conditions, we present in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 5 the worst case costs obtained when implementing both the approximately optimal strategy (Maximum cost with AIS) and the baseline strategy (Maximum cost without AIS) for T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Across 104 simulations with randomly generated uncertainties, we note that using the learned approximate information state consistently improves worst-case performance when compared to the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, learning an approximate information state representation is a viable approach for worst-case re- inforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' In general, we expect our approach to outperform the baseline more for longer time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 5: Worst-case costs for 104 simulations and T = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' CONCLUSION In this paper, we presented a principled approach to worst- case control and learning in partially observed systems us- ing non-stochastic approximate information states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We first presented two sets of properties to characterize information states and used them to construct a DP that yields an optimal control strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, we proposed two sets of properties to characterize approximate information states that can be constructed from output variables with knowledge of the dy- namics or learned from output data with incomplete knowledge of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We proved that approximate information states can be used in a DP to compute an approximate control strategy with a bounded loss in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We also 4-3 ¥-2 1 0 1 2 ¥3 4 4 3 2 1 0 1 2 3 4-4-3 ¥-2 1 0 1 2 ¥3 4 4 3 2 1 0 1 Kpb 2 3 4-4-3 ¥-2 1 0 1 2 ¥3 4 4 3 2 1 0 1 Rpb 2 3 4dt +t Ilt-1 24 2 14 12 24 24 48 56 64 74 Yt tx0 Yo Maximum cost with AlS Maximum cost without AlS (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2) (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='0) 2 3 (-4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 4) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='-4) 12 12 (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='4) (-4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='-2) 11 12 (-2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' -3) (- 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='3) 7 8 (-4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 2) (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 4) 7 7 (-3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 1) (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' -1) 8 913 presented theoretical examples of this bound and numerical examples to illustrate the performance of our approach in both worst-case control and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Our ongoing work is to specialize the approach in this paper to additive cost problems reported in [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Future work should consider extending our results to problems with an infinite time horizon, constructing tighter performance bounds for systems with specific dynamics, and combining these results with other reinforcement learning techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', Q-learning [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' APPENDIX A – L-INVERTIBLE FUNCTIONS In this appendix, we present two classes of functions which are L-invertible: 1) all bi-Lipschitz functions which have a compact domain and a compact co-domain, and 2) all functions with a compact domain and a finite co-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let X and Y be two compact subsets of a metric space (S, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, any bi-Lipischitz function f : X → Y is L-invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We begin by considering the pre-image set for any y ∈ Y under the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that the function f is continuous because it is bi-Lipschitz and the singleton {y} is a compact subset of a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consequently, the pre-image f −1(y) is a bounded subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, let B(X) denote the set of all bounded subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Given the first result, we can consider a set-valued mapping f −1 : Y → B(X) which returns the pre-image for each y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, for any y1, y2 ∈ Y, using the definition of the Hausdorff distance in (1): H � f −1(y1), f −1(y2) � = max � sup x1∈f −1(y1) inf x2∈f −1(y2) η(x1, x2), sup x2∈f −1(y2) inf x1∈f −1(y1) η(x1, x2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (61) In the RHS of (61), the bi-Lipschitz property of f implies that there exist constants Lf, Lf ∈ R>0 such that Lfη(x1, x2) ≤ |f(x1) − f(x2)| ≤ Lfη(x1, x2), for all x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, for all x1 ∈ g−1(y1) and x2 ∈ g−1(y2), we write that η(x1, x2) ≤ L−1 f η(y1, y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (62) The proof is complete by substituting (62) into (61) and defining the constant Lf −1 := L−1 f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let X be a compact subset and Y be a finite subset of (S, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, any function f : X → Y is L-invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let ||Y|| > 0 denote the minimum distance between two distinct elements in the finite, non-empty set Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, for any y1, y2 ∈ Y such that y1 ̸= y2, H � f −1(y1), f −1(y2) � η(y1, y2) ≤ supy1,y2∈Y H � f −1(y1), f −1(y2) � ||Y|| =: Lf −1, where Lf −1 ∈ R≥0 is guaranteed to be finite because the set X is bounded and thus, so is the numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Thus, the function f is L- invertible as defined in (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' APPENDIX B – APPROXIMATION BOUNDS FOR PERFECTLY OBSERVED SYSTEMS In this appendix, we derive the values of ϵt and δt for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T when an approximate information state is constructed using state quantization for a perfectly observed system, as described in Subsection IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We first state a property of the Hausdorff distance which we will use in our derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Lemma 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let X be a metric space with compact subsets A, B, C, D ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, it holds that H � A ∪ B, C ∪ D � ≤ max � H � A, C � , H � B, D �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (63) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The proof for this result is given in [88, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, we state and prove the main result of this appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider a perfectly observed system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=', Yt = Xt, for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let µt : Xt → ˆ Xt such that maxxt∈Xt η(xt, µt(xt)) ≤ γt at each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, ˆΠt = µt(Xt) is an approximate information state which satisfies (22) with ϵt = 2Ldt · γt and (23) with δt = 2γt+1 + 2Lft · γt for all t, where γT +1 = 0, and where Ldt, and Lft are Lipschitz constants for dt and ft, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, let mt = (x0:t, u0:t−1) be the realization of Mt and let the approximate information state be ˆxt = µt(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We first derive the value of ϵt in the RHS of (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' At time t, can expand the conditional ranges to write that [[Xt|mt]] = [[Xt|xt]] = {xt} and [[Xt|ˆxt]] = {xt ∈ X | η(xt, ˆxt) ≤ γt}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' On substituting these into the LHS of (22), we state that �� supct∈[[Ct|mt,ut]] ct − supct∈[[Ct|µt(xt),ut]] ct �� = ��dt(xt, ut) − sup¯xt∈[[Xt|µt(xt)]] dt(¯xt, ut) �� ≤ sup¯xt∈[[Xt|µt(xt)]] |dt(xt, ut) − dt(¯xt, ut)| ≤ Ldt · sup¯xt∈[[Xt|µt(xt)]] η(xt, ¯xt) ≤ Ldt · � η(xt, µt(xt)) + sup¯xt∈[[Xt|µt(xt)]] η(µt(xt), ¯xt) � , ≤ 2Ldt · γt =: ϵt, where, in the third inequality, we use the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, to derive the value of δt, we expand the LHS of (23) as H � [[ ˆXt+1|xt, ut]], [[ ˆXt+1|µt(xt), ut]] � =H �� µt+1(ft(xt, ut, wt))|wt ∈ Wt � , � µt+1(ft(¯xt, ut, wt))|¯xt ∈ [[Xt|µt(xt)]], wt ∈ Wt �� ≤ sup wt∈Wt H({µt+1(ft(xt, ut, wt))}, {µt+1(ft(¯xt, ut, wt))|¯xt ∈ [[Xt|µt(xt)]]}), (64) where, in the inequality, we use (63) from Lemma 10 and the fact that � µt+1(ft(xt, ut, wt))|wt ∈ Wt � = ∪wt∈Wt � µt+1(ft(¯xt, ut, wt)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Once again using (63) in the RHS of (64), we conclude that H � [[ ˆXt+1|xt, ut]], [[ ˆXt+1|µt(xt), ut]] � ≤ supwt∈Wt,¯xt∈[[Xt|µt(xt)]] η � µt+1 � ft(xt, ut, wt) � , µt+1 � ft(¯xt, ut, wt) �� ≤ supwt∈Wt,¯xt∈[[Xt|µt(xt)]] � η � µt+1(ft(xt, ut, wt)), ft(xt, ut, wt) � + η � ft(xt, ut, wt), ft(¯xt, ut, wt) � + η � ft(¯xt, ut, wt), µt+1(ft(¯xt, ut, wt)) �� ≤ γt+1 + 2Lft · γt + γt+1 =: δt, where, in the second inequality, we use the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' 14 APPENDIX C – APPROXIMATION BOUNDS FOR PARTIALLY OBSERVED SYSTEMS In this appendix, we derive the values of ϵt and δt for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, when an approximate information state is constructed using state quantization for a partially observed system, as described in Subsection IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Consider a partially observed system with Yt = ht(Xt, Nt) for all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Let µt : Xt → ˆ Xt such that supxt∈Xt η(xt, µt(xt)) ≤ γt at each t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, ˆΠt = νt(Πt) is an approximate information state with ϵt = 2Ldt · γt and δt = 2γt+1 + 2L ¯ ft · Lht+1 · Lft · γt for all t, where γT +1 = 0, and where Ldt, L ¯ ft, Lht+1, and Lft are Lipschitz constants for the respective functions in the subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' For all t = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' , T, let mt ∈ [[Mt]], Pt = [[Xt|mt]] ∈ Pt, and ˆPt = νt(Pt) ∈ ˆPt be the realizations of the memory Mt, the conditional range Πt and the approximate information state ˆΠt, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Note that the conditional range Pt satisfies (14) and (15) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, to derive the value of ϵt, we write the LHS of (22) using (14) as �� sup ct∈[[Ct|mt,ut]] ct − sup ct∈[[Ct|νt(Pt),ut]] ct �� = �� sup xt∈Pt dt(xt, ut) − sup ¯xt∈[[Xt|νt(Pt)]]) dt(¯xt, ut) �� ≤Ldt · H(Pt, [[Xt|νt(Pt)]]) ≤Ldt · � H(Pt, νt(Pt)) + H(νt(Pt), [[Xt|νt(Pt)]]) � , (65) where, in the equality, we use (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' in the first inequality, we use (34) from Lemma 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and in the second inequal- ity we use the triangle inequality for the Hausdorff dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can expand the first term in the RHS of (65) as H(Pt, νt(Pt)) = H(Pt, {µt(xt) ∈ ˆ Xt | xt ∈ Pt}) = H � ∪xt∈Pt {xt}, ∪xt∈Pt{µt(xt) ∈ ˆ Xt} � ≤ supxt∈Pt η(xt, µt(xt)) ≤ γt, where we use (63) from Lemma 10 in the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can also expand the second term in the RHS of (65) as H(νt(Pt), [[Xt|νt(Pt)]])) = H � νt(Pt), {xt ∈ Xt| inf ¯xt∈νt(Pt) η(xt, ¯xt) ≤ γt} � = supxt∈[[Xt|νt(Pt)]] inf ¯xt∈νt(Pt) η(xt, ¯xt) ≤ γt, where the sec- ond equality holds by expanding the Hausdorff distance and noting that νt(Pt) ⊆ [[Xt|νt(Pt)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The proof is complete by substituting the results for both terms in the RHS of (65).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next, to derive the value of δt, we note that Pt = σt(mt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Then, using the triangle inequality in the LHS of (23), H � [[νt+1(Πt+1)|mt, ut]], [[νt+1(Πt+1)|νt(σt(mt)), ut]] � ≤H � [[νt+1(Πt+1)|mt, ut]], [[Πt+1|mt, ut]] � + H � [[Πt+1|mt, ut]], [[Πt+1|νt(σt(mt)), ut]] � + H � [[Πt+1|νt(σt(mt)), ut]], [[νt+1(Πt+1)|νt(σt(mt)), ut]] � ≤2γt+1 + H � [[Πt+1|mt, ut]], [[Πt+1|νt(σt(mt)), ut]] � , (66) where, in the second inequality we use the fact that H � Pt+1, νt+1(Pt+1) � ≤ γt+1, which was proved above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can write the second term in the RHS of (66) using (15) from Definition 3 as H � [[Πt+1|mt, ut]], [[Πt+1|νt(σt(mt)), ut]] � = H � [[Πt+1|Pt, ut]], [[Πt+1|νt(Pt), ut]] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Furthermore, note that [[Πt+1|νt(Pt), ut]] = � ˜Pt+1 ∈ [[Πt+1| ˜Pt, ut]] | ˜Pt ∈ [[Πt|νt(Pt)]] � = ∪ ˜ Pt∈[[Πt|ν(Pt)]][[Πt+1| ˜Pt, ut]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' we use (63) from Lemma 10 to write that H � [[Πt+1|Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' [[Πt+1|νt(Pt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]] � ≤ sup ˜ Pt∈[[Πt|νt(Pt)]] H � [[Πt+1|Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' [[Πt+1| ˜Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]] � ≤L ¯ ft · sup ˜ Pt∈[[Πt|νt(Pt)]] H � [[Yt+1|Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' [[Yt+1| ˜Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]] � ≤L ¯ ft·Lht+1 · sup ˜ Pt∈[[Πt|νt(Pt)]] H � [[Xt+1|Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' [[Xt+1| ˜Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' (67) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' in the second inequality we use the same arguments as in Lemma 6 and the third inequality can be proven by substituting Yt+1 = ht+1(Xt+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Vt+1) into the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' We can further expand the third term in the RHS of (67) and use (63) from Lemma (10) to write that sup ˜ Pt∈[[Πt|νt(Pt)]] H � [[Xt+1|Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' [[Xt+1| ˜Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut]] � ≤ sup ˜ Pt∈[[Πt|νt(Pt)]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='wt∈Wt H � {ft(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' wt)|xt ∈ Pt},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' {ft(xt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ut,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' wt)|xt ∈ ˜Pt} � ≤ Lft ·sup ˜ Pt∈[[Πt|νt(Pt)]] H � Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ˜Pt � ≤ Lft · sup ˜ Pt∈[[Πt|νt(Pt)]] � H(Pt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' νt(Pt) � + H � νt(Pt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' ˜Pt) � ≤ 2Lft · γt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' in the third inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' we use the triangle inequality and in the fourth inequality we use the fact that for all νt( ˜Pt) = νt(Pt),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' for all ˜Pt ∈ [[Πt|νt(Pt)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' REFERENCES [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Kim and P.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He received a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' in Mechanical Engineering from the Indian Institute of Technology, Bombay, India, in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Prior to his PhD degree, he worked as a Project Manager at Cairn Energy, Gurugram, In- dia from 2016-2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' His current research interests span several areas, including worst-case control, reinforcement learning, decentralized systems, and mechanism design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He is a student member of IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Nishanth Venkatesh S (S’21) is research engineer at the Department of Mechanical Engineering at the University of Delaware, Newark, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He received a Master’s degree in Robotics from the University of Delaware, Newark, USA, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Prior to his Master’s he received a B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' in Mechanical Engineering from the Indian Institute of Technology, Bombay, India, in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' His current research interests span several areas, including worst-case control, rein- forcement learning, and decentralized systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He is a student member of IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Andreas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Malikopoulos (S’06–M’09–SM’17) re- ceived the Diploma in mechanical engineering from the National Technical University of Athens, Greece, in 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He received M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' degrees from the department of mechanical engineering at the University of Michigan, Ann Arbor, Michigan, USA, in 2004 and 2008, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He is the Terri Connor Kelly and John Kelly Career Development Associate Professor in the Department of Mechan- ical Engineering at the University of Delaware, the Director of the Information and Decision Science (IDS) Laboratory, and the Director of the Sociotechnical Systems Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' Prior to these appointments, he was the Deputy Director and the Lead of the Sustainable Mobility Theme of the Urban Dynamics Institute at Oak Ridge National Laboratory, and a Senior Researcher with General Motors Global Research & Development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' His research spans several fields, including analysis, optimization, and control of cyber-physical systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' decentralized systems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' stochastic scheduling and resource allocation problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' and learning in complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' The emphasis is on applications related to smart cities, emerging mobility systems, and sociotechnical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He has been an Associate Editor of the IEEE Transactions on Intelligent Vehicles and IEEE Transactions on Intelligent Transportation Systems from 2017 through 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He is currently an Associate Editor of Automatica and IEEE Transactions on Automatic Control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} +page_content=' He is a member of SIAM, AAAS, and a Fellow of the ASME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8dE4T4oBgHgl3EQfdQww/content/2301.05089v1.pdf'} diff --git a/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf b/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5eba51f26cce89f3c98f23a5b4c28bf6f3d9c1a5 --- /dev/null +++ b/9dFLT4oBgHgl3EQfuS8F/content/2301.12154v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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AstroSat/UVIT study of the open cluster NGC 2818: Blue Stragglers, Yellow Stragglers, +Planetary Nebula, and their membership +Sharmila Rani,1, 2 Gajendra Pandey,1 Annapurni Subramaniam,1 and N. Kameswara Rao1 +1Indian Institute of Astrophysics, Bangalore, 560034, India +2Pondicherry University, R.V. Nagar, Kalapet, 605014, Puducherry, India +ABSTRACT +We present the first far-UV (FUV) imaging results of the intermediate-age Galactic open cluster +NGC 2818 that has a Planetary nebula (PN) within the field using images taken from the Ultra-violet +Imaging Telescope (UVIT) aboard AstroSat. We identify cluster members by combining UVIT-detected +sources with Gaia EDR3 data. We detect four bright and hot blue straggler stars (BSSs) and two +yellow straggler stars (YSSs) based on their location in the optical and FUV-optical color-magnitude +diagrams. Based on the parameters estimated using Spectral Energy Distribution (SED), we infer +that BSSs are either collisional products or might have undetectable white dwarf (WD) companions. +Our photometric analysis of YSSs confirms their binarity, consistent with the spectroscopic results. +We find YSSs to be formed through a mass-transfer scenario and the hot components are likely to be +A-type subdwarfs. A comparison of the radial velocity (RV), Gaia EDR3 proper-motion of the PN +with the cluster, and reddening towards the PN and the cluster does not rule out the membership +of the PN. Comparing the central star’s position with theoretical pAGB models suggest that it has +already entered the WD cooling phase, and its mass is deduced to be ∼ 0.66M⊙. The corresponding +progenitor mass turns out to be ∼ 2.1M⊙, comparable to the turn-off mass of the cluster, implying +that the progenitor could have formed in the cluster. We suggest that the NGC 2818 might be one of +the few known clusters to host a PN, providing a unique opportunity to test stellar evolution models. +Keywords: (Galaxy:) open clusters: individual (NGC 2818) — stars: yellow stragglers — (stars:) blue +stragglers — ultraviolet: stars — (stars:) Hertzsprung–Russell and C–M diagrams +1. INTRODUCTION +Open clusters (OCs) are ideal laboratories to probe +the structure and history of the Galactic disk. +They +are also test-beds to study the formation and evolution +of single and binary stellar populations. Dynamical in- +teractions of stellar populations in star clusters lead to +binaries and the formation of exotic stellar populations +such as blue straggler stars (BSSs), yellow straggler stars +(YSSs), and cataclysmic variables. These systems, as +well as the end products of stellar evolution, such as hot +white dwarfs (WDs), emit the bulk of their energy in +the ultraviolet (UV) regime. UV observations of OCs +are crucial to detect and understand the properties of +Corresponding author: Sharmila Rani +sharmila.rani@iiap.res.in +the hot stellar populations, as highlighted in Landsman +et al. (1997) and Knigge et al. (2008). +One of the intriguing products of stellar interactions +in the OCs are BSSs whose origin and evolution are +still debated (Boffin et al. 2015). +As these stars ap- +pear brighter and bluer than the stars located in the +MS turn-off region of the cluster color-magnitude di- +agram (CMD), they are expected to be more massive +than the turn-off stars. To explain the mass gain and +rejuvenation of these objects, the main formation sce- +narios proposed are, direct collisions or spiraling in of +binary stars resulting in mergers (Hills & Day 1976), or +mass-transfer activity in close-binary systems (McCrea +1964). +The dynamical evolution of hierarchical triple +systems leading to the merger of an inner binary via +the Kozai mechanism (Iben & Tutukov 1999; Perets & +Fabrycky 2009) is another possible mechanism. Obser- +vational studies of BSSs suggest that a combination of +all the formation channels are prevalent, and has a de- +arXiv:2301.01943v1 [astro-ph.SR] 5 Jan 2023 + +2 +Rani et al. +pendence on their environment, as they are found in a +variety of stellar environments such as OCs (Ahumada +& Lapasset 2007; de Marchi et al. 2006), globular clus- +ters (GCs) (Ferraro et al. 2012), the Galactic field (San- +tucci et al. 2015), and dwarf galaxies (Santana et al. +2012). +Thus, studying BSSs can provide information +about the dynamical history of the cluster, the role of +the dynamics on binary evolution, the frequency of bi- +nary systems, and the contribution of binaries to cluster +evolution. Member stars that are redder than the BSSs +and brighter than the sub-giants found in the CMDs +of OCs and GCs are considered as evolved BSSs, and +are known as yellow straggler stars (YSSs) ( See Sindhu +et al. 2018 and references therein). +There are only a few OCs in our Galaxy known to har- +bor Planetary nebulae (PNe). PNe are classically con- +sidered to represent the late stages in the stellar evolu- +tion of all the low as well as intermediate-mass stars with +a mass range of 0.8−8 M⊙ (Weidemann 2000). As the +evolutionary lifetime of PNe are short (around 103 −105 +years, depending on the mass of the progenitor) when +compared to other evolutionary phases, especially when +the number of evolved stars present in OCs are small, +PNe as members of OCs are rare and are not expected +in young OCs. +Objects in this short-lived phase are +critically important to our understanding of the physi- +cal processes and steps that transform stars into their +remnants. They allow us to test the theory of stellar +evolution, including the physics of nucleosynthesis and +the relation between a star’s initial mass and its white +dwarf (WD) remnant (Kwitter et al. 2014). Moreover, +the chemical composition of the PNe can provide infor- +mation about the dredge-up of chemical elements, which +is expected to depend on the star’s initial mass and com- +position. Finding a planetary nebula (PN) as a member +of an OC gives us an excellent opportunity to better +characterize and constrain its crucial parameters, such +as distance, reddening, and age. +NGC 2818, has the unique distinction of being one of +the two galactic OCs probably associated with a PN, and +interestingly, the name NGC 2818 is assigned to both an +OC and a PN. Most importantly, the membership of the +PN to the OC is still debated. In this study, we analyze +both the cluster and the PN, NGC 2818. +Here we present the results of the UV imaging of +NGC 2818 (both PN and OC) in four far-UV (FUV) fil- +ters using the ultraviolet imaging telescope (UVIT) on +AstroSat. Our main aims are: (1) to identify and char- +acterize the blue and yellow straggler stars in the cluster +to shed light on their formation and evolution and (2) to +characterize the central star of the PN (CSPN) to inves- +tigate its association with the cluster. The age of this +cluster is estimated to be ∼800 Myr, and the reddening +of the cluster is E(B−V) = 0.2 mag (Sun et al. 2021). +This cluster is located at a distance of 3250 ± 300 pc +and the metallicity is found to be solar (Sun et al. 2021). +NGC 2818 is one of the OCs that shows an extended +main-sequence turn-off (eMSTO) phenomenon (Bastian +et al. 2018), where the cluster MS is extended in the +CMD more than what is expected from a simple stellar +population with conventional evolutionary history. +It +has been demonstrated that stellar rotation is the most +probable cause of this phenomenon (Bastian & de Mink +2009; Brandt & Huang 2015; Niederhofer et al. 2015; +Cabrera-Ziri et al. 2016; Gossage et al. 2019). A spec- +troscopic study by Bastian et al. (2018) showed that, +in NGC 2818, stellar rotation is indeed linked to the +stars’ position on the MSTO of the CMD made using the +Gaia magnitudes (G) and color (Gbp−Grp), such that +rapidly rotating stars preferentially lie on the red side +of the eMSTO. However, the color range (Gbp−Grp) in +optical CMD is relatively small, whereas a larger color +range is seen in UV colors, and it is expected that the +rotational effects are more prominently displayed in UV +colors mainly because of their sensitivity to surface (ef- +fective) temperature changes. This study also explores +the correlation between the colors derived from UVIT +FUV filters and stellar rotation. +The layout of this paper is as follows. In section 2, +we describe the observations, data reduction, and anal- +ysis methods. In Section 3, we present proper-motion- +based membership information using Gaia EDR3 data +for cluster stars and PN. Section 4 presents the selection +of BSSs and YSSs from the observed UV and Optical +CMDs, including the stellar rotation effects on CMDs. +In Sections 5 and 6, we describe the properties of BSSs, +and YSSs derived from the UVIT photometry along with +GALEX, Gaia and ground-based photometry and their +evolutionary status. A detailed discussion of all results +is provided in Section 7. Finally, in Section 8, we sum- +marize our main results and conclusions. +2. OBSERVATIONAL DATA AND ANALYSIS +2.1. UVIT Data +In order to probe the nature of the exotic stellar pop- +ulations in NGC 2818, we use data acquired with the +UVIT instrument on board the Indian multiwavelength +astronomy satellite AstroSat. UVIT produces images of +the sky in far-UV (FUV), near-UV (NUV), and visible, +simultaneously, over a circular field-of-view of 28′ di- +ameter with a spatial resolution of ∼ 1.′′5 in both FUV +and NUV channels. More details about the telescope, +its initial and new calibration, and its results are de- + +Exotic Stellar Populations in NGC 2818 +3 +scribed in detail by Tandon et al. (2017, 2020). +The +derived magnitudes of the stellar sources observed with +the UVIT filters are in the AB magnitude system. +The observations of NGC 2818 used in this work were +made in two epochs, first on 21st December 2018 (Prop: +A05_196 −P.I: N. K. Rao), and the second on 11th +June 2020 (Prop: A09_047 −P.I: N. K. Rao). In the +first epoch, the observations were carried out in three +FUV filters (F154W, F169M, and F172M), and in the +second, observations were performed with deep expo- +sures in four FUV filters (F148W, F154W, F169M, and +F172M). The observations are carried out in several or- +bits in order to complete the allotted exposure times +in given filters. We utilize a customized software pack- +age, CCDLAB (Postma & Leahy 2017), to correct for +the geometric distortion, flat field, spacecraft drift and +create images for each orbit. Then, the orbit-wise im- +ages were co-aligned and combined to generate science- +ready images in order to get a better signal-to-noise ra- +tio. Further analysis was done using these final science- +ready images to obtain the magnitudes of the sources +detected with UVIT. The details of the UVIT observa- +tions of NGC 2818 used in this analysis are tabulated +in Table 1. In Figure 1, we show the UVIT image of +the cluster taken in the FUV F148W band where the +orange color depicts FUV detections. +This image ex- +hibits an extended structure displaying the beautiful PN +NGC 2818, where the central star can be seen in the +FUV. +2.2. Photometry +To extract the magnitudes of detected stars in all +FUV images, we have carried out the point spread func- +tion (PSF) photometry using the IRAF/NOAO package +DAOPHOT (Stetson 1987). The steps taken to obtain +the magnitude of the sources are as follows: First, the +stars are located in the image using the DAOFIND task +in IRAF. Further, we used the PHOT task to perform +the aperture photometry. To construct the model PSF +using the PSF task, bright and isolated stars are se- +lected in the image using the PSTSELECT task. The +average PSF of the stars in all FUV images is ∼ 1.′′2. +The ALLSTAR task is used to fit the model PSF to +all the detected stars in the image to obtain the PSF- +fitted magnitudes. The PSF magnitudes were converted +to aperture photometry scale using the PSF correction +further followed by aperture correction, estimated us- +ing the curve of growth analysis by choosing isolated +bright stars in the field. +Finally, the saturation cor- +rection, in order to account for more than one photon +per frame, was applied to the obtained magnitudes in +UVIT filters. All steps to perform the saturation cor- +rection are described in detail in Tandon et al. (2017). +The extracted instrumental magnitudes are calibrated +into the AB magnitude system using the zero points +(ZP) reported in the recently published calibration pa- +per (Tandon et al. 2020). Figure 3 shows the PSF-fit +error (median) as a function of magnitude in four FUV +filters for profound observations. We have detected stars +up to ∼ 22 mag with PSF-fit errors less than 0.3 mag in +all FUV filters and considered them for further analysis +in the paper. +To apply the extinction and reddening correction to +the derived UVIT magnitudes of all detected stars, we +adopted the reddening, E(B−V) = 0.2 mag mentioned +in the Sun et al. (2021). The ratio of total-to-selective +extinction, RV = 3.1 for the Milky Way, was taken from +Whitford (1958) to calculate the extinction value in the +visual band (AV ). We used the Fitzpatrick extinction +law (Fitzpatrick 1999) to compute extinction coefficients +Aλ for all UVIT filters, as listed in Table 1. +2.3. Other Catalogs +This cluster was previously observed in UV, optical, +and Infrared (IR) all-sky surveys with GALEX (Bianchi +et al. 2017), SDSS (Alam et al. 2015), APASS (Hen- +den et al. 2015), 2MASS (Cutri et al. 2003), and WISE +(Cutri et al. 2021), respectively. In this work, we com- +bined the UVIT data with the multi-wavelength photo- +metric catalog spanning a wavelength range from UV- +IR. We used the virtual observatory tool in VOSA to +cross-match the UVIT-detected sources with the above- +mentioned photometric catalogs (Bayo et al. 2008). +3. MEMBERSHIP DETERMINATION +We employed the Gaia early data release 3 (EDR3) +catalog that provides data with unprecedented preci- +sion to identify the cluster members. In particular, it +provides the complete 5-parameter astrometric solution +(positions, proper motions, and parallaxes) and mag- +nitudes in its three photometric bands (G, GBP , and +GRP ) with a limiting magnitude of about G∼21 mag. +To assign the proper motion (PM) membership proba- +bility (Pµ) of all stars observed in the cluster, we first +downloaded all detections located within a 30′ radius +from the cluster’s center. To include all possible mem- +bers of the cluster, we opted to use a radius bigger +than that provided by Kharchenko et al. (2013) cata- +log. Then, we applied the data quality criteria to select +the sources with a good astrometric solution. Stars are +selected as follows: (i) we removed those with paral- +laxes that deviate by more than 3σ from the expected +parallax calculated using the previously known distance + +4 +Rani et al. +Figure 1. UVIT color image of OC NGC 2818 in FUV F148W channel. Here orange color depicts the FUV detections. The +extended structure in this image represents the PN NGC 2818. North is up, and east is left in the image. +F154W +N +E +0.5 arcmin +F169M +N +E +0.5 arcmin +N +E +0.5 arcmin +F172M +Figure 2. UVIT/FUV images of PN NGC 2818 in three filters: F154W, F169M, and F172M. +Table 1. List of the FUV observations of NGC 2818 obtained with UVIT in two epochs +used in this work. The last column lists the extinction value computed in each FUV +filter using Fitzpatrick (1999) law of extinction. +Filter +λmean +∆λ +ZP +texp (sec) +Aλ +(Å) +(Å) +(AB mag) +(1st epoch) +(2nd epoch) +(mag) +F148W +1481 +500 +18.09 +- +1736 +1.58 +F154W +1541 +380 +17.77 +1491 +2877 +1.55 +F169M +1608 +290 +17.41 +1715 +1999 +1.54 +F172M +1717 +125 +16.27 +1903 +2878 +1.51 +to the cluster, where σ is the error in parallax given in +Gaia EDR3 catalog, (ii) we also removed the sources +with renormalized unit weight error (RUWE) exceeding +1.2 as larger values of this parameter might lead to an +unreliable astrometric solution (Lindegren et al. 2018; +Riello et al. 2021). +We made use of a probabilistic Gaussian mixture +model (GMM) method to select cluster members and in- +fer the intrinsic parameters of the distributions of both +member and non-member stars. In this method, the dis- +tribution of sources in the vector-point diagram (µα, µδ) +is modeled as a mixture of two Gaussian distributions, +one for the cluster members and another one for the field +sources. The details of this method are well described in + +N +1 arcminExotic Stellar Populations in NGC 2818 +5 +16 +17 +18 +19 +20 +21 +22 +23 +UV mag (AB) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Error (mag) +F148W +F154W +F169M +F172M +Figure 3. +PSF-fit errors (median) as a function of mag- +nitude for our UVIT observations of NGC 2818 in all FUV +bandpasses. +Vasiliev (2019). The Gaussian probability distribution +corresponding to the sum of two distributions is +f(µ|µi, � +i) = +2 +� +i=1 +wi +exp +� +− 1/2(µ − µi)T �−1 +i (µ − µi) +� +2π +� +det � +i +(1) +wi ≥ 0, +2 +� +i=1 +wi = 1 +(2) +where µ is individual PM vector; µi are field and cluster +mean PMs; � is the symmetric covariance matrix; and +wi are weights for the two Gaussian distributions. Full +details of this method for the n-dimensional case are +described in (Vasiliev 2019). +The initial guess for cluster PM µα and µδ values +and internal velocity dispersion are taken from (Cantat- +Gaudin et al. 2020). We utilized GaiaTools1 to maxi- +mize the total log-likelihood of GMM and measure the +mean PM and standard deviation of both the Gaussian +distributions. The membership probabilities (MPs) of +all the selected stars are calculated using the same tech- +nique simultaneously. The equations used to maximize +the log-likelihood of GMM and estimate the MP of the +1 https://github.com/GalacticDynamics-Oxford/GaiaTools +ith star belonging to the kth component are given in +appendix A in Vasiliev (2019). +The PM mean and standard deviations of the clus- +ter distribution are computed to be µα = -4.417 mas/yr +and µδ = 4.540 mas/yr, with σc = 0.045 mas/yr. In +Figure 4, we show the position of stars in the sky, in +the PM space known as vector point diagram (VPD), +and in an optical CMD created using Gaia filters. Cyan +dots in all the plots depict the member stars belonging +to the cluster, and black dots represent the field stars. +718 stars are identified as most likely cluster members +with Pµ>50% and considered for subsequent analysis. +This method works well for a distinguishable distribu- +tion of PM for the field and cluster stars in the VPD. +But, in this case, the PM of cluster stars are located well +within the PM distribution of the field stars, suggesting +a non-trivial identification of cluster members from field +stars. Therefore, it is possible that stars with a lower +membership probability than the above-mentioned limit +might also be members of the cluster. +3.1. Is PN a member of the cluster? +The membership of the PN with OC has been de- +bated in several studies in the past. Tifft et al. (1972) +found that PN NGC 2818 is a member of the OC of +the same name. Dufour (1984) presented the results of +photometric as well as spectroscopic observations of the +nebula to analyze its physical properties and chemical +composition. He suggested that the nebula is probably +associated with the star cluster. +Pedreros (1989) an- +alyzed this cluster using CCD UBV photometric data +and assumed a physical association of the nebula with +the cluster. Surendiranath et al. (1990) also suggested +the association of the PN with the cluster from their +CCD photometry of the cluster. However, Mermilliod +et al. (2001) derived accurate heliocentric radial veloci- +ties for 12 cluster red giants to obtain a mean heliocen- +tric radial velocity of Vhel = +20.7 ± 0.3 kms−1, signif- +icantly different from the PN velocity of −1 ± 3 kms−1 +(Meatheringham et al. 1988), suggesting that they are +unrelated. +Recently, (Vázquez 2012) reanalyzed the +complex kinematics and morphology of the nebula using +high-resolution Hubble Space Telescope (HST) archive +imaging and high-dispersion spectroscopic data and de- +termined a systemic heliocentric velocity of PN to be ++26±2 kms−1 in closer agreement with the OC, sug- +gesting its membership. Moreover, based on its RV, Hα +surface brightness, and radius, Frew et al. (2016) con- +cluded that the PN might be a cluster member. +The Gaia EDR3 trigonometric parallax for the cen- +tral star of the nebula (CSPN) is 0.0319±0.21 mas, but +it can be noted that the uncertainty in it is more than + +6 +Rani et al. +138.4 +138.6 +138.8 +139.0 +139.2 +139.4 +139.6 +RA (deg) +37.0 +36.8 +36.6 +36.4 +36.2 +DEC (deg) +10 +5 +0 +* (mas/yr) +0 +5 +10 + (mas/yr) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +G +GRP +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +G +Figure 4. In three panels from left to right, PM members of the cluster are shown with cyan dots, and the remaining Gaia +EDR3 sample marked with black dots represents field stars. Left Panel: position in the sky; Middle Panel: Vector Point Diagram +(VPD); Right Panel: Gaia Optical CMD. +its value. So, it can not be used to obtain the distance to +the nebula. The best estimate of the statistical distance +is given by (Frew et al. 2016) as 3000±800 pc not too +far from cluster distance of 3250±300 pc estimated by +Sun et al. (2021). (Cantat-Gaudin et al. 2020; Cantat- +Gaudin & Anders 2020) obtained the members of the +several OCs, including NGC 2818, using Gaia DR2 PM +data, and suggested that it is a non-member of the clus- +ter. +In our membership analysis, we have obtained the +membership of the CSPN using the Gaia EDR3 PM +data. The PM in RA and DEC of the CSPN as listed +in Gaia EDR3 catalog is µα = −3.712 ± 0.185 mas/yr +and µδ = 4.94 ± 0.18 mas/yr. Its Pµ is estimated to +be ∼11%, indicating non-membership. Nevertheless, it +can be noted from the location of the CSPN shown with +the red star symbol in the VPD that it is lying close to +the PM distribution of the cluster members (Cyan dots), +implying that it is quite likely a member of the cluster. +Statistically, it is lying within 3σ of the mean PM of +the cluster. We expect that the future Gaia data re- +lease (Gaia DR4) might give more precise and accurate +PM measurements that can re-confirm its association +with the cluster. +Further, assuming both cluster and +nebula at the same distance, we computed their true +velocity using their already available RV and PM infor- +mation. We found that the true velocity of the cluster +and nebula turn out to be approximately the same (VC += 99.7kms−1 & VP N = 98.7kms−1), implying that the +values of the space velocity are similar. +3.1.1. Reddening towards the PN +Several estimates of extinction/reddening towards the +cluster have been made since the initial investigation +by Tifft et al. (1972) of E(B−V) of 0.22 mag, recon- +firmed by Surendiranath et al. (1990) and recently re- +fined by Sun et al. (2021), to 0.20 mag. However, there +are a few independent estimates of extinction towards +the PN NGC 2818. Dufour (1984) estimated it from the +Balmer lines Hα/Hβ ratio as 0.24±0.02 mag. Gathier +& Pottasch (1988) list a value of 0.20 mag, and Frew +et al. (2016) estimated a value of 0.17±0.08 mag. We +presently estimate E(B−V) value using free-free con- +tinuum flux and the nebular Hβ flux. +The flux den- +sity, Sν at 5 GHz of the entire nebula, is measured by +Zhang (1995) as 33 mJy. The total Hβ flux is estimated +by Gathier & Pottasch (1988) as logF(Hβ) as -11.40 +(ergcm−2s−1). Following Pottasch (1984), the expected +ratio of Sν to F(Hβ) is given as +S(ν) +F(Hβ) = 2.51×107×T 0.53 +e +×(ν)−0.1×Y Jy/ergcm−2s−1 +where Te is the electron temperature; ν is frequency in +GHz; Y = (1 + n(He+) +n(H+) ). The value of n(He+) +n(H+) is ∼ 0.13 +assuming all He is in He+ form. Dufour (1984) derived +the Te[OIII] of 14,500±500 K. From the above relation, +the logF(Hβ) expected from the radio continuum is - +11.07. The equation from Milne & Aller (1975) used to +compute the reddening is following: +E(B − V ) = +1 +1.46log F(Hβ)exp +F(Hβ)obs +Inserting the expected and observed logF(Hβ) values +in the above equation, we obtain the value of E(B−V) +∼0.23 mag. Thus, the extinction/reddening towards this +cluster and nebula is of similar value. +From the comparison of distance, RV, PM, and ex- +tinction/reddening values of the cluster and nebula, we +suggest a physical association of the PN with the OC. +4. COLOR MAGNITUDE DIAGRAMS + +Exotic Stellar Populations in NGC 2818 +7 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +Gbp + Grp +12 +14 +16 +18 +20 +G +MS +BSS +YSS +SGB +RGB +FUV detected +775 Myr,[Fe/H]=0.0 dex +Figure 5. +Optical CMD of the NGC 2818, created us- +ing Gaia EDR3 photometry. All filled symbols denote the +stars with Pµ ≥ 50%. +Blue-filled stars and yellow-filled +stars are the selected blue and yellow straggler stars used +for further cross-match with UVIT data, respectively. The +stars detected in all FUV images are outlined with cyan- +colored square and star symbols. +The over-plotted green +solid line represents the non-rotating MIST isochrone of +solar metallicity and an age of 775 Myr, set at redden- +ing, E(B−V)=0.2 mag and distance modulus, (m−M)V = +12.56 mag. +4.1. Classification of Exotic sources +This section describes the classification and identifica- +tion of exotic sources, such as BSSs and YSSs, expected +to emit in the FUV. As mentioned in Section 3, we +considered the probable cluster members with Pµ>50% +and created the PM-cleaned optical CMD (Gbp - Grp +vs. G) using the Gaia filters shown in Figure 5. In this +CMD, stars outlined with cyan color depict the various +identified star populations in FUV images. Rain et al. +(2021) presented a new proper-motion-cleaned catalog +of BSSs in galactic OCs using Gaia DR2 data. +We +cross-matched the Gaia EDR3 cluster members with +the BSS catalog to classify this population in the clus- +ter. Out of five identified BSSs in NGC 2818 by Rain +et al. (2021), we detected four BSSs. The remaining one +BSS, not detected by us, is found to be a non-member +of the cluster in our membership catalog and also falls +outside the FoV of NGC 2818 observed with UVIT in +two epochs. Jadhav & Subramaniam (2021) also pro- +duced a catalog of BSSs in OCs using Gaia DR2 data +with a Pµ>70%, and they found two BSS candidates in +this cluster. The difference in the above-mentioned cat- +alogs could be due to the adopted age criteria, selection +method, and different membership probability cut-offs +used in the two studies. +We obtained the MESA Isochrones & Stellar Tracks +(MIST) for the UVIT and Gaia EDR3 filters from an +updated MIST online database2 to identify and classify +distinct evolutionary sequences in the cluster (Choi et al. +2016; Paxton et al. 2018). +We considered isochrones +with +� +α/Fe +� += +0.0, metallicity, Z = 0.017210 (Sun +et al. 2021), not incorporating initial rotation. Cluster +parameters such as age, extinction, and distance modu- +lus, adopted to fit the isochrone to the observed optical +CMD, are 775 Myr, AV =0.6 mag, and (m−M)V =12.56, +respectively (Sun et al. 2021). The overplotted isochrone +(solid green line) over the observed optical CMD is dis- +played in Figure 5. We notice that the isochrone ap- +pears well-matched to the observed CMD along the +main-sequence, sub-giant branch (SGB), but it is not +reproducing the observed position of the red clump. To +account for this mismatch along the red clump, (Bas- +tian et al. 2018) suggested that there might be a prob- +lem in the calibration of the models for the red clump +or the conversion between theoretical properties of the +isochrones (temperature, gravity, and luminosity) to ob- +servational space in Gaia filters is off. +We also selected the YSSs based on their location in the +optical CMD, as they have colors in between the turn-off +(TO) and RGB and appear brighter than the SGB. We +have chosen two such stars marked with yellow colored +filled symbols shown in Figure 5. +4.2. FUV-optical CMDs +This section presents the FUV-optical CMDs gener- +ated by cross-identifying common stars between optical +and our FUV detections. We cross-matched the sources +detected in the UVIT FUV filters with the Gaia EDR3 +with a maximum separation of 1.′′3, which is the typi- +cal FWHM of the PSF for the UVIT filters. To plot +the FUV-optical CMDs, first, we made the magnitude +system adopted by Gaia similar to that of UVIT. That +is, we transformed the Vega magnitude system used in +the Gaia photometric system to the AB system using +the photometric zero points reported in the Gaia EDR3 +documentation3. +We have created and shown the FUV-optical CMDs +for cluster members in Figure 6 using F148W and +2 https://waps.cfa.harvard.edu/MIST/interp_isos.html +3 https://gea.esac.esa.int/archive/documentation + +8 +Rani et al. +F169M filters. We note that a similar trend of detected +stellar populations is observed in the other two filters +(F154W & F172M). The error bars displayed in all FUV +CMDs are estimated as the median of the stars’ errors at +a chosen magnitude range. The FUV-optical CMDs are +also over-plotted with updated MIST isochrones (Choi +et al. 2016) to compare the locations of the distinct se- +quences predicted by the theoretical models with the +observed ones. In all FUV images, hot and bright stars +such as BSSs, YSSs, and MS are detected. We have de- +tected 4 BSSs out of 5 previously known in the literature +(Rain et al. 2021). Four detected BSSs are confirmed +RV and PM members. +Two YSSs are also identified +in all FUV images. We note that these stars are well- +separated and brighter than the theoretical isochrone +presenting the SGB sequence in all FUV-optical CMDs, +in turn confirming their classification as YSSs. +RGB +and Red clump stars are too faint to be detected in the +FUV. +The FUV-optical CMDs show a large scatter along MS, +as shown in Figure 6, unlike optical CMD. The overlaid +isochrones in all FUV-optical CMDs help to trace the +MS scatter. We note that a few MS stars are brighter +than theoretical MSTO not reproduced by isochrones. +These might have high rotational velocities accounting +for this feature. Some of them may be binaries or poten- +tial BSSs. One BSS is found to be very hot and bright +in all FUV-optical CMDs compared to the other three +BSSs. This BSS can be an exciting candidate to char- +acterize, as it might have a hot WD companion. As two +YSSs are detected in all FUV images and found to be +bright in all FUV-optical CMDs, these stars also might +have a hot companion, which leads to their detection +in the FUV images. These are intriguing targets fur- +ther to understand their formation and evolution in the +clusters. +4.3. Extended MS turn-off in FUV CMDs +In order to check the sensitivity of UVIT colors to +the Teff affected by the rotational velocity, we plot +(Gbp−Grp) vs. (F172M−G) color as shown in Figure 7, +which indicates a linear relation. +The range of Gaia +color is only 0.4 mag whereas F172M−G spans about +3.0 mag, which makes F172M−G color more sensitive +and responsive to rotational velocity. F172M−G color +is preferred over F169M−G because the band F172M +allows only continuum light, and no chromospheric or +transitional emission lines are seen in late-type stars in +FUV. +Comparison of the CMD, F172M−G vs. Gbp (Fig. 8 +upper right) with CMD of Gbp−Grp vs. Gbp (Fig. 8 +upper left) shows the sensitivity of F172M−G color. +The bend in the isochrone in F172M−G vs. Gbp CMD +at a color of 4.0 indicates the beginning of eMSTO +prominently (unlike Fig. 8, left panel), and all the stars +right of the isochrone show high rotational velocity. The +MS comprises stars with both high and low rotational +velocities. However, the CMD of F169M−G vs. Gbp +exhibits some more aspects. +From the comparison of +F169M−G color with F172M−G in Fig. 8, we find that +the former is redder than the latter. It can be due to +the fact that the F169M flux in late-type stars is smaller +than at F172M. Moreover, the predicted colors using +the theoretical isochrones are following the same trend. +It is well known that MS stars later than about F2 +would possess coronal and transitional regions as evi- +denced in the FUV region by emission lines of C IV, +He II, Si IV, N V, N IV, etc. (Linsky & Haisch 1979; +Jordan & Linsky 1987). +Prominent lines like C IV +and He II occur in the F169M band region (unlike the +F172M band). The F154W and F148W would contain +a few more emission lines in addition to C IV and He +II. Thus, the CMD of F169M-G vs. +Gbp shows that +the MS stars are shifted bluewards to the isochrone, +probably suggesting the presence of transitional region +lines. Even in the F169M−F172M vs. Gbp CMD shown +in the lower right panel of Figure 8, it is evident that +most stars have bluer colors than the theoretically ex- +pected ones from isochrones. It is to be noted that all +stars on the blue edge of the MS in CMD of F169M−G +vs. Gbp (15 50% and considered them further to identify their +FUV counterparts with UVIT. FUV-optical and FUV +CMDs were generated for the cluster members and over- +laid with the MIST isochrones to compare the position +of different observed evolutionary sequences with theo- +retically expected ones. MIST isochrones are found to +match well with the observed sequences in FUV-optical +CMDs, but in FUV CMDs, especially F169M−F172M +vs. Gbp, most of the detected stars in both filters are ly- +ing blueward of their expected location from isochrones. + +16 +Rani et al. +3.6 +4.0 +4.4 +4.8 +5.2 +LogTeff +2 +1 +0 +1 +2 +3 +4 +LogL/L +1 +2 +1 +2 +775Myr +WD (0.5M ) +WD (0.2M ) +pAGB (0.528M ) +pAGB (0.576M ) +pAGB (0.580M ) +pAGB (0.657M ) +PNe +MS +YSSs +sdA +BSSs +Field_ELM_WDs +Field_sdA +Figure 12. HR diagram of the bright stars identified with UVIT. Various evolutionary tracks are presented from the beginning +of the MS to the moment when a star has entered to the stage, followed by the WD cooling sequences. All these tracks are +generated for cluster metallicity and age. The pAGB sequences with different final masses are shown here to compare the +location of the CSPN marked with a red star symbol. BSSs and YSSs are displayed with blue-filled circles and yellow star +symbols, respectively. The hotter companions of YSSs are shown with magenta star symbols. In addition, Field ELM WDs and +A-type subdwarfs represented with cyan and purple symbols are also placed in the HR diagram to compare the position of the +hot companions of both YSSs. Green color solid and dashed lines correspond to the DA-WD tracks with different masses. +In all FUV images, we have identified four BSSs, two +YSSs, and MS based on their location in the optical +as well as FUV-optical CMDs. +Then, we performed +the SED analysis to deduce their physical properties to +evaluate their nature. The Teff of BSSs estimated from +SED fit ranges from 8500−11500 K, hinting that they +are quite hot, consistent with the young age (700−800 +Myr) of the cluster. +In the previous studies of BSSs +in other OCs conducted using UVIT data, the Teff +range varies from cluster to cluster depending upon its +age. +The temperature range of BSSs in OC M67 (4 +Gyr) is 6250−9000 K (Jadhav et al. 2019), in King 2 +(6 Gyr) 5750−8500 K (Jadhav et al. 2021), in OC +NGC 188 (7 Gyr) 6100−6800 K (Gosnell et al. 2015). +In intermediate-age OCs such as NGC 7789 (1.6 Gyr) +(Vaidya et al. 2022) and NGC 2506 (2.2 Gyr) (Pan- +thi et al. 2022), BSSs span a temperature range from +7250−10250 K, and 7750−9750 K, respectively. +The +SEDs of all BSSs are well-fitted with a single model, +and we suggest that collisions leading to the mergers +might explain their formation in this cluster. Another +plausible possibility is that they might have a faint WD +companion undetectable with UVIT. If this is the case, +then the second prominent scenario to explain their +existence in star clusters, i.e., mass transfer in close bi- +naries, will dominate over the previous one. Moreover, +mass transfer in binaries will dominate in OCs as they +are less dense and compact than GC systems. Further, +spectroscopic analysis of these stars will help to confirm +their nature. + +Exotic Stellar Populations in NGC 2818 +17 +Two YSSs, from their SED fits, are found to be bina- +ries, and the location of YSSs and their hot components +in the HR diagram suggests that cool components are +already in the RGB phase. +In contrast, hot compo- +nents most plausibly belong to sdA class. We infer from +here that these two stars are post-mass-transfer systems +where BSS (accretor) has evolved into a giant stage and +became YSS, and the donor star into a sdA. In addition, +a spectroscopic study performed by Mermilliod et al. +(2001) of RGB stars, including these two stars, found +that they are spectroscopic binaries, confirming our +result. Their radial velocities estimated by them also +verify their membership. Hence, we suggest that these +two stars to be formed via a mass transfer scenario in +the cluster. +From the comparison of the distance, extinction, RV +and PM values of the PN with the cluster, it turns out +that it is a most likely member of the cluster. +Bohi- +gas (2003, 2008) estimated the Teff from the ionization +modeling of the nebula as Teff 149,000 K and log g of +7.1 (however, this might also be dependent on the dis- +tance assumed). Mata et al. (2016) gives the Teff as +160,000 K. +Gathier & Pottasch (1988) estimate the +HI Zanstra temp 175,000K and HeII Zanstra temp of +215,000K. Kohoutek et al. (1986) derived the luminos- +ity (L∗ = 851L⊙) and radius (R∗ = 0.038R⊙) of CSPN +using optical observations, and adopting the identical +distance to the nebula as that of the cluster (d=3.5 kpc). +The atmospheric parameters of CSPN determined using +the SED fitting technique are more or less in agreement +with the previous estimations. Based on the compari- +son of the central star’s location with the predicted ones +from the theoretical models in the HR diagram, the cen- +tral star’s mass turns out to be 0.66 M⊙. Cummings +et al. (2018) presented the WD initial–final mass rela- +tion (IFMR) for progenitor stars of Minitial from 0.85 to +7.5 M⊙. In their Figure 5, they displayed the compari- +son of the Initial–Final Mass Relation (IFMR) estimated +for the observed sample with the theoretical isochrones. +For a WD with a mass of 0.66 M⊙, the initial mass of the +progenitor is estimated to be ∼2.1 M⊙ (From their Fig. +5). In this work, the MSTO mass of this cluster deter- +mined using isochrone fit is ∼2 M⊙. The previously re- +ported turn-off mass for this cluster and the initial mass +of the nebula’s progenitor are ∼2.1 M⊙, and 2.2 ± 0.3 +M⊙, respectively (Dufour 1984). Our estimations are +consistent with the previous ones. From the comparison +of the cluster turn-off mass and progenitor mass, we in- +fer that PN is quite likely a cluster member. Thus, this +study showcases the significance of using the FUV data +to study the exotic populations and late stages of the +evolution of intermediate-mass stars in OCs. +8. SUMMARY AND CONCLUSIONS +The main results from this work can be summarized +as follows: +• In this study, we employed UVIT observations on- +board AstroSat to identify BSSs and YSSs in the +open cluster NGC 2818, and also characterize the +CSPN. We further created the optical and UV- +optical CMDs of member stars co-detected using +UVIT and Gaia EDR3 data in this cluster. +• The PM members of the cluster are obtained us- +ing Gaia EDR3 data, and we found that PN +NGC 2818 might be a member of this cluster, con- +sistent with the previous studies. +• As this cluster is young, hot and bright stars such +as BSSs, YSSs, and MS are detected in all FUV +images. +• To compare the observations with theoretical pre- +dictions, optical and UV-optical CMDs are over- +laid with non-rotating MIST isochrones generated +for respective UVIT and Gaia filters. The theoret- +ical isochrones reproduce the features of all CMDs +quite well. +• The FUV-optical CMDs prominently show the +eMSTO phenomenon already reported in this clus- +ter, consistent with the previous studies. +• We characterized the four detected BSSs in the +cluster, and a single model fits well to all the ob- +served SEDs. We suggest from the single model +fits that these stars might have a faint WD com- +panion that could not be detected with UVIT’s +detection limit or result from the merger of two +close binaries. +• We suggest the presence of two YSSs in this cluster +based on their location in the CMDs. Both YSSs +were found to have excess flux in the UV, con- +nected to its binarity. They are confirmed spec- +troscopic binaries, and their hot companions are +compact objects, likely to be sdA stars. Based on +these results, we conclude that they are products +of the binary mass transfer. +• From comparing the position of the CSPN with +the theoretical pAGB evolutionary tracks, we +found that it has entered the WD cooling phase, +and its mass is found to be ∼ 0.66M⊙. The mass + +18 +Rani et al. +of the progenitor corresponding to the WD of mass +0.66M⊙ would be ∼ 2.1M⊙, similar to the turn-off +mass of the cluster, further confirming its member- +ship. +ACKNOWLEDGEMENTS +We thank the anonymous referee for the valuable +comments and suggestions. AS acknowledges support +from SERB Power Fellowship. S. Rani wants to thank +Vikrant Jadhav for providing the field ELM WDs SED +fit parameters catalog. S. Rani thanks Sonith L. S. for +the fruitful discussions. +This publication utilizes the +data from AstroSat mission’s UVIT, which is archived +at the Indian Space Science Data Centre (ISSDC). +The UVIT project is a result of collaboration between +IIA, Bengaluru, IUCAA, Pune, TIFR, Mumbai, sev- +eral centers of ISRO, and CSA. This research made +use of VOSA, developed under the Spanish Virtual Ob- +servatory project supported by the Spanish MINECO +through grant AyA2017-84089. 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Y. 1995, ApJS, 98, 659, doi: 10.1086/192173 + diff --git a/9tAzT4oBgHgl3EQf-_7r/content/tmp_files/load_file.txt b/9tAzT4oBgHgl3EQf-_7r/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..530a57079dc02f2e96a13a7ad740f613d471e167 --- /dev/null +++ b/9tAzT4oBgHgl3EQf-_7r/content/tmp_files/load_file.txt @@ -0,0 +1,1486 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf,len=1485 +page_content='Draft version January 6, 2023 Typeset using LATEX twocolumn style in AASTeX631 UOCS-IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' AstroSat/UVIT study of the open cluster NGC 2818: Blue Stragglers, Yellow Stragglers, Planetary Nebula, and their membership Sharmila Rani,1, 2 Gajendra Pandey,1 Annapurni Subramaniam,1 and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Kameswara Rao1 1Indian Institute of Astrophysics, Bangalore, 560034, India 2Pondicherry University, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Nagar, Kalapet, 605014, Puducherry, India ABSTRACT We present the first far-UV (FUV) imaging results of the intermediate-age Galactic open cluster NGC 2818 that has a Planetary nebula (PN) within the field using images taken from the Ultra-violet Imaging Telescope (UVIT) aboard AstroSat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We identify cluster members by combining UVIT-detected sources with Gaia EDR3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We detect four bright and hot blue straggler stars (BSSs) and two yellow straggler stars (YSSs) based on their location in the optical and FUV-optical color-magnitude diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Based on the parameters estimated using Spectral Energy Distribution (SED), we infer that BSSs are either collisional products or might have undetectable white dwarf (WD) companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Our photometric analysis of YSSs confirms their binarity, consistent with the spectroscopic results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We find YSSs to be formed through a mass-transfer scenario and the hot components are likely to be A-type subdwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' A comparison of the radial velocity (RV), Gaia EDR3 proper-motion of the PN with the cluster, and reddening towards the PN and the cluster does not rule out the membership of the PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Comparing the central star’s position with theoretical pAGB models suggest that it has already entered the WD cooling phase, and its mass is deduced to be ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='66M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The corresponding progenitor mass turns out to be ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1M⊙, comparable to the turn-off mass of the cluster, implying that the progenitor could have formed in the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We suggest that the NGC 2818 might be one of the few known clusters to host a PN, providing a unique opportunity to test stellar evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Keywords: (Galaxy:) open clusters: individual (NGC 2818) — stars: yellow stragglers — (stars:) blue stragglers — ultraviolet: stars — (stars:) Hertzsprung–Russell and C–M diagrams 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' INTRODUCTION Open clusters (OCs) are ideal laboratories to probe the structure and history of the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' They are also test-beds to study the formation and evolution of single and binary stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Dynamical in- teractions of stellar populations in star clusters lead to binaries and the formation of exotic stellar populations such as blue straggler stars (BSSs), yellow straggler stars (YSSs), and cataclysmic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' These systems, as well as the end products of stellar evolution, such as hot white dwarfs (WDs), emit the bulk of their energy in the ultraviolet (UV) regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' UV observations of OCs are crucial to detect and understand the properties of Corresponding author: Sharmila Rani sharmila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='rani@iiap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='in the hot stellar populations, as highlighted in Landsman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (1997) and Knigge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' One of the intriguing products of stellar interactions in the OCs are BSSs whose origin and evolution are still debated (Boffin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' As these stars ap- pear brighter and bluer than the stars located in the MS turn-off region of the cluster color-magnitude di- agram (CMD), they are expected to be more massive than the turn-off stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' To explain the mass gain and rejuvenation of these objects, the main formation sce- narios proposed are, direct collisions or spiraling in of binary stars resulting in mergers (Hills & Day 1976), or mass-transfer activity in close-binary systems (McCrea 1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The dynamical evolution of hierarchical triple systems leading to the merger of an inner binary via the Kozai mechanism (Iben & Tutukov 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Perets & Fabrycky 2009) is another possible mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Obser- vational studies of BSSs suggest that a combination of all the formation channels are prevalent, and has a de- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='01943v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='SR] 5 Jan 2023 2 Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' pendence on their environment, as they are found in a variety of stellar environments such as OCs (Ahumada & Lapasset 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' de Marchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2006), globular clus- ters (GCs) (Ferraro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2012), the Galactic field (San- tucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2015), and dwarf galaxies (Santana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Thus, studying BSSs can provide information about the dynamical history of the cluster, the role of the dynamics on binary evolution, the frequency of bi- nary systems, and the contribution of binaries to cluster evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Member stars that are redder than the BSSs and brighter than the sub-giants found in the CMDs of OCs and GCs are considered as evolved BSSs, and are known as yellow straggler stars (YSSs) ( See Sindhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2018 and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' There are only a few OCs in our Galaxy known to har- bor Planetary nebulae (PNe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' PNe are classically con- sidered to represent the late stages in the stellar evolu- tion of all the low as well as intermediate-mass stars with a mass range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='8−8 M⊙ (Weidemann 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' As the evolutionary lifetime of PNe are short (around 103 −105 years, depending on the mass of the progenitor) when compared to other evolutionary phases, especially when the number of evolved stars present in OCs are small, PNe as members of OCs are rare and are not expected in young OCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Objects in this short-lived phase are critically important to our understanding of the physi- cal processes and steps that transform stars into their remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' They allow us to test the theory of stellar evolution, including the physics of nucleosynthesis and the relation between a star’s initial mass and its white dwarf (WD) remnant (Kwitter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Moreover, the chemical composition of the PNe can provide infor- mation about the dredge-up of chemical elements, which is expected to depend on the star’s initial mass and com- position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Finding a planetary nebula (PN) as a member of an OC gives us an excellent opportunity to better characterize and constrain its crucial parameters, such as distance, reddening, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' NGC 2818, has the unique distinction of being one of the two galactic OCs probably associated with a PN, and interestingly, the name NGC 2818 is assigned to both an OC and a PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Most importantly, the membership of the PN to the OC is still debated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In this study, we analyze both the cluster and the PN, NGC 2818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Here we present the results of the UV imaging of NGC 2818 (both PN and OC) in four far-UV (FUV) fil- ters using the ultraviolet imaging telescope (UVIT) on AstroSat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Our main aims are: (1) to identify and char- acterize the blue and yellow straggler stars in the cluster to shed light on their formation and evolution and (2) to characterize the central star of the PN (CSPN) to inves- tigate its association with the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The age of this cluster is estimated to be ∼800 Myr, and the reddening of the cluster is E(B−V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 mag (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' This cluster is located at a distance of 3250 ± 300 pc and the metallicity is found to be solar (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' NGC 2818 is one of the OCs that shows an extended main-sequence turn-off (eMSTO) phenomenon (Bastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2018), where the cluster MS is extended in the CMD more than what is expected from a simple stellar population with conventional evolutionary history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' It has been demonstrated that stellar rotation is the most probable cause of this phenomenon (Bastian & de Mink 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Brandt & Huang 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Niederhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Cabrera-Ziri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gossage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' A spec- troscopic study by Bastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2018) showed that, in NGC 2818, stellar rotation is indeed linked to the stars’ position on the MSTO of the CMD made using the Gaia magnitudes (G) and color (Gbp−Grp), such that rapidly rotating stars preferentially lie on the red side of the eMSTO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' However, the color range (Gbp−Grp) in optical CMD is relatively small, whereas a larger color range is seen in UV colors, and it is expected that the rotational effects are more prominently displayed in UV colors mainly because of their sensitivity to surface (ef- fective) temperature changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' This study also explores the correlation between the colors derived from UVIT FUV filters and stellar rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The layout of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In section 2, we describe the observations, data reduction, and anal- ysis methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In Section 3, we present proper-motion- based membership information using Gaia EDR3 data for cluster stars and PN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Section 4 presents the selection of BSSs and YSSs from the observed UV and Optical CMDs, including the stellar rotation effects on CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In Sections 5 and 6, we describe the properties of BSSs, and YSSs derived from the UVIT photometry along with GALEX, Gaia and ground-based photometry and their evolutionary status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' A detailed discussion of all results is provided in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Finally, in Section 8, we sum- marize our main results and conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' OBSERVATIONAL DATA AND ANALYSIS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' UVIT Data In order to probe the nature of the exotic stellar pop- ulations in NGC 2818, we use data acquired with the UVIT instrument on board the Indian multiwavelength astronomy satellite AstroSat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' UVIT produces images of the sky in far-UV (FUV), near-UV (NUV), and visible, simultaneously, over a circular field-of-view of 28′ di- ameter with a spatial resolution of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='′′5 in both FUV and NUV channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' More details about the telescope, its initial and new calibration, and its results are de- Exotic Stellar Populations in NGC 2818 3 scribed in detail by Tandon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2017, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The derived magnitudes of the stellar sources observed with the UVIT filters are in the AB magnitude system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The observations of NGC 2818 used in this work were made in two epochs, first on 21st December 2018 (Prop: A05_196 −P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='I: N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Rao), and the second on 11th June 2020 (Prop: A09_047 −P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='I: N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Rao).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In the first epoch, the observations were carried out in three FUV filters (F154W, F169M, and F172M), and in the second, observations were performed with deep expo- sures in four FUV filters (F148W, F154W, F169M, and F172M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The observations are carried out in several or- bits in order to complete the allotted exposure times in given filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We utilize a customized software pack- age, CCDLAB (Postma & Leahy 2017), to correct for the geometric distortion, flat field, spacecraft drift and create images for each orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Then, the orbit-wise im- ages were co-aligned and combined to generate science- ready images in order to get a better signal-to-noise ra- tio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Further analysis was done using these final science- ready images to obtain the magnitudes of the sources detected with UVIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The details of the UVIT observa- tions of NGC 2818 used in this analysis are tabulated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In Figure 1, we show the UVIT image of the cluster taken in the FUV F148W band where the orange color depicts FUV detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' This image ex- hibits an extended structure displaying the beautiful PN NGC 2818, where the central star can be seen in the FUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Photometry To extract the magnitudes of detected stars in all FUV images, we have carried out the point spread func- tion (PSF) photometry using the IRAF/NOAO package DAOPHOT (Stetson 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The steps taken to obtain the magnitude of the sources are as follows: First, the stars are located in the image using the DAOFIND task in IRAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Further, we used the PHOT task to perform the aperture photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' To construct the model PSF using the PSF task, bright and isolated stars are se- lected in the image using the PSTSELECT task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The average PSF of the stars in all FUV images is ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='′′2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The ALLSTAR task is used to fit the model PSF to all the detected stars in the image to obtain the PSF- fitted magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The PSF magnitudes were converted to aperture photometry scale using the PSF correction further followed by aperture correction, estimated us- ing the curve of growth analysis by choosing isolated bright stars in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Finally, the saturation cor- rection, in order to account for more than one photon per frame, was applied to the obtained magnitudes in UVIT filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' All steps to perform the saturation cor- rection are described in detail in Tandon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The extracted instrumental magnitudes are calibrated into the AB magnitude system using the zero points (ZP) reported in the recently published calibration pa- per (Tandon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Figure 3 shows the PSF-fit error (median) as a function of magnitude in four FUV filters for profound observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We have detected stars up to ∼ 22 mag with PSF-fit errors less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='3 mag in all FUV filters and considered them for further analysis in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' To apply the extinction and reddening correction to the derived UVIT magnitudes of all detected stars, we adopted the reddening, E(B−V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 mag mentioned in the Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The ratio of total-to-selective extinction, RV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1 for the Milky Way, was taken from Whitford (1958) to calculate the extinction value in the visual band (AV ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We used the Fitzpatrick extinction law (Fitzpatrick 1999) to compute extinction coefficients Aλ for all UVIT filters, as listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Other Catalogs This cluster was previously observed in UV, optical, and Infrared (IR) all-sky surveys with GALEX (Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2017), SDSS (Alam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2015), APASS (Hen- den et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2015), 2MASS (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2003), and WISE (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2021), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In this work, we com- bined the UVIT data with the multi-wavelength photo- metric catalog spanning a wavelength range from UV- IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We used the virtual observatory tool in VOSA to cross-match the UVIT-detected sources with the above- mentioned photometric catalogs (Bayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' MEMBERSHIP DETERMINATION We employed the Gaia early data release 3 (EDR3) catalog that provides data with unprecedented preci- sion to identify the cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In particular, it provides the complete 5-parameter astrometric solution (positions, proper motions, and parallaxes) and mag- nitudes in its three photometric bands (G, GBP , and GRP ) with a limiting magnitude of about G∼21 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' To assign the proper motion (PM) membership proba- bility (Pµ) of all stars observed in the cluster, we first downloaded all detections located within a 30′ radius from the cluster’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' To include all possible mem- bers of the cluster, we opted to use a radius bigger than that provided by Kharchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2013) cata- log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Then, we applied the data quality criteria to select the sources with a good astrometric solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Stars are selected as follows: (i) we removed those with paral- laxes that deviate by more than 3σ from the expected parallax calculated using the previously known distance 4 Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' UVIT color image of OC NGC 2818 in FUV F148W channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Here orange color depicts the FUV detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The extended structure in this image represents the PN NGC 2818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' North is up, and east is left in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' F154W N E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='5 arcmin F169M N E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='5 arcmin N E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='5 arcmin F172M Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' UVIT/FUV images of PN NGC 2818 in three filters: F154W, F169M, and F172M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' List of the FUV observations of NGC 2818 obtained with UVIT in two epochs used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The last column lists the extinction value computed in each FUV filter using Fitzpatrick (1999) law of extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Filter λmean ∆λ ZP texp (sec) Aλ (Å) (Å) (AB mag) (1st epoch) (2nd epoch) (mag) F148W 1481 500 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='09 1736 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='58 F154W 1541 380 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='77 1491 2877 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='55 F169M 1608 290 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='41 1715 1999 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='54 F172M 1717 125 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='27 1903 2878 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='51 to the cluster, where σ is the error in parallax given in Gaia EDR3 catalog, (ii) we also removed the sources with renormalized unit weight error (RUWE) exceeding 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 as larger values of this parameter might lead to an unreliable astrometric solution (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Riello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We made use of a probabilistic Gaussian mixture model (GMM) method to select cluster members and in- fer the intrinsic parameters of the distributions of both member and non-member stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In this method, the dis- tribution of sources in the vector-point diagram (µα, µδ) is modeled as a mixture of two Gaussian distributions, one for the cluster members and another one for the field sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The details of this method are well described in N 1 arcminExotic Stellar Populations in NGC 2818 5 16 17 18 19 20 21 22 23 UV mag (AB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='40 Error (mag) F148W F154W F169M F172M Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' PSF-fit errors (median) as a function of mag- nitude for our UVIT observations of NGC 2818 in all FUV bandpasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Vasiliev (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The Gaussian probability distribution corresponding to the sum of two distributions is f(µ|µi, � i) = 2 � i=1 wi exp � − 1/2(µ − µi)T �−1 i (µ − µi) � 2π � det � i (1) wi ≥ 0, 2 � i=1 wi = 1 (2) where µ is individual PM vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' µi are field and cluster mean PMs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' � is the symmetric covariance matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' and wi are weights for the two Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Full details of this method for the n-dimensional case are described in (Vasiliev 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The initial guess for cluster PM µα and µδ values and internal velocity dispersion are taken from (Cantat- Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We utilized GaiaTools1 to maxi- mize the total log-likelihood of GMM and measure the mean PM and standard deviation of both the Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The membership probabilities (MPs) of all the selected stars are calculated using the same tech- nique simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The equations used to maximize the log-likelihood of GMM and estimate the MP of the 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='com/GalacticDynamics-Oxford/GaiaTools ith star belonging to the kth component are given in appendix A in Vasiliev (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The PM mean and standard deviations of the clus- ter distribution are computed to be µα = -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='417 mas/yr and µδ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='540 mas/yr, with σc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='045 mas/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In Figure 4, we show the position of stars in the sky, in the PM space known as vector point diagram (VPD), and in an optical CMD created using Gaia filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Cyan dots in all the plots depict the member stars belonging to the cluster, and black dots represent the field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 718 stars are identified as most likely cluster members with Pµ>50% and considered for subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' This method works well for a distinguishable distribu- tion of PM for the field and cluster stars in the VPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' But, in this case, the PM of cluster stars are located well within the PM distribution of the field stars, suggesting a non-trivial identification of cluster members from field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Therefore, it is possible that stars with a lower membership probability than the above-mentioned limit might also be members of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Is PN a member of the cluster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The membership of the PN with OC has been de- bated in several studies in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Tifft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (1972) found that PN NGC 2818 is a member of the OC of the same name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Dufour (1984) presented the results of photometric as well as spectroscopic observations of the nebula to analyze its physical properties and chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' He suggested that the nebula is probably associated with the star cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Pedreros (1989) an- alyzed this cluster using CCD UBV photometric data and assumed a physical association of the nebula with the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Surendiranath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (1990) also suggested the association of the PN with the cluster from their CCD photometry of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' However, Mermilliod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2001) derived accurate heliocentric radial veloci- ties for 12 cluster red giants to obtain a mean heliocen- tric radial velocity of Vhel = +20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='3 kms−1, signif- icantly different from the PN velocity of −1 ± 3 kms−1 (Meatheringham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 1988), suggesting that they are unrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Recently, (Vázquez 2012) reanalyzed the complex kinematics and morphology of the nebula using high-resolution Hubble Space Telescope (HST) archive imaging and high-dispersion spectroscopic data and de- termined a systemic heliocentric velocity of PN to be +26±2 kms−1 in closer agreement with the OC, sug- gesting its membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Moreover, based on its RV, Hα surface brightness, and radius, Frew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2016) con- cluded that the PN might be a cluster member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The Gaia EDR3 trigonometric parallax for the cen- tral star of the nebula (CSPN) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0319±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='21 mas, but it can be noted that the uncertainty in it is more than 6 Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='4 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='6 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='8 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='4 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='6 RA (deg) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='8 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='4 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 DEC (deg) 10 5 0 (mas/yr) 0 5 10 (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 G GRP 10 11 12 13 14 15 16 17 18 19 20 G Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In three panels from left to right, PM members of the cluster are shown with cyan dots, and the remaining Gaia EDR3 sample marked with black dots represents field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Left Panel: position in the sky;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Middle Panel: Vector Point Diagram (VPD);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Right Panel: Gaia Optical CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' its value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' So, it can not be used to obtain the distance to the nebula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The best estimate of the statistical distance is given by (Frew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2016) as 3000±800 pc not too far from cluster distance of 3250±300 pc estimated by Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (Cantat-Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Cantat- Gaudin & Anders 2020) obtained the members of the several OCs, including NGC 2818, using Gaia DR2 PM data, and suggested that it is a non-member of the clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In our membership analysis, we have obtained the membership of the CSPN using the Gaia EDR3 PM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The PM in RA and DEC of the CSPN as listed in Gaia EDR3 catalog is µα = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='712 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='185 mas/yr and µδ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='18 mas/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Its Pµ is estimated to be ∼11%, indicating non-membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Nevertheless, it can be noted from the location of the CSPN shown with the red star symbol in the VPD that it is lying close to the PM distribution of the cluster members (Cyan dots), implying that it is quite likely a member of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Statistically, it is lying within 3σ of the mean PM of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We expect that the future Gaia data re- lease (Gaia DR4) might give more precise and accurate PM measurements that can re-confirm its association with the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Further, assuming both cluster and nebula at the same distance, we computed their true velocity using their already available RV and PM infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We found that the true velocity of the cluster and nebula turn out to be approximately the same (VC = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='7kms−1 & VP N = 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='7kms−1), implying that the values of the space velocity are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Reddening towards the PN Several estimates of extinction/reddening towards the cluster have been made since the initial investigation by Tifft et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (1972) of E(B−V) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='22 mag, recon- firmed by Surendiranath et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (1990) and recently re- fined by Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2021), to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='20 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' However, there are a few independent estimates of extinction towards the PN NGC 2818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Dufour (1984) estimated it from the Balmer lines Hα/Hβ ratio as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='24±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='02 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gathier & Pottasch (1988) list a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='20 mag, and Frew et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2016) estimated a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='17±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='08 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We presently estimate E(B−V) value using free-free con- tinuum flux and the nebular Hβ flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The flux den- sity, Sν at 5 GHz of the entire nebula, is measured by Zhang (1995) as 33 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The total Hβ flux is estimated by Gathier & Pottasch (1988) as logF(Hβ) as -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='40 (ergcm−2s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Following Pottasch (1984), the expected ratio of Sν to F(Hβ) is given as S(ν) F(Hβ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='51×107×T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='53 e ×(ν)−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1×Y Jy/ergcm−2s−1 where Te is the electron temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' ν is frequency in GHz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Y = (1 + n(He+) n(H+) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The value of n(He+) n(H+) is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='13 assuming all He is in He+ form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Dufour (1984) derived the Te[OIII] of 14,500±500 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' From the above relation, the logF(Hβ) expected from the radio continuum is - 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The equation from Milne & Aller (1975) used to compute the reddening is following: E(B − V ) = 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='46log F(Hβ)exp F(Hβ)obs Inserting the expected and observed logF(Hβ) values in the above equation, we obtain the value of E(B−V) ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='23 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Thus, the extinction/reddening towards this cluster and nebula is of similar value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' From the comparison of distance, RV, PM, and ex- tinction/reddening values of the cluster and nebula, we suggest a physical association of the PN with the OC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' COLOR MAGNITUDE DIAGRAMS Exotic Stellar Populations in NGC 2818 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='8 Gbp Grp 12 14 16 18 20 G MS BSS YSS SGB RGB FUV detected 775 Myr,[Fe/H]=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 dex Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Optical CMD of the NGC 2818, created us- ing Gaia EDR3 photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' All filled symbols denote the stars with Pµ ≥ 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Blue-filled stars and yellow-filled stars are the selected blue and yellow straggler stars used for further cross-match with UVIT data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The stars detected in all FUV images are outlined with cyan- colored square and star symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The over-plotted green solid line represents the non-rotating MIST isochrone of solar metallicity and an age of 775 Myr, set at redden- ing, E(B−V)=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2 mag and distance modulus, (m−M)V = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='56 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Classification of Exotic sources This section describes the classification and identifica- tion of exotic sources, such as BSSs and YSSs, expected to emit in the FUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' As mentioned in Section 3, we considered the probable cluster members with Pµ>50% and created the PM-cleaned optical CMD (Gbp - Grp vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' G) using the Gaia filters shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In this CMD, stars outlined with cyan color depict the various identified star populations in FUV images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Rain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2021) presented a new proper-motion-cleaned catalog of BSSs in galactic OCs using Gaia DR2 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We cross-matched the Gaia EDR3 cluster members with the BSS catalog to classify this population in the clus- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Out of five identified BSSs in NGC 2818 by Rain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (2021), we detected four BSSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The remaining one BSS, not detected by us, is found to be a non-member of the cluster in our membership catalog and also falls outside the FoV of NGC 2818 observed with UVIT in two epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Jadhav & Subramaniam (2021) also pro- duced a catalog of BSSs in OCs using Gaia DR2 data with a Pµ>70%, and they found two BSS candidates in this cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The difference in the above-mentioned cat- alogs could be due to the adopted age criteria, selection method, and different membership probability cut-offs used in the two studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We obtained the MESA Isochrones & Stellar Tracks (MIST) for the UVIT and Gaia EDR3 filters from an updated MIST online database2 to identify and classify distinct evolutionary sequences in the cluster (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We considered isochrones with � α/Fe � = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0, metallicity, Z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='017210 (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2021), not incorporating initial rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Cluster parameters such as age, extinction, and distance modu- lus, adopted to fit the isochrone to the observed optical CMD, are 775 Myr, AV =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='6 mag, and (m−M)V =12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='56, respectively (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The overplotted isochrone (solid green line) over the observed optical CMD is dis- played in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We notice that the isochrone ap- pears well-matched to the observed CMD along the main-sequence, sub-giant branch (SGB), but it is not reproducing the observed position of the red clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' To account for this mismatch along the red clump, (Bas- tian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2018) suggested that there might be a prob- lem in the calibration of the models for the red clump or the conversion between theoretical properties of the isochrones (temperature, gravity, and luminosity) to ob- servational space in Gaia filters is off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We also selected the YSSs based on their location in the optical CMD, as they have colors in between the turn-off (TO) and RGB and appear brighter than the SGB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We have chosen two such stars marked with yellow colored filled symbols shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' FUV-optical CMDs This section presents the FUV-optical CMDs gener- ated by cross-identifying common stars between optical and our FUV detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We cross-matched the sources detected in the UVIT FUV filters with the Gaia EDR3 with a maximum separation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='′′3, which is the typi- cal FWHM of the PSF for the UVIT filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' To plot the FUV-optical CMDs, first, we made the magnitude system adopted by Gaia similar to that of UVIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' That is, we transformed the Vega magnitude system used in the Gaia photometric system to the AB system using the photometric zero points reported in the Gaia EDR3 documentation3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We have created and shown the FUV-optical CMDs for cluster members in Figure 6 using F148W and 2 https://waps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='cfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='edu/MIST/interp_isos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='html 3 https://gea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='int/archive/documentation 8 Rani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' F169M filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We note that a similar trend of detected stellar populations is observed in the other two filters (F154W & F172M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The error bars displayed in all FUV CMDs are estimated as the median of the stars’ errors at a chosen magnitude range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The FUV-optical CMDs are also over-plotted with updated MIST isochrones (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2016) to compare the locations of the distinct se- quences predicted by the theoretical models with the observed ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' In all FUV images, hot and bright stars such as BSSs, YSSs, and MS are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We have de- tected 4 BSSs out of 5 previously known in the literature (Rain et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Four detected BSSs are confirmed RV and PM members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Two YSSs are also identified in all FUV images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We note that these stars are well- separated and brighter than the theoretical isochrone presenting the SGB sequence in all FUV-optical CMDs, in turn confirming their classification as YSSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' RGB and Red clump stars are too faint to be detected in the FUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The FUV-optical CMDs show a large scatter along MS, as shown in Figure 6, unlike optical CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The overlaid isochrones in all FUV-optical CMDs help to trace the MS scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' We note that a few MS stars are brighter than theoretical MSTO not reproduced by isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' These might have high rotational velocities accounting for this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Some of them may be binaries or poten- tial BSSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' One BSS is found to be very hot and bright in all FUV-optical CMDs compared to the other three BSSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' This BSS can be an exciting candidate to char- acterize, as it might have a hot WD companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' As two YSSs are detected in all FUV images and found to be bright in all FUV-optical CMDs, these stars also might have a hot companion, which leads to their detection in the FUV images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' These are intriguing targets fur- ther to understand their formation and evolution in the clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Extended MS turn-off in FUV CMDs In order to check the sensitivity of UVIT colors to the Teff affected by the rotational velocity, we plot (Gbp−Grp) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (F172M−G) color as shown in Figure 7, which indicates a linear relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The range of Gaia color is only 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='4 mag whereas F172M−G spans about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 mag, which makes F172M−G color more sensitive and responsive to rotational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' F172M−G color is preferred over F169M−G because the band F172M allows only continuum light, and no chromospheric or transitional emission lines are seen in late-type stars in FUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Comparison of the CMD, F172M−G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gbp (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 8 upper right) with CMD of Gbp−Grp vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gbp (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 8 upper left) shows the sensitivity of F172M−G color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The bend in the isochrone in F172M−G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gbp CMD at a color of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content='0 indicates the beginning of eMSTO prominently (unlike Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 8, left panel), and all the stars right of the isochrone show high rotational velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The MS comprises stars with both high and low rotational velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' However, the CMD of F169M−G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gbp exhibits some more aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' From the comparison of F169M−G color with F172M−G in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' 8, we find that the former is redder than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' It can be due to the fact that the F169M flux in late-type stars is smaller than at F172M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Moreover, the predicted colors using the theoretical isochrones are following the same trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' It is well known that MS stars later than about F2 would possess coronal and transitional regions as evi- denced in the FUV region by emission lines of C IV, He II, Si IV, N V, N IV, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' (Linsky & Haisch 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Jordan & Linsky 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Prominent lines like C IV and He II occur in the F169M band region (unlike the F172M band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' The F154W and F148W would contain a few more emission lines in addition to C IV and He II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Thus, the CMD of F169M-G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gbp shows that the MS stars are shifted bluewards to the isochrone, probably suggesting the presence of transitional region lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Even in the F169M−F172M vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gbp CMD shown in the lower right panel of Figure 8, it is evident that most stars have bluer colors than the theoretically ex- pected ones from isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' It is to be noted that all stars on the blue edge of the MS in CMD of F169M−G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tAzT4oBgHgl3EQf-_7r/content/2301.01943v1.pdf'} +page_content=' Gbp (15 0 and all f ∈ Cǫ +b(R2), +� +R2[nX(z)]kf(z)dz +converges as k → ∞. +Furthermore, if η = η1dx1 +η2dx2 with η1, η2 ∈ C1+ǫ(R2) is such that f = ∂1η2 −∂2η1, almost +surely, +lim +k→∞ +� +R2[nX(z)]kf(z)dz = +� T +0 +η ◦ dX + +� +[XT ,X0] +η, +where the stochastic integral in the right hand side is to be understood in the sense of Stratonovich. +Corollary 2. For all x and y in R2, the same result holds if the planar Brownian motion is +replaced with a planar Brownian loop or a planar Brownian bridge between distinct points. +We will denote this limit as −� +R2 nX(z)f(z)dz, since we want to think of it as to the integral +of nX with respect to the measure f(z)dz. +1.2. Magnetic impurities. In the theory of weak localization in 2 dimensional crystals, for +which we refer to [2], one studies quasiclassical electrons moving inside a metal with magnetic +impurities, in the presence of a magnetic fields which induces an Aharonov–Bohm effect on +the electrons. In some regime of the parameters, the electron is usually modeled by a planar +Brownian trajectory. +In particular, for the computation of the weak-localization correction +to the Drude conductivity, the electron is modeled by a Brownian loop (see e.g. +[7]). +The +impurities are modeled by a Poisson process of points P with intensity ρdz in the plane, and +the Aharonov–Bohm effect is described by a phase shift exp(iα � +z∈P nX(z)). +In [4], the authors study the limit ρ → +∞ with κ = αρ constant, and derive a formula for +the phase shift averaged over both P and X. +For an integrable function f ∈ L1(R2), 1 +ρ +� +z∈P f(z) is a Monte–Carlo estimation for +� +R2 f(z)dz, +and therefore +eiκ +� +R2 f(z)dz = lim +ρ→∞ EP� +ei κ +ρ +� +z∈P f(z)� +. +However, as it is noticed in [5], for a Brownian loop X, +EX� +eiκ−� +R2 nX(z)dz� +̸= lim +ρ→∞ EX,P� +ei κ +ρ +� +z∈P nX(z)� +, +which is due to the lack of integrability of the function nX. +As we proved in [11], the Monte–Carlo method fails in this situation: it is true that X-almost +surely, 1 +ρ +� +z∈P nX(z) converges in distribution (with respect to P) as ρ → ∞, but the limit is +not deterministic –or should we say, not measurable with respect to X. It is instead equal to +the sum of −� +R2 nX(z)dz with a centered Cauchy distribution independent from X. From this +result, one can rigorously prove the formula obtained first in [5] for +lim +ρ→∞ EX,P[ei κ +ρ +� +z∈P nX(z)]. + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +3 +However, for the scales at play, the magnetic field which induces the Aharonov-Bohm effect +cannot be considered as homogeneous in general [8]. Our second goal in this paper is to derive +an asymptotic formula for the functional of X given by +lim +ρ→∞ EP[ei 1 +ρ +� +z∈P f(z)nX(z)], +for a non homogeneous magnetic field f and a non homogeneous density of impurities. +Theorem 3. Let f, g ∈ Cǫ +b(R2), with g ≥ 0. For ρ > 0, let P be Poisson process on R2 with +intensity ρg(z)dz, and let X be either a Brownian motion or a Brownian bridge with duration +1, independent from P. Then, X-almost surely, +lim +ρ→∞ EP[ei 1 +ρ +� +z∈P f(z)nX(z)] = exp +� +iα− +� +nX(z)f(z)g(z)dz − |α| +2 +� 1 +0 +|f(Xt)|g(Xt)dt +� +where EP is the expectation over P (conditional on X). +Although this formula is suited to the problem of magnetic impurities, the following alternative +formulation might be more appealing to the reader. +Corollary 4. Let g ∈ Cǫ +b(R2), with g ≥ 0. For ρ > 0, let P be Poisson process on R2 with +intensity ρg(z)dz, and X be either a Brownian motion or a Brownian bridge with duration 1, +independent from P. +Let also Γ : [0, 1] → R be a standard Cauchy process. +Then, for all +(f1, . . . , fn) ∈ Cǫ(R2), X-almost surely, the n-uple +�1 +ρ +� +z∈P +f1(z)nX(z), . . . , 1 +ρ +� +z∈P +fn(z)nX(z) +� +converges in distribution toward (ξ(f1), . . . , ξ(fn)) where +ξ(f) = − +� +nX(z)f(z)g(z)dz + 1 +2 +� 1 +0 +f(Xt)g(Xt)dΓt. +Remark 5. Given f, g ∈ Cǫ +b(R2), there always exists a differential 1-form η with regularity C1+ǫ +such that ∂1η2 − ∂2η1 = fg, so that −� +nX(z)f(z)g(z)dz can always be written as a stochastic +integral. +Since all the results hold X-almost surely, the assumptions that the functions are bounded +can easily be lifted, but some of the intermediate results come with a quantitative version which +depends upon the L∞ norms. +This paper is built in the continuity of two former papers from the same author, [11] and [9]. +It is not necessary to read them to understand the present paper, but we will use some results +from those papers, as well as from [10]. +2. Notations +2.1. Differential forms and integrals. For α ∈ (0, 1), we define Cα(R2) as the set of functions +f : R2 → R such that the semi-norm +|f|Cα := sup +x,y∈R2 +x̸=y +f(x) − f(y) +|x − y| +is finite. We also define Cα +b (R2) = Cα(R2) ∩ L2(R2), which we endow with the norm +∥f∥Cα +b = ∥f∥∞ + |f|Cα. +For a differential 1-form η = η1dx1 + η2dx2 and α ∈ [0, 1), we write η ∈ C1+α(T ∗R2) if +∂iηj ∈ Cα(R2) for all i, j ∈ {1, 2}. +Given a curve X : [0, T] → R2, we write +� +X +η := +� T +0 +η1(Xt)dX1 +t + +� T +0 +η2(Xt)dX2 +t , + +4 +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +where these integrals are to be understood either as classical integrals or as Stratonovich inte- +grals, depending on the regularity of X. No Itô integral will be involved in this paper, and all +the stochastic integrals are to be understood in the sense of Stratonovich. +For η ∈ C1+α(T ∗R2), we identify the 2-form dα = (∂1η2 − ∂2η1)dx1 ∧ dx2 with the signed +measure (∂1η2 − ∂2η1)dx, where dx is the Lebesgue measure on R2. +For a bounded set D ⊂ R2 and f ∈ L1 +loc(R2), we use the unconventional notation +f(D) = +� +D +f(z)dz, +and |D| for the Lebesgue measure of D. +2.2. Winding. Given a curve X on R2, that is a continuous function from [0, T] to R2 for some +T > 0, we write ¯X for the concatenation of X with a straight line segment from XT to X0. +Although the parameterisation of this line segment does not matter in the following, we will +assume it is parameterized by [T, T + 1] at constant speed, unless X is a loop (that is, a curve +with XT = X0), in which case we set ¯X = X. +Given a curve X and a point z outside the range of ¯X, we write nX(z) for the winding number +of ¯X around z. +For a relative integer k, we define +AX +k = {z ∈ R2 \ Range( ¯X) : nX(z) = k}. +For n > 0, we also define +DX +n = {z ∈ R2 \ Range( ¯X) : nX(z) ≥ n} = +� +n≤k<+∞ +AX +k , +and +DX +−n = {z ∈ R2 \ Range( ¯X) : nX(z) ≤ −n} = +� +−∞ 0 is the probability distribution on R which has a density with respect to the +Lebesgue measure given at x by +1 +πσ +σ2 +σ2 + (x − p)2 . +A Cauchy random variable with position parameter p and scale parameter σ is a random variable +distributed according to C(p, σ). In ordre to unify some results, we will also write C(p, 0) for a +random variable which is actually deterministic and equal to p. + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +5 +Following [6, Definition 5.2]1, we will say that a random variable Z on R lies in the strong +domain of attraction of a Cauchy distribution if there exists σ ≥ 0, δ > 0 such that +P(Z ≥ x) += +x→+∞ +σ +πx + o(x−(1+δ)), +P(Z ≤ −x) += +x→+∞ +σ +πx + o(x−(1+δ)). +It then follows from Lemma 5.1 and Theorem 1.2 in [6] that Z follows a central limit theorem: +if (Zi)i∈N are i.i.d. copies of Z, then there exists a unique p such that +1 +N +N +� +i=1 +Zi =⇒ Y ∼ C(p, σ). +Notice that the same assumptions with δ = 0 are not sufficient for such a central limit theorem +to hold. +The parameters p and σ such that Y ∼ C(p, σ) are uniquely defined. We call them respectively +the position parameter pZ of Z, and the scale parameter σZ of Z.2 +3. Former results +We will use the following results from [11], [9] and [10]. +Lemma 3.1 (Lemma 5.2 in [11] ). Assume Z belongs to the strong attraction domain of a +Cauchy distribution. Then, its position parameter pZ is equal to +lim +n→∞ E[[Z]n]. +When Y and Z lie in the strong attraction domain of Cauchy distributions, or even when they +are Cauchy random variables, but they are not independent, Y + Z does not necessarily belong +to the strong attraction domain of a Cauchy distribution. What might be even more surprising +is that, even if Y , Z, and Y + Z are Cauchy random variables, pY +Z can differ from pY + pZ +(see e.g. [3] for an explicit counter-example). Yet, the following lemma offers conditions weaker +then independence under which additivity is restored. +Lemma 3.2 ( Lemma 5.3 in [11] ). Let n ≥ 1 and Z1, . . . , Zn be random variables which each lie +in the strong attraction domain of a Cauchy distribution. Assume that there exists δ > 0 such +that, for all i, j ∈ {1, . . . , n}, i ̸= j, +P(|Zi| ≥ x and |Zj| ≥ x) += +x→+∞ o(x−(1+δ)). +Then, Z = �n +i=1 Zi lies in the strong attraction domain of a Cauchy distribution, and pZ = +�n +i=1 pZi. +The following lemma should be compared with the definition of the strong domain of attrac- +tion, where the random variable Z is given by nX(P), with X fixed and P a random point +distributed according to +1 +Z +1K(z)f(z)dz (when f ≥ 0), where K is a convex set containing +Range(X). +Lemma 3.3 (Lemma 5 in [9]). Let X : [0, 1] → R2 be a planar Brownian motion. For all β < 1 +2, +there exists δ > 0 such that almost surely, there exists C such that for all bounded continuous +function f ∈ Cb(R2), for all n ≥ 1, +���2πnf(Dn) − +� 1 +0 +f(Xu)du +��� ≤ C(ωf(2∥X∥Cβn−δ) + ∥f∥∞n−δ), +where ωf is the continuity modulus of f, i.e. ωf(r) = supx,y:|x−y|≤r |f(x) − f(y)|. +From symmetry of the Brownian motion, Lemma 3.3 also holds with Dn replaced with D−n. +We will also need some Lp control. +1As opposed to [6], we include the trivial case σ = 0 in our definition. +2When Z is a Cauchy random variable, it belongs to the strong domain of attraction of a Cauchy distribution. +There is thus two definitions of its position parameter, and two definitions of its scale parameter. Of course, the +two definitions of its position parameter agree, and the two definitions of its scale parameter agree as well. + +6 +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +Lemma 3.4 ( Theorem 6.2 in [11] ). For all δ < 1 +2 and p ≥ 2, there exists a constant C such +that for all N ≥ 1, +E +���DN − +1 +2πN +��p� 1 +p ≤ CN −1−δ. +Finally, the following lemma will be used to check the condition inside Lemma 3.2. +Lemma 3.5 (Theorem 1 in [10]). Let X, X′ : [0, 1] → R2 be two independent Brownian motions, +starting from equal or different points in the plane. Then, n2|DX +n ∩DX′ +n | almost surely converges +as n → ∞. +A few more results will be used, but will be easier to formulate later. +4. Stokes formula +In this section, X : [0, 1] → R2 is a standard Brownian motion under P. +4.1. Existence of a limit. We will first prove the first part of Theorem 1: +Lemma 4.1. Let ǫ > 0. P-almost surely, for all f ∈ Cǫ +b(R2), the limits +− +� +nX(x)f(x)dx := lim +N→∞ +� +R2[nX(z)]Nf(z)dz +and +lim +N→∞ +� +R2 nX(z)1|nX(z)|≤N f(z)dz +exist and are equal. Almost surely, the application f �→ nX(f) from Cǫ +b(R2) to R is continuous. +Proof. We fix β ∈ +� +0, 1 +2 +� +. +Let δ > 0 be such that Lemma 3.3 holds, and let E be the full +probability event on which ∥X∥Cβ < ∞ and Lemma 3.3 holds both for the sequence Dn and the +sequence D−n, with a corresponding random constant C. +On E, for all ǫ > 0, with C′ = 4πC, C′′ = C′(1 + |X|ǫ +Cβ), for all f ∈ Cǫ(R2), +���f(Dn) − f(D−n) +��� ≤ C′n−1(ωf(2|X|Cβn−δ) + ∥f∥∞n−δ) +≤ C′′n−1(|f|Cǫn−δǫ + ∥f∥∞n−δ). +(1) +Thus, on E, the sum +� +n≥1 +(f(Dn) − f(D−n)) +is absolutely convergent. By applying an Abel summation, we obtain +N +� +n=1 +(f(Dn) − f(D−n)) = +� +R2[nX(z)]Nf(z)dz, +so that the right-hand side is convergent on the event E. +Besides, +��� +� +R2[nX(z)]Nf(z)dz − +� +R2 nX(z)1|nX(z)|≤Nf(z)dz +��� = N|f(DN+1) − f(D−N−1)|, +which, on the almost sure event E, converges toward 0 as N goes to infinity (by (1)). +The only thing that remains to be shown is the almost sure continuity of the application +f �→ −� +nX(x)f(x)dx. Since it is clearly linear, it suffices to show that it is almost surely a +bounded operator. By (1), +N +� +n=1 +|f(Dn) − f(D−n)| +is bounded by C(3)∥f∥Cǫ +b for a random constant C(3) which depends on ǫ, β and δ, but not on f +nor N. Thus, |−� +nX(x)f(x)dx| ≤ C(3)∥f∥Cǫ +b, which concludes the proof. +□ + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +7 +4.2. Strategy for the Stokes’ formula. In order to conclude the proof of Theorem 1, we +now need to identify −� +nX(x)f(x)dx with the Stratonovich integral +� +X η + +� +[X1,X0] η, when +f = ∂1η2 − ∂2η1. +To this end, we decompose the trajectory X into several pieces. First, we denote by X(n) the +dyadic piecewise-linear approximation of X with 2n pieces: for λ ∈ [0, 1], i ∈ {0, . . . , 2n − 1}, +and t = (i + λ)2−n, +X(n) +t += Xi2−n + λ(X(i+1)2−n − Xi2−n). +For i ∈ {0, . . . , 2n − 1}, we also set Xi, the restriction of X to the interval [i2−n, (i + 1)2−n]. +Finally, set −� +nXi(x)f(x)dx the almost sure limit +− +� +nXi(z)f(z)dz = lim +N→∞ +� +R2[nXi(z)]Nf(z)dz. +By Lemma 4.1, scale invariance, and translation invariance of the Brownian motion, almost +surely, −� +nXi(x)f(x)dx is well -defined for all n ≥ 0, for all i ∈ {0, . . . , 2n−1}, for all f ∈ Cǫ(R2).3 +Let us first sketch the strategy of our proof. First, notice that for all z ∈ R2 which does not +belong to Range(X) nor to Range(X(n)), +nX(z) = +2n−1 +� +i=0 +nXi(z) + nX(n)(z), +which essentially comes from the additivity of the winding index, with respect to the concate- +nation of loops. Thus, it is reasonable to expect that +− +� +nX(z)f(z)dz = +2n−1 +� +i=0 +− +� +nXi(z)f(z)dz + +� +R2 nX(n)(z)f(z)dz. +By applying the standard Stokes’ formula on the last integral, we get +− +� +nX(z)f(z)dz = +2n−1 +� +i=0 +− +� +nXi(z)f(z)dz + +� +X(n) η + +� +[X1,X0] +η. +As n goes to infinity, we will see that the contribution from the small pieces (i.e. the sum over i) +vanishes, whilst the integral along X(n) converges toward the Stratonovich integral +� +X η, which +gives the expected formula. +We will decompose the actual proof into the three following lemma, which we will prove in +the three following subsections. Let f ∈ Cǫ +b(R2), and η ∈ C1+ǫ(T ∗R2) such that f = ∂1η2 − ∂2η1. +Lemma 4.2. For all n, almost surely, +− +� +nX(z)f(z)dz = +2n−1 +� +i=0 +− +� +nXi(z)f(z)dz + +� +R2 nX(n)(z)f(z)dz. +(2) +Lemma 4.3. As n goes to infinity, +2n−1 +� +i=0 +− +� +nXi(z)f(z)dz +converges almost surely toward zero. +Lemma 4.4. As n goes to infinity, +� +X(n) η converges almost surely toward +� +η ◦ dX. +Of course, the conclusion that almost surely, +− +� +nX(z)f(z)dz = +� +X +η + +� +[X1,X0] +η, +and therefore that Theorem 1 holds, follows directly from these three lemma. +3Since we use the translation invariance, the function f is replaced with the random function z �→ f(z+Xi2−n). +This is not an issue, because Lemma 4.1 holds almost surely for all f, and not the other way around. + +8 +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +4.3. Additivity. Intuitively, the equality in Lemma 4.2 follows from integration of the almost- +everywhere equality +nX(z) = +2n−1 +� +i=0 +nXi(z) + nX(n)(z), +applied together with the Stokes formula for X(n). However, neither nX nor nXi are integrable, +we have to deal with the cut-offs that allow to define −� +nX(z)f(z)dz and the −� +nXi(z)f(z)dz : +in general, for a finite k, +[nX(z)]k ̸= +2n−1 +� +i=0 +[nXi(z)]k + [nX(n)(z)]k. +Proof of Lemma 4.2. From linearity with respect to f, we can and we do assume f ≥ 0. In the +event that that the restriction of f to B(0, ∥X∥∞) is identically vanishing, the result is trivial, +and we thus assume that +Z := +� +B(0,∥X∥∞) +f(z)dz +is strictly positive. +Let P be a random point in R2 those distribution conditional on X admits a density with +respect to the Lebesgue measure, given by +f(z)1B(0,∥X∥∞)(z) +Z +. +Then, X-almost surely, P-almost surely, +nX(P) = +2n−1 +� +i=0 +nXi(P) + nX(n)(P). +Notice that, for N ≥ 0, for ˜X equal to either X, or to one of the Xi, or to X(n), it holds that +P(n ˜ +X(P) ≥ N|X) = 1 +Z f(D +˜ +X +N), +P(n ˜ +X(P) ≤ −N|X) = 1 +Z f(D +˜ +X +−N). +Thus, Lemma 3.3 ensures that X-almost surely, the random variable n ˜ +X(P) belong to the strong +attraction domain of a Cauchy distribution for either ˜X = X or ˜X = Xi. As for ˜X = X(n), +|n ˜ +X| is bounded by 2n and therefore n ˜ +X(P) also belong to the strong attraction domain of a +(degenerate, σ = 0) Cauchy distribution. +Let us check that, X-almost surely, we can apply Lemma 3.2 to the set of variables +(Z0, . . . , Z2n−1, Z2n) = (nX0(P), . . . , nX2n−1(P), nX(n)(P)). +First, for i ∈ {0, . . . , 2n − 1}, for x ≥ 2n, +P(|nXi(P)| ≥ x and |nX(n)(P)| ≥ x) = 0 = o(x−(1+δ)). +Besides, for i, j ∈ {0, . . . , 2n − 1}, i ̸= j, +P(|nXi(P)| ≥ N and |nXj(P)| ≥ N) = 1 +Z f +�� +DXi +N ∪ DXi +−N +� +∩ +� +DXj +N ∪ DXj +−N +�� +≤ C∥f∥∞ +Z +|N|2, +for a random constant C. The last equality follows from Lemma 3.5, applied to the independent +Brownian motions +ˆXi : t �→ X(i+1−t)2−n − X(i+1)2−n, +ˆXj : t �→ X(j+t)2−n − X(i+1)2−n. +Notice that the constant C = C(n, i, j) depends upon i and j, but we can replace it with +C(n) = maxi,j C(n, i, j) so that it only depends on n. Furthermore, since there is only countably +many couples (i, j), the previous inequality holds almost surely for all (i, j) simultaneously. + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +9 +Thus, we can indeed apply Lemma 3.2 to deduce that the, X-almost surely, the position +parameters add up: +pnX(P ) = +2n−1 +� +i=0 +pnXi(P ) + pnX(n)(P ). +(3) +Furthermore, since |nX(n)(P)| is bounded, pnX(n)(P ) is quickly checked to be equal EP[nX(n)(P)|X], +that is +pnX(n)(P ) = 1 +Z +� +R2 nX(n)(z)f(z)dz. +Finally, from Lemma 3.1, we deduce that X-almost surely, +pnX(P) = lim +N→∞ EP� +[nX(P)]N +��X +� += lim +N→∞ +1 +Z +� +R2[nX(z)]Nf(z)dz = 1 +Z − +� +nX(z)f(z)dz, +and similarly +pnXi(P) = 1 +Z − +� +nXi(z)f(z)dz. +Thus, Equation 3 turns into +− +� +nX(z)f(z)dz = +2n−1 +� +i=0 +− +� +nXi(z)f(z)dz + +� +R2 nX(n)(z)f(z)dz, +as announced. +□ +4.4. Contribution from the small loops. We now prove that almost surely, +2n−1 +� +i=0 +− +� +nXi(z)f(z)dz −→ +n→∞ 0. +We will first need the following result, which should be compared with Lemma 3.3. +Lemma 4.5. Let ǫ > 0 and p ≥ 1. There exists a constant C and δ > 0 such that for all +f ∈ Cǫ(R2) and all N ≥ 1, +E +���f(DX +N ) − +1 +2πN +� 1 +0 +f(Xt)dt +��p� 1 +p ≤ CN −1−δ∥f∥Cǫ. +Proof. The proof is largely inspired from [9]. +Let T ≥ 1, which we will later take to be a function of N. For i ∈ {0, . . . , T −1}, let Xi be the +restriction of X to the interval [iT −1, (i+1)T −1]. Let Xpl be the piecewise linear approximation +of X with T pieces, +Xpl +(i+λ)T −1 = XiT −1 + λ(X(i+1)T −1 − XiT −1), +i ∈ {0, . . . , T − 1}, λ ∈ [0, 1]. +For i, j ∈ {0, . . . , T − 1}, let +Di +N = DXi +N , +Di,j +N = +� +DXi +N ∪ DXi +−N +� +∩ +� +DXj +N ∪ DXj +−N +� +. +For z outside Range(X) ∪ Range(Xpl), we have +nX(z) = +T−1 +� +i=0 +nXi(z) + nXpl(z), +|nXpl(z)| ≤ T. +It follows4 that, for all T, M, N ≥ 1 such that T(M + 1) < N, +DX +N ⊆ +T−1 +� +i=0 +Di +N−T−M(T−1) ∪ +T−1 +� +i,j=0 +i̸=j +Di,j +M ∪ Range(X) ∪ Range(Xpl), +4See Section 3.2 in [11] for more details. + +10 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +and therefore +f(DX +N ) ≤ +T−1 +� +i=0 +f(Di +N−T−M(T−1)) + +T−1 +� +i,j=0 +i̸=j +f(Di,j +M ). +We set t ∈ (0, 1 +3), m ∈ (1+t +2 , 1 − t), α < 1 +2, T = ⌊N t⌋, M = ⌊N m⌋, and we assume that N is +large enough for the inequality T(M + 1) < N to hold. We also set N ′ = N − T − M(T − 1) to +ease notations. +Using the fact that Di +N′ is contained inside the convex hull of Xi, hence in the ball centered +at X i +T with radius ∥X∥CαT −α, we deduce that f is bounded above by f(X i +T )+|f|Cǫ∥X∥ǫ +CαT −ǫα +on Di +N′. Thus, +f(DN) ≤ +T−1 +� +i=0 +f(X( i +T ))|Di +N′| + |f|Cǫ∥X∥ǫ +CαT −ǫα +T−1 +� +i=0 +|Di +N′| + ∥f∥∞ +� +i̸=j +|Di,j +M |. +≤ +1 +2πNT +T−1 +� +i=0 +f(X( i +T )) + ∥f∥∞ +T−1 +� +i=0 +�� +1 +2πNT − |Di +N′| +�� + |f|Cǫ∥X∥ǫ +CαT −ǫα +T−1 +� +i=0 +|Di +N′| ++ ∥f∥∞ +� +i̸=j +|Di,j +M | +≤ +1 +2πN +� 1 +0 +f(Xt)dt + |f|Cǫ∥X∥ǫ +CαT −ǫα +2πN ++ ∥f∥∞ +T−1 +� +i=0 +�� +1 +2πNT − |Di +N′| +�� ++ |f|Cǫ∥X∥ǫ +CαT −ǫα +T−1 +� +i=0 +|Di +N′| + ∥f∥∞ +� +i̸=j +|Di,j +M |. +Writing (f)p ++ for the positive part of f, to the power p, and using the triangle inequality in +Lp(P), we obtain +E +�� +f(DN)− +1 +2πN +� 1 +0 +f(Xt)dt +�p ++ +� 1 +p ≤ |f|CǫT −ǫα +2πN +E[∥X∥ǫp +Cα] +1 +p + ∥f∥∞E +���� +1 +2πN − |DN′| +��� +p� 1 +p ++ |f|CǫT −ǫαE[|DN′|2p] +1 +2p E[∥X∥2pǫ +Cα ] +1 +2p + ∥f∥∞E +�� � +i̸=j +|Di,j +M | +�p� 1 +p . +We now use the asymptotic equivalence N ′ ∼N→∞ N and +1 +N − +1 +N′ ∼N→∞ N t+m−2, as well +as Lemma 3.4, and the following estimations ([11, Lemma 2.4]): for all p ≥ 1, there exists a +constant C such that for all N ≥ 1, +E +�� � +i̸=j +|Di,j +M | +�p� 1 +p ≤ C log(N + 1)3+ 2 +p M−2T 1− 1 +p . +We end up with +E +�� +f(DN) − +1 +2πN +� 1 +0 +f(Xt)dt +�p ++ +� 1 +p ≤ C +� +|f|CǫN −1−tǫα + ∥f∥∞N m+t−2 + ∥f∥∞N −1−δ ++ |f|CǫN −1−tǫα + ∥f∥∞ log(N + 1)3+ 2 +p N −2m+t− t +p � +, +for an arbitrary but fixed δ ∈ (0, 1 +2). The conditions on t and m ensures that all the exponents +of N are smaller than −1, so that there exists δ′ and C such that +E +�� +f(DN) − +1 +2πN +� 1 +0 +f(Xt)dt +�p ++ +� 1 +p ≤ C∥f∥Cα +b N −1−δ′. +The negative part is treated in a similar way, and the lemma follows. +□ +Corollary 4.6. Let ǫ > 0 and p ≥ 1. There exists a constant C such that for all f ∈ Cǫ +b(R2), +E[(−� +nX(z)f(z)dz)p] +1 +p ≤ C∥f∥Cǫ +b. + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES 11 +Proof. Let C and δ be the constants of Lemma 4.5. Then, for all f ∈ Cǫ +b and n, +E[|f(Dn) − f(D−n)|p] +1 +p ≤ 2Cn−1−δ∥f∥Cǫ +b. +By triangle inequality in Lp, +E +���� +∞ +� +n=1 +(f(Dn) − f(D−N)) +��� +p� 1 +p ≤ 2C∥f∥Cǫ +b +∞ +� +n=1 +N −1−δ ≤ C′∥f∥Cǫ +b, +as expected. +□ +With this estimation in hand, we can now prove Lemma 4.3. +Proof of Lemma 4.3. For i ∈ {0, . . . , 2n −1}, we define ¯f i : R2 → R the constant function whose +unique value is equal to f(Xi2−n), and ˜f i = f − ¯f i. Since for all i, f �→ −� +Xi nX(z)f(z)dz is +linear, it suffices to show that both +2n +� +i=1 +− +� +Xi nX(z) ¯f i(z)dz = +2n +� +i=1 +f(Xi2−n)− +� +Xi nX(z)dz +and +2n +� +i=1 +− +� +Xi nX(z) ˜f i(z)dz +almost surely converge toward 0 as n → ∞. +From symmetry, for all i, E +� +−� +Xi nX(z)dz|(Xs)s≤ i +2n +� += 0. It follows that, for i < j, +E +� +f(Xi2−n)f(Xj2−n)− +� +Xi +nX(z)dz− +� +Xj +nX(z)dz +� += 0. +Besides, from a simple scaling argument, +E +�� +− +� +Xi nX(z)dz +�2� += 2−2nE +�� +− +� +X +nX(z)dz +�2� +. +Notice E[(−� +X nX(z)dz)2] < ∞, which follows for example from the previous corollary. +We deduce that +E +�� 2n +� +i=1 +− +� +Xi nX(z) ¯f i(z)dz +�2� += +2n +� +i=1 +E +� +f(Xi2−n)2� +− +� +Xi nX(z)dz +�2� +≤ 2−n∥f∥2 +∞E +�� +− +� +X +nX(z)dz +�2� +. +This L2 convergence rate is sufficient to conclude to the almost sure convergence: for all ǫ′ > 0, +P +� +∃n ≥ n0 : +��� +2n +� +i=1 +− +� +Xi nX(z) ¯f i(z)dz +��� ≥ ǫ′� +≤ 1 +ǫ′2 E +� +sup +n≥n0 +� 2n +� +i=1 +− +� +Xi nX(z) ¯f i(z)dz +�2� +≤ 21−n0 +ǫ′2 +∥f∥2 +∞E +�� +− +� +X +nX(z)dz +�2� +−→ +n0→∞ 0. +In order to deal with the sum involving ˜f i, one must be a bit careful about the way we use +the translation invariance and scale invariance of the Brownian motion. We set α < 1 +2 and we +define the event +R = {∥X∥Cα ≤ R}, +for a fixed R ≥ 1. Let ˆf i be the (random) function defined by +ˆf i(Xi2−n + z) = +� ˜f i(Xi2−n + z) +if |z| ≤ R2−αn, +˜f i(Xi2−n + R2−αn +|z| +z) +otherwise. +In particular, ˆf i satisfies the following properties: +⋄ ˆf i = ˜f i on B = B(Xi2−n, R2−αn), so that, in the event R, ˆf i(Di +n) = ˜f i(Di +n), +⋄ | ˆf i|Cǫ ≤ |f|Cǫ, and ∥ ˆf i∥∞ ≤ Rǫ2−ǫαn|f|Cǫ, + +12 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +⋄ As a random variable, ˆf i is measurable with respect to σ(Xi2−n). +Set also ˇf i(z) = ˆf i(Xi2−n+2− n +2 z), ˇXi : s ∈ [0, 1] �→ 2 +n +2 (X(i+s)2−n−Xi2−n), which is a standard +planar Brownian motion started from 0, independent from Xi2−n. Notice that ∥ ˇf i∥∞ = ∥ ˆf i∥∞ ≤ +Rǫ2−ǫαn|f|Cǫ and | ˇf i|Cǫ = 2− ǫn +2 | ˆf i|Cǫ ≤ 2− ǫn +2 |f|Cǫ, so that +∥ ˇf i∥Cǫ +b ≤ 21−ǫαn|f|Cǫ. +On the event R, we have +− +� +nXi(z) ˜f i(z)dz = 2−n− +� +n ˇ +Xi(w) ˇf i(2− n +2 w)dw. +Using Corollary 4.6 with p = 1, we deduce +E +� +1R +���− +� +nXi(z) ˜f i(z)dz +��� +� += 2−nE +� +E +���− +� +n ˇ +Xi(w) ˇf i(2− n +2 w)dw +�� +����Xi2−n +�� +≤ 2−nE +� +C∥ ˇf i∥Cǫ +b +� +≤ C21−n−ǫαn|f|Cǫ. +Thus, +P +� +R and ∃n ≥ n0 : +��� +2n−1 +� +i=0 +− +� +nXi(z) ˜f i(z)dz +��� ≥ ǫ′� +≤ 1 +ǫ′ +∞ +� +n=n0 +2n−1 +� +i=0 +E +� +1R +���− +� +nXi(z) ˜f i(z)dz +��� +� +≤ Cǫ,ǫ′,α,R2−ǫαn0|f|Cǫ +−→ +n0→∞ 0. +Since this holds for all R, we deduce that �2n−1 +i=0 +−� +nXi(z) ˜f i(z)dz almost surely converges toward +0 as n → ∞, which concludes the proof. +□ +4.5. Stratonovich integral as a limit of integrals along piecewise-linear paths. It only +remains to prove lemma 4.4 which for η ∈ C1+ǫ(T ∗R2) identifies the limit +lim +n→∞ +� +X(n) η +with the Stratonovich integral of η along X, which is fairly classical. It is for example a direct +consequence of the following lemma. +Lemma 4.7. For a given dissection ∆ = (t0 = 0, t1, . . . , tn = 1), and X : [0, 1] → R2 a +Brownian motion, let X∆ be the piecewise-linear approximation of X associated with ∆: for +λ ∈ [0, 1] and t = λti + (1 − λ)ti+1, +X∆(t) = λXti + (1 − λ)Xti+1. +For f ∈ C1(R2), let +I1 +∆(f) = +� +[ti,ti+1]∈∆ +f +�Xti+1 + Xti +2 +� +(X1(ti+1) − X1(ti)), +I2 +∆(f) = +� +[ti,ti+1]∈∆ +f(Xti+1) + f(Xti) +2 +(X1(ti+1) − X1(ti)), +I3 +∆(f) = +� 1 +0 +f(X∆(t))dX∆(t). +Then, almost surely, for all f ∈ C1+ǫ(R2), as |∆| → 0, +I2 +∆(f) − I1 +∆(f) → 0 +and +I3 +∆(f) − I1 +∆(f) → 0. + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES 13 +Proof. Let α ∈ +� 1 +2+ǫ, 1 +2 +� +. On the almost sure event ∥X∥Cα < ∞, we have +��� +� +[ti,ti+1]∈∆ +�f(Xti+1) + f(Xti) +2 +− f +�Xti+1 + Xti +2 +�� +(X1 +ti+1 − X1 +ti) +��� +≤ +� +[ti,ti+1]∈∆ +1 +2 +���f(Xti+1) − f +�Xti+1 + Xti +2 +� ++ f(Xti) − f +�Xti+1 + Xti +2 +���� +���X1 +ti+1 − X1 +ti +��� +≤ +� +[ti,ti+1]∈∆ +1 +4 +��� ∇Xti+1−Xtif +�Xti+1 + Xti +2 +� ++ ∇Xti−Xti+1f +�Xti+1 + Xti +2 +� +� +�� +� +=0 +��� +���X1 +ti+1 − X1 +ti +��� ++ +� +[ti,ti+1]∈∆ +2 +22+ǫ ∥∇f∥Cǫ|Xti+1 + Xti|2+ǫ +≤ 2−1−ǫ∥f∥C1+ǫ∥X∥2+ǫ +Cα +� +[ti,ti+1]∈∆ +|ti+1 − ti|α(2+ǫ) −→ +|∆|→0 0. +The second convergence is proved in a similar way: +��� +� +[ti,ti+1]∈∆ +� � ti+1 +ti +f(X∆(s))dX∆(s) − f +�Xti+1 + Xti +2 +� +(X1 +ti+1 − X1 +ti) +���� +≤ +� +[ti,ti+1]∈∆ +|X1 +ti+1 − X1 +ti| +��� +� 1 +1 +2 +� +f(λXti + (1 − λ)Xti+1) + f((1 − λ)Xti + λXti+1) − 2f +�Xti+1 + Xti +2 +�� +dλ +��� +≤ 2−1−ǫ∥f∥C1+ǫ∥X∥2+ǫ +Cα +� +[ti,ti+1]∈∆ +|ti+1 − ti|α(2+ǫ) −→ +|∆|→0 0. +□ +This concludes the proof of Lemma 4.4, and therefore the proof of Theorem 1 as well. Before +we conclude this section, we will shortly prove Corollary 2. +Proof of Corollary 2. To keep the proof simple, we treat the case when X : [0, 1] → R2 is a +Brownian loop started from 0. To deal with the case when X is a Brownian bridge from x to +y ̸= x, one must also take into account the winding function of the triangle between x, y, and +X 1 +2 , but this is done in a straightforward way. +From linearity, it suffices to prove the result when restricted to functions f ≥ 0. Furthermore, +since the result is trivial in the event f|B(0,∥X∥∞) = 0, we assume +� +B(0,∥X∥∞) f(z)dz > 0. +Let X1 be the restriction of X to [0, 1 +2] , X2 its restriction to [1 +2, 1], and ˆX2 : t ∈ [0, 1 +2] �→ X1−t. +Then, the distribution of X1 (resp. ˆX2) admits a density with respect to the density of a standard +planar Brownian motion defined on [0, 1 +2]. Using scale invariance, we can apply Theorem 1 to +both X1 and ˆX2. We deduce that, for i ∈ {1, 2}, for all ǫ > 0, almost surely, for all f ∈ Cǫ(R2), +� +R2[nXi(z)]kf(z)dz +converges as k → ∞, and the limits are almost surely equal to respectively +� +X1 η + +� +[X 1 +2 ,0] η and +� +X2 η − +� +[X 1 +2 ,0] η, where η is such that ∂1η2 − ∂2η1 = f. +Now we need to show that almost surely, for all f ∈ Cǫ(R2), −� +nX1(z)f(z)dz and −� +nX2(z)f(z)dz +add up properly, for which we proceed as in Lemma 4.2, introducing again a random point P. +Going through the same arguments as in the proof of Lemma 4.2, we see that it suffices to show +that, X-almost surely, +|DX1 +±N ∩ DX2 +±N| = o(N −1−δ), +(4) + +14 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +for the four possible couple of signs in front of N, and for some δ > 0. +To prove (4), we further decompose X1 and X2 by setting X11 (resp. X12, X21,X22) the +restriction of X to the interval [0, 1 +4] (resp. [1 +4, 1 +2], [1 +2, 3 +4], [3 +4, 1]). Then, DXi +±N ⊆ DXi1 +±N′ ∪ DXi2 +±N′ +where N ′ = ⌊N/2⌋. +We show that almost surely, |DX11 +N′ +∩ DX21 +N′ | = O(N −2), the 15 other intersections are treated +either similarly. Conditionally on X 1 +2, X11 and X21 are independen. Furthermore, both their +distribution, conditional on X 1 +2 , have a density with respect to the distribution of a standard +Brownian motion with duration 1 +4, started respectively from 0 and X 1 +2 . Thus, it suffices to show +that for all y, |DX11 +N′ +∩ DX21 +N′ | = O(N −2) when X11 and X21 are independent Brownian motions +started respectively from 0 and y. This follows directly from 3.5, with a scaling of 1 +2. +□ +5. Magnetic impurities +In this section, we fix a function g ∈ Cǫ +b(R2). For all λ > 0, we define Pλ a Poisson process on +R2 with intensity λg(z)dz, independent from X, and Γ : [0, T] → R a standard Cauchy process, +independent from X. We write EP the expectation with respect to Pλ, EX the one with respect +to X, EΓ the expectation with respect to Γ and E = EX ⊗ EP ⊗ EΓ the expectation on the +product space (although none of the variables we consider depend on both P and Γ, so truly +E = EX ⊗ EP or E = EX ⊗ EΓ, whichever is relevant). +For a function f ∈ Cǫ +b(R2), we define +ξλ(f) = 1 +λ +� +z∈Pλ +f(z)nX(z), +as well as +ξ(f) = − +� +nX(z) f · g(z)dz + 1 +2 +� 1 +0 +f · g(Xt)dΓt. +Notice that Γ almost surely has a finite p-variation for all p > 1 (see [1, Theorem 4.1]). Since +X-almost surely, (f · g) ◦ X ∈ C +ǫ +4([0, 1]), the integral +� 1 +0 fg(Xt)dΓt is well-defined as a Young +integral. +The main result from this section is the following +Lemma 5.1. Let f, g ∈ Cǫ +b(R2) be continuous and bounded functions. Assume that g takes +non-negative values. Let +Gβ,f,g := +� +k̸=0 +� +Ak +(eikβf(z) − 1)g(z)dz. +Then, X-almost surely, as β → 0, +Gβ,f,g = +β→0 iβ− +� +nX(z)fg(z)dz − |β| +2 +� 1 +0 +|f(Xt)|g(Xt)dt + o(β). +(5) +Before we dive into the proof of this lemma, we first explain with it implies both Theorem 3 +and Corollary 4. +Lemma 5.1 implies Theorem 3 and Corollary 4. Since the function min(|nX · f|, 1) is integrable +against the intensity measure λgdz of Pλ, we can use Campbell’s theorem, which gives +EP[eiαξλ(f)] = exp +� � +k̸=0 +� +Ak +(eik α +λ f(z) − 1)λg(z)dz +� += exp(λGβ,f,g), +where β = α +λ. +Besides, conditional on X, +� 1 +0 f(Xt)g(Xt)dΓ(t) is a centered Cauchy random variable with +scale parameter +� 1 +0 |f(Xt)|g(Xt)dt, whilst −� +nX(z)fg(z)dz is deterministic. It follows that +EΓ[eiαξ(f)] = eiα−� +nX(z)fg(z)dz− |α| +2 +� 1 +0 |f(Xt)|g(Xt)dt, +Thus, Lemma 5.1 implies Theorem 3. + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES 15 +Furthermore, since both ξλ(f) and ξ(f) are linear in f, one can use the Cramér-Wold device +to deduce Corollary 4 from its special case n = 1. By Lévy’s continuity theorem, this specific +case is equivalent to the statement that X-almost surely, for all α ∈ R, +EP[eiαξλ(f)] −→ +λ→∞ EΓ[eiαξ(f)]. +From our previous computation, this amount to show that X almost surely, for all α ∈ R, +exp(λGβ,f,g) −→ +λ→∞ exp +� +iα− +� +nX(z)fg(z)dz − |α| +2 +� 1 +0 +|f(Xt)|g(Xt)dt +� +, +which follows again from Lemma 5.1. +□ +Proof of Lemma 5.1. From symmetry, we can assume β > 0. Performing an Abel summation, +we obtain +Gβ,f,g = +∞ +� +k=1 +� � +Dk +eiβkf(1 − e−iβf)gdz + +� +D−k +e−iβkf(1 − eiβf)gdz +� += +∞ +� +k=1 +(φk,β(Dk) + φ−k,β(D−k)), +where +φk,β = eiβkf(1 − e− sgn(k)iβf)g. +The two terms in (5) comes from two different parts in this last sum: the term iβ−� +nX(z)f(z)g(z)dz +comes from the bulk of the sum, that is the part with k of the order of 1. The second term +comes from the tail of the sum, or more precisely from the part of the sum when k is of the +order of β−1. We will split the sum into several parts. For n, N ∈ N ∪ {∞} with n < N, we set +Gn,N +β,f,g = +N +� +k=n+1 +(φk,β(Dk) + φ−k,β(D−k)). +For N1 = N1(β) and N2 = N2(β) which will be set later on, we decompose Gβ,f,g into three +parts, +Gβ,f,g = G0,N1 +β,f,g +� �� � +bulk ++ GN1,N2 +β,f,g +� �� � +tail ++ GN2,∞ +β,f,g +� �� � +end +. +As β → 0, both N1 and βN2 will slowly diverge toward ∞. In particular, N1(β)<< β−1<< N2(β). +The reason why we need to treat the end part in a separate way is that its convergence toward +0 is not absolute, in the sense that the +∞ +� +k=N2+1 +|φk,β(Dk) + φ−k,β(D−k)| +does not converge toward zero as β → 0, and one must be a bit careful when dealing with this +term. The general term (without the absolute values) slowly oscillates between positive and +negative values, and we must take advantage of compensations. +For a given k ̸= 0, as β → 0, uniformly in z, +φk,β(z) = sgn(k)iβf(z)g(z) + O(β2), +and it follows that +φk,β(Dk) + φ−k,β(D−k) = iβ((fg)(Dk) − (fg)(D−k)) + O(β2). +For k ≥ 1, let Ck be such that for all β ∈ (0, 1), +|φk,β(Dk) + φ−k,β(D−k) − iβ((fg)(Dk) − (fg)(D−k))| ≤ Ckβ2, +and set N1(β) = min(⌊β− 1 +3⌋, sup{N : ∀k ≤ N, Ck ≤ β− 1 +3 }). + +16 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +Then, +���G0,N +β,f,g − iβ +N1 +� +k=1 +((fg)(Dk) − (fg)(D−k)) +��� ≤ +N1 +� +k=1 +Ckβ2 ≤ β +4 +3 = o(β). +Besides, N1 −→ +β→0 +∞, and Theorem 1 implies that +N1 +� +k=1 +((fg)(Dk) − (fg)(D−k)) −→ +β→0 − +� +nX(z)f(z)g(z)dz. +Therefore, +G0,N +β,f,g = iβ− +� +nX(z)f(z)g(z)dz + o(β). +(6) +We now look at the tail part of Gβ,f,g. Let δ > 0 and C (random) be such that for all N ̸= 0 +and φ ∈ Cǫ +b, +���φ(DN) − +1 +2π|N| +� 1 +0 +φ(Xu)du +��� ≤ C∥φ∥Cǫ +bN −1−δ. +Recall that the existence of such a couple (δ, C) is provided by Lemma 3.3. Let N2 = N2(β) be +any integer-valued function such that βN2 −→ +β→0 +∞ and βN 1−δ +2 +−→ +β→0 0. +For all φ, ψ ∈ Cǫ +b, |φψ|Cǫ ≤ |φ|Cǫ∥ψ∥∞ + ∥φ∥∞|ψ|Cǫ. We deduce that for all k and β, +∥φk,β∥∞ ≤ ∥eiβkf∥∞∥1 − eiβf∥∞∥g∥∞ ≤ β∥f∥∞∥g∥∞, +|φk,β|Cǫ ≤ |eiβkf|Cǫ∥1 − eiβf∥∞∥g∥∞ + ∥eiβkf∥∞|1 − eiβf|Cǫ∥g∥∞ + ∥eiβkf∥∞∥1 − eiβf∥∞|g|Cǫ +≤ kβ2|f|Cǫ∥f∥∞∥g∥∞ + β|f|Cǫ∥g∥∞ + β∥f∥∞|g|Cǫ, +so that +∥φk,β∥Cǫ +b ≤ β(1 + kβ)(1 + ∥f∥Cǫ +b)∥f∥Cǫ +b∥g∥Cǫ +b. +We deduce that, for all k > 0, +���φk,β(Dk)+φ−k,β(D−k)− 1 +2πk +� 1 +0 +(φk,β(Xu)+φ−k,β(Xu))du +��� ≤ 2C(1+∥f∥Cǫ)∥f∥Cǫ∥g∥Cǫβ(1+kβ)k−1−δ, +and there exists constants C′ = C′(f, g), C′′ = C′′(f, g) such that for all N2 ≥ N1, +���GN1,N2 +β,f,g − 1 +2π +N2 +� +k=N1+1 +1 +k +� 1 +0 +(φk,β(Xu) + φ−k,β(Xu))du +��� +≤ C′ +N2 +� +k=N1+1 +β(1 + kβ)k−1−δ ≤ C′′β(N −δ +1 ++ βN 1−δ +2 +) = o(β). +The remaining part of the analysis is standard calculus. Set +ψk,β = eiβkf sgn(k)iβfg. +Then, for β ≤ ∥f∥∞, +��� +N2 +� +k=N1+1 +φk,β − ψk,β +k +��� = |g| +��� +N2 +� +k=N1+1 +1 +keiβkf(1 − e− sgn(k)iβf − sgn(k)iβf) +��� +≤ |g| +N2 +� +k=N1+1 +1 +k +β2f 2 +2 +≤ Cf,g| log(β)|β2 = o(β). + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES 17 +It follows that +GN1,N2 +β,f,g += 1 +2π +N2 +� +k=N1+1 +1 +k +� 1 +0 +(ψk,β(Xu) + ψ−k,β(Xu))du + o(β) += −β +π +N2 +� +k=N1+1 +� 1 +0 +f(Xu)g(Xu)sin(kβf(Xu)) +k +du + o(β) += −β +π +N2 +� +k=1 +� 1 +0 +f(Xu)g(Xu)sin(kβf(Xu)) +k +du + o(β). +The last line follows from the fact that +���β +π +N1 +� +k=1 +� 1 +0 +f(Xu)g(Xu)sin(kβf(Xu)) +k +du +��� ≤ ∥f∥2 +∞∥g∥∞β2N1 = o(β). +For s ≤ 0, let +Φ(s) = +� � 1 +0 f(Xu)g(Xu)sin(sf(Xu)) +s +du +for s ̸= 0 +� 1 +0 f(Xu)2g(Xu)du +for s = 0, +so that Φ is continuous on [0, ∞) and +GN1,N2 +β,f,g += −β2 +π +N2 +� +k=1 +Φ(βk) + o(β). +(7) +For all R > 0, +���β +⌊Rβ−1⌋ +� +k=1 +Φ(βk) − +� R +0 +Φ(s)ds +��� ≤ β∥f∥2 +∞∥g∥∞ + ωΦ,[0,R](β), +where ωΦ,[0,R](β) = sups,t∈[0,R] |Φ(s) − Φ(t)| is the continuity modulus of Φ. +Since β + ωΦ,[0,R](β) → 0 for all R > 0, there exists a function Rβ such that Rβ → ∞ as +β → 0 and β + ωΦ,[0,Rβ](β) → 0. We fix such a function, and set N2 = β− +2 +2−δ ∧ (β−1Rβ). This +way, we do have βN2 −→ +β→0 +∞ and βN 1−δ +2 +−→ +β→0 0. +We obtain +���β +N2 +� +k=1 +Φ(βk) − +� β−1N2 +0 +Φ(s)ds +��� = o(1). +(8) +To estimate this last integral, there is two things we must be careful about. First, because of +the sinc function in the definition of Φ, the function Φ is not integrable on [0, +∞) so we cannot +naively replace the bound β−1N2 with its limit. Secondly, when manipulating the integral, we +must be extra careful at the vicinity of f(Xu) = 0. +Recall that for x ̸= 0, limC→∞ +� C +0 +sin(sx) +s +ds = sgn(x)π +2 . Performing an integration by part, we +deduce that for all x and C > 0, +��� +� C +0 +sin(sx) +s +ds − sgn(x)π +2 +��� = +��� lim +C′→∞ +� C′ +C +sin(sx) +s +ds +��� += +���cos(Cx) +Cx +− lim +C′→∞ +� C′ +C +cos(sx) +s2x +ds +��� +≤ +2 +C|x|. + +18 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES +It follows that +��� +� β−1N2 +0 +Φ(s)ds − π +2 +� 1 +0 +|f(Xu)|g(Xu)du +��� += +��� +� 1 +0 +f(Xu)g(Xu) +� � β−1N2 +0 +sin(sf(Xu)) +s +ds − sgn(f(Xu))π +2 +� +du +��� +≤ +� 1 +0 +|f(Xu)|g(Xu) +2 +β−1N2|f(Xu)|du += O(βN −1 +2 ) = o(1). +(9) +Combining (7), (8) and (9), we obtain +GN1,N2 +β,f,g += −β +2 +� 1 +0 +|f(Xu)|g(Xu)du + o(β). +(10) +We finally look at the end part of Gβ,f,g. +Since the Cǫ norm of φk,β becomes arbitrarily +large as k goes to infinity, one cannot directly rely on Lemma 3.3. For a positive integer j, we +decompose Gj2N2,(j+1)2N2 +β,f,g +into +Gj2N2,(j+1)2N2 +β,f,g += +(j+1)2N2 +� +k=j2N2+1 +(φk,β(D(j+1)2N2) − φ−k,β(D−(j+1)2N2)) +� +�� +� +Hj +β,f,g ++ +(j+1)2N2 +� +k=j2N2+1 +(φk,β(Dk) − φk,β(D(j+1)2N2) − φk,β(D−k) + φ−k,β(D−(j+1)2N2) +� +�� +� +Kj +β,f,g +. +We have +��� +(j+1)2N2 +� +k=j2N2+1 +φk,β(D(j+1)2N2) +��� = +��� +� +D(j+1)2N2 +(j+1)2N2 +� +k=j2N2+1 +e−iβkf(z)(1 − e−iβf(z))g(z)dz +��� += +��� +� +D(j+1)2N2 +e−iβ(j2N2+1)f(z)(1 − e−iβ((j+1)2N2−j2N2)f(z))g(z)dz +��� +≤ +� +D(j+1)2N2 +2|g(z)|dz +≤ 2∥g∥∞D(j+1)2N2. +Using again Lemma 3.3 with f = 1, we deduce that almost surely, there exists C such that +for all N, DN ≤ C +N . It follows that +|Hj +β,f,g| ≤ +4C∥g∥∞ +(j + 1)2N2 +, +which yields +��� +∞ +� +j=1 +Hj +β,f,g +��� ≤ 4C∥g∥∞ +N2 +∞ +� +j=2 +1 +j2 = o(β). + +BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO. MAGNETIC IMPURITIES 19 +As for Kj +β,f,g, using the fact that the sequences (Dk)k≥1 and (D−k)k≥1 are nested, we have +Kj +β,f,g = +(j+1)2N2 +� +k=j2N2+1 +|φx,β|(Dk − D(j+1)2N2 + D−k − D−(j+1)2N2) +≤ +(j+1)2N2 +� +k=j2N2+1 +β∥f∥∞∥g∥∞(Dk − D(j+1)2N2 + D−k − D−(j+1)2N2). +Let C, δ > 0 such that for all N ̸= 0, +��DN − +1 +2π|N| +�� ≤ CN −1−δ. +Then, for all k ∈ {j2N2 + 1, . . . , (j + 1)2N2}, +0 ≤ Dk − D(j+1)2N2 ≤ +1 +2πk − +1 +2π(j + 1)2N2 ++ 2Ck−1−δ ≤ C′� +1 +j3N 2 +2 ++ (j2N2)−1−δ� +. +We deduce +|Kj +β,f,g| ≤ C′′∥f∥∞∥g∥∞N −1 +2 j−2, +and it follows that +∞ +� +j=1 +|Kj +β,f,g| = o(β). +Finally, we have +|GN2,∞ +β,f,g | ≤ +∞ +� +j=1 +|Gj2N2,(j+1)2N2 +β,f,g +| ≤ +∞ +� +j=1 +|Kj +β,f,g| + +∞ +� +j=1 +|Hj +β,f,g| = o(β). +(11) +We conclude the proof by putting together (6), (10) and (11). +□ +6. Funding +I am pleased to acknowledge support from the ERC Advanced Grant 740900 (LogCorRM), +and later from the EPSRC grant EP/W006227/1 . +References +[1] Robert M. Blumenthal and Ronald Getoor. 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Appl., +57(2):225–245, 1995. + diff --git a/ANAyT4oBgHgl3EQfq_kk/content/tmp_files/load_file.txt b/ANAyT4oBgHgl3EQfq_kk/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ca18f3d1f2e89c85e8e130d81dc671107755c6d --- /dev/null +++ b/ANAyT4oBgHgl3EQfq_kk/content/tmp_files/load_file.txt @@ -0,0 +1,587 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf,len=586 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='00551v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='PR] 2 Jan 2023 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMOGENEOUS MAGNETIC IMPURITIES ISAO SAUZEDDE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' We give a general Green formula for the planar Brownian motion, which we apply to study the Aharonov–Bohm effect induced by Poisson distributed magnetic impurities on a Brownian electron in the presence of an inhomogeneous magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Stochastic Green’s formula 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Magnetic impurities 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Notations 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Differential forms and integrals 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Winding 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Cauchy variables 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Former results 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Stokes formula 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Existence of a limit 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Strategy for the Stokes’ formula 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Additivity 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Contribution from the small loops 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Stratonovich integral as a limit of integrals along piecewise-linear paths 12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Magnetic impurities 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Funding 19 References 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Stochastic Green’s formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For a smooth loop X = (X1, X2) : [0, T] → R2 and a point z outside the range of X, let nX(z) ∈ Z be the winding index of X around z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For any smooth differential 1-form η = η1dx1 + η2dx2, the Green formula states that � X η = � R2 nXdη, where dη is the exterior derivative of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' In other words, for two smooth functions η1, η2 : R2 → R, � T 0 η1(Xt)dX1 t + � T 0 η2(Xt)dX2 t = � R2 nX(z)(∂1η2(z) − ∂2η1(z))dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' When the smooth loop is replaced with a Brownian one, such a formula cannot be written down directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For its left-hand side, we do have a genuine candidate provided by the Stratonovich integrale of η along X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' However, the index function nX fails from being integrable on the vicinity of X [12], and we need to use some kind of regularization in order to define the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' In such a framework, the Green formula is thus a convergence result rather than an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' University of Warwick E-mail address: isao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='sauzedde@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Primary 60J65;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' 60K37 Secondary 60G17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' 1 2 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' MAGNETIC IMPURITIES In [13], Wendelin Werner proposed two alternative regularizations, for which he was able to prove that the Green formula holds with a convergence in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' In [11], I proposed two more regularizations, for which I proved that the Green formula holds with an almost sure limit, but only in the case ∂1η2 − ∂2η1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' The first goal of this paper is to extend such a formula to any differential 1-form η with regularity C1+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For an integer x and a positive integer k, let [x]k be equal to either x1|x|≤k or max(min(x, k), −k) (the following theorem holds for both choice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Let X : [0, T] → R2 be a Brownian motion, and let nX be the winding function associated with the loop obtained by concatenation of X with the straight line segment [XT , X0] between its endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Then, almost surely, for all ǫ > 0 and all f ∈ Cǫ b(R2), � R2[nX(z)]kf(z)dz converges as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Furthermore, if η = η1dx1 +η2dx2 with η1, η2 ∈ C1+ǫ(R2) is such that f = ∂1η2 −∂2η1, almost surely, lim k→∞ � R2[nX(z)]kf(z)dz = � T 0 η ◦ dX + � [XT ,X0] η, where the stochastic integral in the right hand side is to be understood in the sense of Stratonovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For all x and y in R2, the same result holds if the planar Brownian motion is replaced with a planar Brownian loop or a planar Brownian bridge between distinct points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' We will denote this limit as −� R2 nX(z)f(z)dz, since we want to think of it as to the integral of nX with respect to the measure f(z)dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Magnetic impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' In the theory of weak localization in 2 dimensional crystals, for which we refer to [2], one studies quasiclassical electrons moving inside a metal with magnetic impurities, in the presence of a magnetic fields which induces an Aharonov–Bohm effect on the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' In some regime of the parameters, the electron is usually modeled by a planar Brownian trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' In particular, for the computation of the weak-localization correction to the Drude conductivity, the electron is modeled by a Brownian loop (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' The impurities are modeled by a Poisson process of points P with intensity ρdz in the plane, and the Aharonov–Bohm effect is described by a phase shift exp(iα � z∈P nX(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' In [4], the authors study the limit ρ → +∞ with κ = αρ constant, and derive a formula for the phase shift averaged over both P and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For an integrable function f ∈ L1(R2), 1 ρ � z∈P f(z) is a Monte–Carlo estimation for � R2 f(z)dz, and therefore eiκ � R2 f(z)dz = lim ρ→∞ EP� ei κ ρ � z∈P f(z)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' However, as it is noticed in [5], for a Brownian loop X, EX� eiκ−� R2 nX(z)dz� ̸= lim ρ→∞ EX,P� ei κ ρ � z∈P nX(z)� , which is due to the lack of integrability of the function nX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' As we proved in [11], the Monte–Carlo method fails in this situation: it is true that X-almost surely, 1 ρ � z∈P nX(z) converges in distribution (with respect to P) as ρ → ∞, but the limit is not deterministic –or should we say, not measurable with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' It is instead equal to the sum of −� R2 nX(z)dz with a centered Cauchy distribution independent from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' From this result, one can rigorously prove the formula obtained first in [5] for lim ρ→∞ EX,P[ei κ ρ � z∈P nX(z)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' MAGNETIC IMPURITIES 3 However, for the scales at play, the magnetic field which induces the Aharonov-Bohm effect cannot be considered as homogeneous in general [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Our second goal in this paper is to derive an asymptotic formula for the functional of X given by lim ρ→∞ EP[ei 1 ρ � z∈P f(z)nX(z)], for a non homogeneous magnetic field f and a non homogeneous density of impurities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Let f, g ∈ Cǫ b(R2), with g ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For ρ > 0, let P be Poisson process on R2 with intensity ρg(z)dz, and let X be either a Brownian motion or a Brownian bridge with duration 1, independent from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Then, X-almost surely, lim ρ→∞ EP[ei 1 ρ � z∈P f(z)nX(z)] = exp � iα− � nX(z)f(z)g(z)dz − |α| 2 � 1 0 |f(Xt)|g(Xt)dt � where EP is the expectation over P (conditional on X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Although this formula is suited to the problem of magnetic impurities, the following alternative formulation might be more appealing to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Let g ∈ Cǫ b(R2), with g ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For ρ > 0, let P be Poisson process on R2 with intensity ρg(z)dz, and X be either a Brownian motion or a Brownian bridge with duration 1, independent from P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Let also Γ : [0, 1] → R be a standard Cauchy process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Then, for all (f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' , fn) ∈ Cǫ(R2), X-almost surely, the n-uple �1 ρ � z∈P f1(z)nX(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' , 1 ρ � z∈P fn(z)nX(z) � converges in distribution toward (ξ(f1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' , ξ(fn)) where ξ(f) = − � nX(z)f(z)g(z)dz + 1 2 � 1 0 f(Xt)g(Xt)dΓt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Given f, g ∈ Cǫ b(R2), there always exists a differential 1-form η with regularity C1+ǫ such that ∂1η2 − ∂2η1 = fg, so that −� nX(z)f(z)g(z)dz can always be written as a stochastic integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Since all the results hold X-almost surely, the assumptions that the functions are bounded can easily be lifted, but some of the intermediate results come with a quantitative version which depends upon the L∞ norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' This paper is built in the continuity of two former papers from the same author, [11] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' It is not necessary to read them to understand the present paper, but we will use some results from those papers, as well as from [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Notations 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Differential forms and integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For α ∈ (0, 1), we define Cα(R2) as the set of functions f : R2 → R such that the semi-norm |f|Cα := sup x,y∈R2 x̸=y f(x) − f(y) |x − y| is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' We also define Cα b (R2) = Cα(R2) ∩ L2(R2), which we endow with the norm ∥f∥Cα b = ∥f∥∞ + |f|Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For a differential 1-form η = η1dx1 + η2dx2 and α ∈ [0, 1), we write η ∈ C1+α(T ∗R2) if ∂iηj ∈ Cα(R2) for all i, j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Given a curve X : [0, T] → R2, we write � X η := � T 0 η1(Xt)dX1 t + � T 0 η2(Xt)dX2 t , 4 BROWNIAN WINDINGS, STOCHASTIC GREEN’S FORMULA AND INHOMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' MAGNETIC IMPURITIES where these integrals are to be understood either as classical integrals or as Stratonovich inte- grals, depending on the regularity of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' No Itô integral will be involved in this paper, and all the stochastic integrals are to be understood in the sense of Stratonovich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For η ∈ C1+α(T ∗R2), we identify the 2-form dα = (∂1η2 − ∂2η1)dx1 ∧ dx2 with the signed measure (∂1η2 − ∂2η1)dx, where dx is the Lebesgue measure on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For a bounded set D ⊂ R2 and f ∈ L1 loc(R2), we use the unconventional notation f(D) = � D f(z)dz, and |D| for the Lebesgue measure of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Winding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Given a curve X on R2, that is a continuous function from [0, T] to R2 for some T > 0, we write ¯X for the concatenation of X with a straight line segment from XT to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Although the parameterisation of this line segment does not matter in the following, we will assume it is parameterized by [T, T + 1] at constant speed, unless X is a loop (that is, a curve with XT = X0), in which case we set ¯X = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' Given a curve X and a point z outside the range of ¯X, we write nX(z) for the winding number of ¯X around z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For a relative integer k, we define AX k = {z ∈ R2 \\ Range( ¯X) : nX(z) = k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ANAyT4oBgHgl3EQfq_kk/content/2301.00551v1.pdf'} +page_content=' For n > 0, we also define DX n = {z ∈ R2 \\ Range( ¯X) : nX(z) ≥ n} = � n≤k<+∞ AX k , and DX −n = {z ∈ R2 \\ Range( ¯X) : nX(z) ≤ −n} = � −∞0.05 +OS risk +o p> 0.05 +OS risk +o p>0.05 +100 +1e+00 +1e+06 +Hazard ratios +1e+00 +1e+03 +Hazard ratios +Hazard ratios +B +Points +0 +20 +40 +60 +80 +100 +Grade risk +0.8 0.4 0 +Grade +1.5 +2 +2.5 +3 +3.5 +4 +OS risk +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Stage +V +A +1 +1.5 +2 +2.5 +3 +3.5 +4 +Total points +40 +60 +80 +100 +120 +140 +160 +180 +200 +220 +240 +260 +Pr( time < 3 years ) +0.1 +0.14 +0.2 +0.3 +0.4 +0.6 +0.8 +0.9 +0.97 +0.99 +Pr( time < 5 years) +0.04 +0.06 +0.1 +0.14 +0.2 +0.3 +0.5 +0.7 +0.84 +0.9218 + + +Figure 6 Evaluations of the CRN model. (A) Kaplan-Meier survival analysis of overall survival in +the Cancer Genome Atlas cohort. (B) Kaplan-Meier survival analysis of overall survival in the +General cohort. (C) Kaplan-Meier survival analysis of overall survival in the Clinical Proteomic +Tumor Analysis Consortium cohort. (D, E, F) ROC curves of 1-, 3-, and 5-year overall survival +prediction for the CRN model and comprehensive clinicopathology features in the Cancer Genome +Atlas cohort. (G, H, I) ROC curves of 1-, 3-, and 5-year overall survival prediction for the CRN +model and comprehensive clinicopathology feature in the General cohort. (J, K, L) ROC curves of +1-, 3-, and 5-year overall survival prediction for the CRN model and comprehensive +clinicopathology features in the Clinical Proteomic Tumor Analysis Consortium cohort. CRN, +competing-risk nomogram, ROC, receiver operator characteristics; AUC, area under curve. + + + + + +A +B +c +100 +100- +100 +Survival (%) +80 +Survival (%) +80 +Survival (%) +80. + 60 +60 +60. +40 - +40 +40. +Overall +Overall +overall +20 +20, +HR = 5.664 (3.893-8.239) +HR = 35.74 (5.889-216.9) +HR = 6.107 (1.815-20.54) +Log-rank P < 0.0001 +ro +Log-rank P < 0.0001 +0 +Log-rank P < 0.0001 +Lo +3 +6 +9 +12 +2 +6 +80 +2 +Time (months) +Time (months) +Time (months) +Favorable 374 +222 +LL +2g +1 +Favorable 271 +253 +160 +62 +0 +Favorable 164 +115 +88 +54 +10 +Worse +129 +52 +13 +6 +0 +Worse +35 +24 +10 +3 +0 +Worse +31 +20 +12 +5 +D +E +F +0 +8 +(%) , +Sensitivity ( +8 +Sensitivity ( +09 +1-year +3-year +5-year +4 +4 +AUC +AUC +AUC +2 +CRN: 86.9% +2 +CRN: 84.1% +2 +CRN: 82.5% +Clinicopathology: 83.2% +Clinicopathology: 81.5% +Clinicopathology: 78.7% +100 +80 +60 +40 +20 +0 +100 +80 +60 +40 +20 +0 +100 +80 +60 +40 +20 +0 +Specificity (%) +Specificity (%) +Specificity (%) +G +H +100 +100 +80 +8 +Sensitivity (%) +Sensitivity (%) +09 +09 +Sensitivity ( +8 +1-year +3-year +5-year +40 +4 +4 +AUC +AUC +AUC +CRN: 96.9% +2 +CRN: 92.4% +CRN:81.4% +Clinicopathology: 95.1% +Clinicopathology: 89.4% +Clinicopathology: 83.5% +100 +80 +60 +40 +20 +0 +100 +80 +60 +40 +20 +0 +100 +80 +60 +40 +20 +0 +Specificity (%) +Specificity (%) +Specificity (%) +J +K +L +10 +8 +8 +8 +Sensitivity (%) +Sensitivity (%) +(%) +8 +8 +Sensitivity ( +8 +1-year +3-year +5-year +40 +4 +4 +AUC +AUC +AUC +2 +CRN: 75.4% +CRN: 80.9% +2 +CRN:80.3% +Clinicopathology: 73.0% +Clinicopathology: 80.0% +Clinicopathology: 79.5% +T +100 +80 +60 +40 +20 +0 +100 +80 +60 +40 +20 +0 +100 +80 +60 +40 +20 +0 +Specificity (%) +Specificity (%) +Specificity (%)19 + + + + + + + +Figure S1 Accurate diagnosis of ccRCC, pRCC, and ChRCC from normal renal tissues. Left, ROC +curves for distinguishing RCC from normal renal tissues; Middle, original slide images; Right, +visualization of detected tumor regions for each type of RCC; RCC, renal cell carcinoma; ccRCC, +clear cell renal cell carcinoma; pRCC, papillary renal cell carcinoma; ChRCC, chromophobe renal +cell carcinoma; ROC, receiver operator characteristics; AUC, area under the curve (with 95% +confidence interval). + + + + +ROC curve +Original slide +Tumorous heatmap +0 +8 +CCRCC +Sensitivity (%) +4 +2 +AUC:0.987(0.979-0.993) +Sensitivity:0.907(0.970-0.988) +Specificity:0.909(0.854-0.948) +5mm +100 +80 +60 +40 +0 +Specificity(%) +8 +pRCC +Sensitivity (%) +g +4 +2 +AUC:0.939(0.913-0.960） +Sensitivity:0.962(0.934-0.981) +Specificity:0.872(0.811-0.919) +5mm +100 +80 +60 +40 +20 +Specificity (%) +8 +ChRCC +Sensitivity (%) +4 +2 +AUC:0.984(0.961-0.995) +Sensitivity:0.982(0.935-0.998) +Specificity:0.902(0.846-0.943) +5mm +100 +80 +60 +40 +20 +0 +Specificity (%)20 + + + + + + + + + +Figure S2 Differential diagnosis of renal cell carcinoma from renal oncocytoma. AUC, area under +the curve (with 95% confidence interval). + + + + + + + + + + + + + +AUC: 0.951(0.922-0.972) +Sensitivity: 0.821(0.772-0.862) +Specificity: 0.962(0.804-0.999)21 + + +Figure S3 Prediction of high tumor grade for patients with clear cell renal cell carcinoma. (A, C, E) +ROC curves for the prediction of high tumor grade for ccRCC in the TCGA cohort, the General +cohort, and the CPTAC cohort, respectively. (B, D, F) Comparations of the graderisk among patients +with different tumor grades in the TCGA cohort, the General cohort, and the CPTAC cohort, +respectively. ROC, receiver operator characteristics; AUC, area under the curve; TCGA, the Cancer +Genome Atlas; CPTAC, Clinical Proteomic Tumor Analysis Consortium; CI, confidence interval. + + + + + + + +A +B +< 0.001 +8 +1.0 +TCGA cohort +risk +Grade +0.5 +4 +0.0 - +AUC +95%CI +TCGA: 0.840 (0.805-0.871) +G1 +G2 +80 +60 +40 +20 +0 +G3 +G4 +100 +Specificity (%) +c +D +< 0.001 +p +1.00 +General cohort +8 +0.75 +(%) Asue +risk +0.25 +AUC +95%CI +General: 0.857 (0.813-0.894) +0.00 +100 +80 +60 +40 +20 +0 +G1 +G2 +G3 +G4 +Specificity (%) +E +F +d +<0.001 +CPTAC cohort +1.2 +8 +0.8 +Grade risk +4 +0.4 +2 +AUC +95%CI +0.0 - +CPTAC: 0.894 (0.842-0.933) +80 +60 +40 +/ +100 +20 +0 +G1 +G2 +G3 +G4 +Specificity (%)22 + + +Figure S4 Prediction of the 5-year OS status for patients with clear cell renal cell carcinoma. (A, D, +G) ROC curves for the prediction of the 5-year OS status for ccRCC in the TCGA cohort, the +General cohort, and the CPTAC cohort, respectively. (B, E, H) Comparations of the OSrisk among +patients with different tumor grades in the TCGA cohort, the General cohort, and the CPTAC cohort, +respectively. (C, F, I) Comparations of the OSrisk among patients with different tumor stages in the +TCGA cohort, the General cohort, and the CPTAC cohort, respectively. OS, overall survival; ROC, +receiver operator characteristics; AUC, area under the curve; TCGA, the Cancer Genome Atlas; +CPTAC, Clinical Proteomic Tumor Analysis Consortium; CI, confidence interval. + + + + + + + + + +A +B +C +1.00 +<0.001 +1.00 +<0.001 +TCGA cohort +0.75 +0.75 +os +0.25 +0.25 +2 +AUC +95%CI +TCGA: 0.784 (0.746-0.819) +0.00 +0.00 +T +100 +08 +60 +40 +20 +0 +G1 +G2 +G3 +G4 +Stagei Stage ii Stage ili Stage iv +Specificity (%) +D +E +F +8 +<0.001 +p <0.001 +I cohort +1.0 +0.8 +Sensitivity ( +risk +General +so +0.0 +AUC +95%C +0.0 +General:0.774(0.723-0.820 +T +100 +80 +60 +40 +20 +0 +G1 +G2 +G3 +G4 +Stagei +Stage ii +Stage ili +Specificity (%) +G +H +<0.001 +1.001 +d +<0.001 +0.75 +CPTAC cohort +8 +0.75 +0.50 +8 +risk +risk +0.50 +SO +0.25 +AUC +95%CI +0.00 +0.00- +CPTAC: 0.702 (0.632-0.765) +100 +80 +60 +40 +20 +G1 +0 +G2 +G3 +G4 +Stagei Stage ii Stage ili Stage iv +Specificity (%)23 + + + + +Supplemental Table + +Table S1 Basic clinical characteristics of patients recruited for this study. + +General Cohort (401) +TCGA Cohort (820) +CPTAC Cohort (195) +Age(years) + + + + ≥65 +139(34.7%) +263(32.1%) +78(40.0%) + ＜65 +262(65.3%) +557(67.9%) +117(60.0%) +Sex + + + + Male +288(71.8%) +548(66.8%) +138(70.8%) + Female +113(28.2%) +272(33.2%) +57(29.2%) +Stage + + + + i +362(90.3%) +434(52.9%) +100(51.3%) + ii +25(6.2%) +105(12.8%) +20(10.3%) + iii +14(3.5%) +182(22.2%) +54(27.7%) + iv +0 +99(12.1%) +21(10.7%) +WSI +401 +847 +195 +Subtype + + + + ChRCC +44(11.0%) +65(7.9%) +/ + pRCC +51(12.7%) +244(29.8%) +/ + ccRCC +306(76.3%) +511(62.3%) +195(100%) + Nuclear grade + + + + High (iii/iv) +56(18.3%) +275(53.8%) +81(41.5%) + Low (i/ii) +250(81.7%) +228(44.6%) +114(58.5%) + Unknown +0 +8(1.6%) +0 + Status + + + +Dead +14(5.6%) +170(33.3%) +23(11.8%) +Alive +292(95.4%) +333(65.2%) +172(88.2%) +Unknown +0 +8(1.5%) +0 + + + + + + diff --git a/C9E4T4oBgHgl3EQfFwy_/content/tmp_files/load_file.txt b/C9E4T4oBgHgl3EQfFwy_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac4abac216e9cdd7962779ab2b303dbdf4b7069c --- /dev/null +++ b/C9E4T4oBgHgl3EQfFwy_/content/tmp_files/load_file.txt @@ -0,0 +1,911 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf,len=910 +page_content='1 Artificial intelligence for diagnosing and predicting survival of patients with renal cell carcinoma: Retrospective multi- center study Siteng Chen1*, Xiyue Wang2*, Jun Zhang3*, Liren Jiang4*, Ning Zhang1, Feng Gao4, Wei Yang3, Jinxi Xiang3, Sen Yang3, Junhua Zheng5#, Xiao Han3# 1 Department of Urology, Shanghai General Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200080, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 2 College of Computer Science, Sichuan University, Chengdu 610065, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 3 Tencent AI Lab, Shenzhen 518057, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 4 Department of Pathology, Shanghai General Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200080, China 5 Department of Urology, Renji Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200135, China Equal contributors and co first authors #Corresponding authors: Junhua Zheng, Department of Urology, Renji Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200080, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' E-mail: zhengjh0471@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Tel: 86-021-63240090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Xiao Han, Tencent AI Lab, Shenzhen 518057, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' E-mail: haroldhan@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Tel: 86- 075586013388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 2 Abstract Background: Clear cell renal cell carcinoma (ccRCC) is the most common renal-related tumor with high heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' There is still an urgent need for novel diagnostic and prognostic biomarkers for ccRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Methods: We proposed a weakly-supervised deep learning strategy using conventional histology of 1752 whole slide images from multiple centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Our study was demonstrated through internal cross-validation and external validations for the deep learning-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Results: Automatic diagnosis for ccRCC through intelligent subtyping of renal cell carcinoma was proved in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Our graderisk achieved aera the curve (AUC) of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='840 (95% confidence interval: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='805-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='871) in the TCGA cohort, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='840 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='805-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='871) in the General cohort, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='840 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='805- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='871) in the CPTAC cohort for the recognition of high-grade tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The OSrisk for the prediction of 5-year survival status achieved AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='784 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='746-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='819) in the TCGA cohort, which was further verified in the independent General cohort and the CPTAC cohort, with AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='774 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='723-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='820) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='702 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='632-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='765), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Cox regression analysis indicated that graderisk, OSrisk, tumor grade, and tumor stage were found to be independent prognostic factors, which were further incorporated into the competing-risk nomogram (CRN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Kaplan-Meier survival analyses further illustrated that our CRN could significantly distinguish patients with high survival risk, with hazard ratio of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='664 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='893-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='239, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001) in the TCGA cohort, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='740 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='889-216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='900, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001) in the General cohort and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='107 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='815 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='540, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001) in the CPTAC cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Comparison analyses conformed that our CRN outperformed current prognosis indicators in the prediction of survival status, with higher concordance index for clinical prognosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Conclusion: Deep learning-based pathology signature could be used for the diagnosis and prognosis prediction for ccRCC, which might provide intelligent advice to improve the process of individualized treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Background Renal-related malignant tumor is one of the most common malignant tumors worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In 2015, the incidence rate of renal cancer arrived at 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8 per 100,000 in China [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In the United States, renal cancer is estimated to have 76,080 new cases and 13,780 associated deaths in 2021 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Among all of the solid lesion within the kidney, renal cell carcinoma (RCC) is the most common renal-related tumor, accounting for approximately 90% of all kidney malignancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' According to cellular morphological characteristics, RCC is mainly divided into three subtypes, including clear cell RCC (ccRCC), papillary RCC (pRCC), and chromophobe RCC (ChRCC) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' However, some reports for ccRCC by experienced pathologists might miss essential elements and lack appropriate information associated prognosis [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In addition, traditional diagnosis of ccRCC by pathologist is still time- consuming and labor-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Recently, the pathology ecosystem has been gradually challenged by the emergence of digital pathology, which has also catalyzed the popularization and application of computer-aided diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Deep learning, which can be performed as a representation-learning method, has been successful used in medical image analysis with massive amounts of well-annotated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' For gigapixel whole- slide images (WSIs), they are usually annotated at the slide-level without considering the detailed internal cellular composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Due to the gigapixel size and heterogeneity tissue distribution within the WSI, usually only a tiny region could be matched with the corresponding slide-level label, which 3 makes the WSI-level classification problem a weakly supervised learning scenario [5-7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Some studies have preliminarily demonstrated the utility of weakly-supervised deep learning in kidney segmentation and tumor classification from single center [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' It is also widely recognized that nuclear grading of cancer cell could act as a prognostic factor for patients with ccRCC [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' However, traditional assessment with manual observation of nuclear grading may lead to inconsistent judgement between pathologists [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Moreover, there are still limitations in current TNM staging system, resulting in an urgent need for novel diagnostic and prognostic biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In this study, we developed deep learning strategies to conduct automatic diagnosis, tumor grading, and prognosis prediction for RCC based on multi-source patient cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Our study suggested that deep learning-based pathology signature could be used for the diagnosis and prognosis prediction for RCC, which might provide intelligent advice to improve the process of individualized treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Materials and methods Patient cohorts and data sources In this study, three independent patient cohorts from different sources, including Shanghai General Hospital, Clinical Proteomic Tumor Analysis Consortium (CPTAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='cancerimagingarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='net) [12, 13], and the Cancer Genome Atlas (TCGA, https://portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='gdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='gov) [12] were included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' All included patients should meet the following selection criteria: (i) pathologically diagnosed as RCC without other types of malignant tumors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (ii) with corresponding clinical and pathological information (ground-truth label in slide-level);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (iii) with access to corresponding H&E slides or their scanned WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The General cohort recruited 401 patients from the Shanghai General Hospital, who underwent partial or radical nephrectomy and were pathologically diagnosed as RCC from January 2012 to September 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In addition, 26 patients with renal oncocytoma were also enrolled from Shanghai General Hospital for differential diagnosis analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The hematoxylin-eosin staining (H&E)-stained slides were scanned with Leica Aperio AT2 scanners at 20× equivalent magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Furthermore, 820 patients from the TCGA cohort with diagnostic WSIs, and 195 patients from the CPTAC cohort with WSIs, who met the inclusion criteria mentioned above, were also included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The basic clinical characteristics of the included patients in this study were shown in Supplementary Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Another 400 H&E-stained WSIs of RCC from the Pathology AI Platform (PAIP, wisepaip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='org/paip), which were manually annotated in pixel-level by the pathologist of the Seoul National University Hospital were collected for the training and internal validation of RCC segmentation (PAIP cohort).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Hybrid neural network for RCC segmentation We proposed a hybrid network for RCC segmentation as shown in Figure 1, which combined a U- net and a multi-task learning strategy to capture representative features by sharing the encoder in three task-specific branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' There are two pixel-level RCC region segmentation branches with shared five decoder layers, which are trained using the whole dataset (RCC whole-seg) and the positive data (RCC tumor-seg), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The RCC whole-seg branch aims to learn distinctive features for normal and cancerous regions, which helps to reduce the rate of false positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The RCC tumor-seg branch tagetes for the more robust features to recognize tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The third classification branch (RCC class) adopts the idea of deep supervision, which acts as an auxiliary binary classifier to determine whether an input image is positive or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 4 SE-ResNeXt-50 is employed as our encoder, which is a combination of ResNeXt architecture and squeeze-and-excitation (SE) module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The ResNeXt aggregates parallel residual structures to build a wider and complex network, and the SE applies the channel attention to enhance informative feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' For each decoder layer, the trainable transposed convolution operator (TransConv) is used to up-sample feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' These features are further connected with features in its corresponding encoder layer via skip-concatenations to preserve the consistently spatial information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Then, two convolutional layers with a batch normalization (BN) layer, a rectified linear unit (ReLU), and a selective kernel module (SKM) [14] are utilized to adaptively learn the multi-scale features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The output of the decoder is a segmentation map (256 ×256 ×1), indicating the probability of being tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The loss function is the combination of segmentation loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=', Dice) and classification loss (binary cross-entropy) in the three branches, which was defined in our previous report [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Attention-based weakly-supervised deep learning strategy As illustrated in Figure 2, our classification procedure can be classified into two parts: patch-level feature extraction based on self-supervised learning (SSL) and WSI-level feature aggregation based on a deep attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' For the detailed procedure, we first crop the entire WSI into small image patches (1024*1024) and then feed these patches into the pretrained SSL feature extractor [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' to obtain a descriptive 1024-dimensional feature vector for each patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' These obtained patch- level feature vectors are assembled by deep-attention-based pooling to represent the WSI-level feature information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Referring to the attention weight of each patch, the attention pooling would average the representative features of a WSI for prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Two fully-connected layers following rectified linear unit (ReLU) are used to conduct WSI-level classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In the interpretability analysis process, heatmaps, which generated by the attention weights, are used to visualize the possible disease regions that are highlighted in warm colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Binary variable definition For patients with ccRCC, binary classification (high or low) was used for the prediction of nuclear grade, in which high grade was defined as the collection of grade III and grade IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The overall survival (OS) status at 5-year follow up was used for the training of the prognosis-related models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Statistical analysis Continuous variants among different groups were analyzed and compared by analysis of variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The Dice score was set as the evaluation metrics evaluate the performance of our hybrid network in tumor segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Survival analysis was performed via Kaplan–Meier (KM) curve with hazard ratio (HR) and 95% confidence interval (CI) to compare different OS outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' We also carried out receiver operating characteristic curve (ROC) analysis with area under curve (AUC) to evaluate the accuracy of the prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Results Pixel-wise segmentation of RCC in the PAIP cohort A total of 400 H&E-stained WSIs of RCC with pixel-level manual annotations from the PAIP cohort was randomly divided at the patient level for the training (80%) and internal validation (20%) of the tumor segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Evaluated by five-fold cross-validation in the PAIP cohort, our hybrid network achieved a mean Dice score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='796 in the cross-validation cohort, exhibiting satisfactory 5 performance of our novel hybrid architecture for pixel-wise RCC segmentation from of H&E- stained WSIs, which was independent of a classification model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As shown in Figure 3A, our segmentation model could accuracy distinguished tumor region, which included attentional regions with high diagnostic importance while ignoring regions of low diagnostic relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Our hybrid network was generally capable of delineating the boundary between tumor and normal renal tissue with smooth mask (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Insight into the magnifying representation of histopathology images indicated that the marked regions principally included tissues with dyskaryosis and structure invasion, which were also the typically morphology recognized by pathologists in clinical practices, while the normal renal tissue and other tissue, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' fiber texture and stroma tissue, were not included in the attentional region (Figure 3B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Intelligent diagnosis of RCC in the external validation cohort We further verified our model in an external validation cohort, which combined 928 WSIs (RCC slide: 916, normal renal slide: 12) from the TCGA cohort and 757 WSIs (RCC slide: 504, normal renal slide: 253) from the CPTAC cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Since the validation dataset comprised both tumor and normal images without pixel-level annotations, which was more in accordance with the clinical practices, we assigned a probability value of RCC to a test image if the area of segmentation occupied more than 5% of the WSI after removing the white space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Based on the strategy, the AUC for distinguishing RCC from normal renal tissue achieved 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='977 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='969-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='984, Figure 3C) in the in an external validation cohort, which borne comparison with an experienced pathologist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Further subgroup analysis based on the subtypes of RCC revealed that our diagnosis model could diagnosis clear cell RCC, papillary RCC, and chromophobe RCC from normal renal tissues, with AUCs of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='987 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='979-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='993), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='939 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='913-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='960), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='984 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='961-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='995), respectively (Figure S1), which indicating the robust generalization performance of our model when applied to different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Interpretability and whole-slide attention visualization Readable interpretability of deep learning-based clinical models plays important role in further clinical applications [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' To gain insight into the potential interpretability of our model, we visualized the learned feature space in two dimensions to generate pixel-level heatmaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As shown in the Figure S1 (right column), the most attended regions recognized by our model were considered to be highly associated with RCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Areas with red color of the heatmap represented the regions with predicted RCC tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The pixel-level visualization by our model presented the spatial distributions of diverse tissues, which also helped to provide human-in-the-loop interaction to optimize the current diagnostic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Differential diagnosis of RCC from renal oncocytoma Renal oncocytoma was one of the most common benign tumors in renal, which had several features that overlapped with RCC with a preponderance of granular cytoplasm [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Misconceptions could be reviewed out in clinical practice due to the Review out spectrum of eosinophilic renal neoplasms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Therefore, we further explored whether our hybrid network could be used for the differential diagnosis of RCC from renal oncocytoma in clinical practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As shown in the Figure S2, our diagnosis model exhibited excellent performance in the differential diagnosis of RCC, which achieved an AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='951 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='922-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='972), a sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='821 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='772-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='862), and a specificity of 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='962 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Intelligent subtyping of RCC through deep learning Clinical outcomes differ remarkably among patients with different subtypes of RCC, and ccRCC causes worse prognosis than pRCC and ChRCC [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Since the identification of different subtypes plays a vital role in clinical practices, we proposed a novel neural network for the intelligent subtyping of RCC based on a weakly-supervised deep learning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As shown in Figure 4A, our subtyping model performed well in the subtype prediction of RCC, with an average AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='990 (95% CI: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='981-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='996) in distinguishing ccRCC from pRCC and ChRCC in the TCGA cross-validation cohort, which could be used for the automatic diagnosis of ccRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The classification accuracy was further verified in the General cohort, with AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='970 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='957-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='980, Figure 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Visualization of the subtyping model revealed that our diagnosis model could recognize the tumor regions with transparent and gelatinous material, which contributed to the accurate diagnosis ccRCC (Figure 4C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Recognition of high-grade tumor through deep learning The prognostic value of the nuclear grading has been widely recognized for patients with ccRCC [3, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Therefore, we further applied the weakly-supervised learning strategy to predict high-grade tumors for the grade-classification of ccRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The model was trained and cross-verified from the TCGA cohort and was based on the hypothesis that some microscopic features associated with high- grade tumors could be identified and integrated to calculate the graderisk for the automatic recognition of high-grade ccRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As shown in Figure S3A, our graderisk achieved an average AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='840 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='805-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='871) in the TCGA cross-validation cohort for distinguishing high-grade tumors, which was further verified in the independent General cohort and the CPTAC cohort, with AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='840 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='805-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='871, Figure S3C) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='840 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='805-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='871, Figure S3E), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Comparation analyses indicated that the graderisk distributed differently among patients with different tumor grades (Figure S3B, D, F), which further confirmed the potential for clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Intelligent risk quantitation for five-year survival follow-up Clear cell RCC accounts for most of the adverse prognosis related to renal malignancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Therefore, it is of great importance to accurately predict the 5-year OS status and quantify the survival risk for patients in clinical follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Based on the weakly-supervised learning strategy, we assembled patch-level feature vectors with attention weight to conduct WSI-level classification of 5-year OS status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The survival risk for 5-year follow-up (OSrisk) was then calculated based on the prediction possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As illustrated in Figure S4A, our OSrisk achieved an average AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='784 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='746-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='819) in the TCGA cross-validation cohort for identifying patient with adverse clinical outcome in 5-year follow-up, which was further verified in the independent General cohort and the CPTAC cohort, with AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='774 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='723-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='820, Figure S4D) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='702 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='632-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='765, Figure S4G), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Further comparation analyses revealed that our OSrisk distributed differently among patients with different tumor grades (Figure S4B, D, G) and different tumor grades (Figure S4C, E, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Patients with higher tumor grades or stages seemed to have higher OSrisk, which was consistent with the clinical observations that patients with higher tumor grades/stages might suffer from more survival risk and less likely to get a five-year survival follow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 7 Development of the competing-risk nomogram Integration of multiple biomarkers might improve predictive value over single-scale counterpart [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' We had proved that deep learning-based pathology signatures, including the graderisk and the OSrisk, were significantly associated with high-grade tumor and 5-year survival status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Therefore, we next to explore whether our deep learning-based pathology signatures could cooperate with traditional clinicopathological characteristics to improve the prognosis prediction for clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' We firstly carried out cox regression analysis to identify prognostic indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As shown in Figure 5A, the graderisk, the OSrisk, tumor grade, and tumor stage were found to be independent prognostic factors for patient with ccRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' These four factors were further incorporated into the construction of the competing-risk nomogram (CRN, Figure 5B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ROC analysis revealed that when the cut-off value was set as 103, our CRN achieved the best performance in predicting the OS status in 5-year follow-up, with the highest AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='825 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='789-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='858), specificity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='902, and sensitivity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' With the same cut-off value, patients in the TCGA cohort were classified into the worse group or the favorable group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Kaplan-Meier survival analyses further illustrated that our CRN could significantly distinguish patients with high survival risk (Figure 6A), with HR of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='664 (95% CI 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='893-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='239, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Verification of our CRN in the independent General cohort (Figure 6B) and the CPTAC cohort (Figure 6C) further confirmed the robust prognostic power, with HR of 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='740 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='889-216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='900, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='107 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='815 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='540, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Comparison with current prognosis indicators To further identify the superiority of our CRN in prognosis prediction of ccRCC, we compared the CRN with current prognosis indicators through multiple indexes, including AUCs for 5-year, 3-year, 1-year OS status and the concordance index (C-index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' As shown in Table 1, our CRN outperformed current prognosis indicators in the prediction of 5-year, 3-year, 1-year OS status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The CRN achieved the highest C-index value from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='770 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='846, which overmatched current prognosis indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In addition, CRN also achieved higher AUCs in the prediction of 5-year, 3-year, 1-year OS status when compared to the comprehensive clinicopathology feature (Figure 6D-L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Discussion Traditional visual inspection of pathological images can be distinguished by the nuclear shape, size, nucleolus, and chromatin features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' For renal carcinoma with high tumor heterogeneity, traditional microscope vision may miss a lot of important information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Furthermore, the shortage of pathologists has aggravated the presence of overwork in pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In the United States, the absolute pathologist workforce had decreased from 2007 to 2017, which resulted in the increase of the diagnostic workload by about 42% [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' There is still an urgent need to develop novel technologies to prevent potential diagnostic error from traditional pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The application of deep neural networks in digital pathology has greatly catalyzed the intelligent analysis of pathological image, otherwise it cannot be analyzed by human-based image interrogation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' DL with CNN demonstrates consummate performance in multiple prediction task from pathological WSI, including tissue segmentation [24], cancer diagnosis [25], cancer prognosis [26], and mutation prediction [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Excellent performance of DL has also been reported in displaying distinct immunogenomic landscape and potential response to immunotherapy [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Based on the full landscapes of WSIs, a deep CNN was reported to identify different subtypes 8 of RCC [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' A histopathology image classifier could also distinguish TFE3 Xp11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2 translocation RCC from ccRCC, which contributed to overcome the difficulties that could not be easily solved in traditional analysis through naked eye [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Benefiting from the increasing number of image datasets, AI-based approaches are now defining integrated and clinically classification of RCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' However, most of the AI-based models were trained from comparatively small samples, without sufficient additional validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Currently, the diagnosis reports of WSIs are usually at the global level (slide-level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' However, the slide-level labels are often associated with tiny/small regions from the gigapixel WSI, which turns the WSI-level classification problem into a weakly-supervised learning scene (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=', inexact supervision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' To tackle this problem, we performed the multiple-instance learning to achieve WSI- level classification in view of the entire information from the slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Since the gigapixel WSIs could not be directly feed into network, we segmented the WSI into non-overlapped patches with 1024*1024 pixels at 20× magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' All patches extracted from the same slide were then identified as the instances of a specific WSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' It is noted that WSI were labeled in slide-level annotations of tumor region, and thus, these extracted patches have no annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Through the application of CNN, we proposed an end-to-end neural network for the diagnosis and prognosis prediction of RCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' With a WSI input, the network could achieve automatic and rapid diagnosis, grading, and survival prediction for the patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' To our knowledge, this is the largest cohort used in our neural network for the classification of ccRCC using H&E-stained WSIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The subtype identification performed well in the internal and external validation cohorts, with the matched sensitivity and specificity of an experienced pathologist, but substantial workload had been saved through our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In addition, we also provided convincing predictions survival status, which might facilitate clinical decision-making but could not be provided through traditional pathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' In this study, we proposed a data-efficient weakly-supervised learning strategy to address the annotation lack problems in the field of histopathological images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Recently, a clustering- constrained-attention multiple-instance learning framework (CLAM) was also proposed to improve the weakly-supervised learning [16], which was further applied to AI-based assessment of tumor origins [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' There are two major differences between this study and ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' First, CLAM adopts pretrained model based on natural images as the feature extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The huge domain shift between natural and histopathological images may decrease the model generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' We encode the semantic content of each patch using our previous pretrained feature extractor on large-scale and diverse histopathological images in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Second, we conduct a multi-task learning for comprehensive RCC stratifications, including cancer/nuclei subtyping and prognosis/mutation prediction, whereas CLAM performs a single task for cancer subtype classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Several strengths could be found in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Firstly, adequate WSIs from three independent patient cohorts were recruited for training and testing the deep neural network, which improved the generalization performance of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Secondly, with only a WSI input, the weakly-supervised network makes it possible for automatic and rapid classification for ccRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Thirdly, based on the importance scores of sub-regions in the WSI, an interpretable probability map can be generated to point out the diagnostically relevant regions for pathologists, making it more practical to clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' There are also some limitations waiting for solution in our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Firstly, part of the images 9 analyzed in this study were acquired for public databases, which might be affected by the potential population bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Secondly, batch effect might be involved in this analysis since different H&E- staining protocols might be performed among different patient cohorts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Thirdly, this is a retrospective study, which might need further validations in prospective clinical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Conclusions In summary, we proposed a weakly-supervised deep learning strategy for the diagnosis and prognosis prediction of RCC with interpretable probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Using conventional histology, our method could achieve automatic diagnosis, tumor grading, and prognosis prediction for patients with ccRCC, thereby providing intelligent advice to improve the process of individualized treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Acknowledgements We appreciate the partial image data from Clinical Proteomic Tumor Analysis Consortium, the Cancer Genome Atlas, and the Cancer Imaging Archive used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Authors’ contributions JHZ and XH conceptualized and supervised the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' STC, XYW, and JZ performed data curation, formal analysis, investigation, visualization, and writing original draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' LRJ, NZ, FG, WY, SY and JXX performed data curation, and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' All authors involved manuscript editing and manuscript review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Funding This study was supported by the National Natural Science Foundation of China (81972393).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The funders had no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Availability of data and materials Primary data are available from Atlas (https://portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='gdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='cancer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='gov/) and the Clinical Proteomic Tumor Analysis Consortium (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='cancerimagingarchive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='net/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Other private data could only be reasonably requested from the corresponding author according to the Research Ethics Committee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Declarations Ethics approval and consent to participate Our study was approved by the Research Ethics Committee of Shanghai General.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Consents were acquired form the participates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Consent for publication Not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Competing interests The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' References 10 [1] Chen W, Zheng R, Baade PD, Zhang S, Zeng H, Bray F, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Cancer statistics in China, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CA Cancer J Clin.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' FC, fully connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' s extraction % Subtype diagnosis Image process and features Grade staging Survival 1 × 2048 1 × 256 × 256 × 3 Interaction analysis Feature profile of digital pathology Clinical profile of patients Factor 1 Factor2 Clinical data Factor 3 abe subtype grade Factor n survival Training and verifying of the models Training cohort Validation cohort Validation cohort External validation External validation Subtype model Grade model Survival model Competing-risk nomogram 0<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='000115 Figure 3 Accurate segmentation of RCC for intelligent diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (A) Example RCC segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Left, original slide image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Right, recognized slide image with green curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (B) Example of segmentation on different kinds of tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (C) ROC curve for distinguishing RCC from normal renal tissues in the independent verification cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' RCC, renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ROC, receiver operator characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC, area under the curve (with 95% confidence interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' A C Sensitivity (%) 4 2 AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='977(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='984) Sensitivity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='973(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='981) Specificity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='902(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='943) 5mm 100 80 60 40 20 0 Specificity(%) B RCC tissue Normalrenaltissue Othertissue16 Figure 4 Intelligent subtyping of renal cell carcinoma from weakly-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (A, B) ROC curves for intelligent subtyping of RCC in the cross-validation cohort (The Cancer Genome Atlas cohort) and the validation cohort (General cohort), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (C) Visualizations of the diagnosis for ccRCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' The detected tumor regions were shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ccRCC, clear cell renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' pRCC, papillary renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ChRCC, chromophobe renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ROC, receiver operator characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC, area under the curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CI, confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' A B 100 100 8 8 Sensitivity (%) Sensitivity (%) 09 4 4 AUC 95%CI 2 AUC 95%CI 2 CcRCC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='990(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='996) CcRCC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='970(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='980） pRCC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='995 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='999) pRCC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='995 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='997) 0 ChRCC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='992(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='998) 0 ChRCC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='935（0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content="999） 100 80 60 40 20 0 100 80 60 40 20 0 Specificity(%) Specificity(%) C Visualization of thediagnosisfor ccRCC Originalslide 20 8'0 5mm CCRCC17 Figure 5 Construction of the competing-risk nomogram." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (A) Cox regression analyses of the deep learning-based pathology signature and clinicopathological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (B) The competing-risk nomogram for the construction of the prognosis prediction model combining the deep learning- based pathology signature and clinicopathological features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' TCGA, the Cancer Genome Atlas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CPTAC, Clinical Proteomic Tumor Analysis Consortium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' OS, overall survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' A TCGA:cohort Age General:cohort Age CPTACicohort Age Sex Sex Sex Grade Grade Grade Stage Stage Stage Grade risk Grade risk Grade risk ●p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='05 ●p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='05 ●p<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='05 OS risk o p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='05 OS risk o p> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='05 OS risk o p>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='05 100 1e+00 1e+06 Hazard ratios 1e+00 1e+03 Hazard ratios Hazard ratios B Points 0 20 40 60 80 100 Grade risk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4 0 Grade 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 4 OS risk 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8 Stage V A 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 4 Total points 40 60 80 100 120 140 160 180 200 220 240 260 Pr( time < 3 years ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='99 Pr( time < 5 years) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9218 Figure 6 Evaluations of the CRN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (A) Kaplan-Meier survival analysis of overall survival in the Cancer Genome Atlas cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (B) Kaplan-Meier survival analysis of overall survival in the General cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (C) Kaplan-Meier survival analysis of overall survival in the Clinical Proteomic Tumor Analysis Consortium cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (D, E, F) ROC curves of 1-, 3-, and 5-year overall survival prediction for the CRN model and comprehensive clinicopathology features in the Cancer Genome Atlas cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (G, H, I) ROC curves of 1-, 3-, and 5-year overall survival prediction for the CRN model and comprehensive clinicopathology feature in the General cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (J, K, L) ROC curves of 1-, 3-, and 5-year overall survival prediction for the CRN model and comprehensive clinicopathology features in the Clinical Proteomic Tumor Analysis Consortium cohort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CRN, competing-risk nomogram, ROC, receiver operator characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC, area under curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' A B c 100 100 100 Survival (%) 80 Survival (%) 80 Survival (%) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 60 60 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' 40 40 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Overall Overall overall 20 20, HR = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='664 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='893 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='239) HR = 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='74 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='889 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9) HR = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='107 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='815 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='54) Log rank P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001 ro Log rank P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001 0 Log rank P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0001 Lo 3 6 9 12 2 6 80 2 Time (months) Time (months) Time (months) Favorable 374 222 LL 2g 1 Favorable 271 253 160 62 0 Favorable 164 115 88 54 10 Worse 129 52 13 6 0 Worse 35 24 10 3 0 Worse 31 20 12 5 D E F 0 8 (%) , Sensitivity ( 8 Sensitivity ( 09 1 year 3 year 5 year 4 4 AUC AUC AUC 2 CRN: 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9% 2 CRN: 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='1% 2 CRN: 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5% Clinicopathology: 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2% Clinicopathology: 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5% Clinicopathology: 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='7% 100 80 60 40 20 0 100 80 60 40 20 0 100 80 60 40 20 0 Specificity (%) Specificity (%) Specificity (%) G H 100 100 80 8 Sensitivity (%) Sensitivity (%) 09 09 Sensitivity ( 8 1 year 3 year 5 year 40 4 4 AUC AUC AUC CRN: 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9% 2 CRN: 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4% CRN:81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4% Clinicopathology: 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='1% Clinicopathology: 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4% Clinicopathology: 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5% 100 80 60 40 20 0 100 80 60 40 20 0 100 80 60 40 20 0 Specificity (%) Specificity (%) Specificity (%) J K L 10 8 8 8 Sensitivity (%) Sensitivity (%) (%) 8 8 Sensitivity ( 8 1 year 3 year 5 year 40 4 4 AUC AUC AUC 2 CRN: 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4% CRN: 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9% 2 CRN:80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='3% Clinicopathology: 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0% Clinicopathology: 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0% Clinicopathology: 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5% T 100 80 60 40 20 0 100 80 60 40 20 0 100 80 60 40 20 0 Specificity (%) Specificity (%) Specificity (%)19 Figure S1 Accurate diagnosis of ccRCC, pRCC, and ChRCC from normal renal tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Left, ROC curves for distinguishing RCC from normal renal tissues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Middle, original slide images;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' Right, visualization of detected tumor regions for each type of RCC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' RCC, renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ccRCC, clear cell renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' pRCC, papillary renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ChRCC, chromophobe renal cell carcinoma;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ROC, receiver operator characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC, area under the curve (with 95% confidence interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ROC curve Original slide Tumorous heatmap 0 8 CCRCC Sensitivity (%) 4 2 AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='987(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='993) Sensitivity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='907(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='988) Specificity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='909(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='948) 5mm 100 80 60 40 0 Specificity(%) 8 pRCC Sensitivity (%) g 4 2 AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='939(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='960） Sensitivity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='962(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='981) Specificity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='872(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='919) 5mm 100 80 60 40 20 Specificity (%) 8 ChRCC Sensitivity (%) 4 2 AUC:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='984(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='995) Sensitivity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='982(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='998) Specificity:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='902(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='943) 5mm 100 80 60 40 20 0 Specificity (%)20 Figure S2 Differential diagnosis of renal cell carcinoma from renal oncocytoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC, area under the curve (with 95% confidence interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='951(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='972) Sensitivity: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='821(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='862) Specificity: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='962(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='999)21 Figure S3 Prediction of high tumor grade for patients with clear cell renal cell carcinoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (A, C, E) ROC curves for the prediction of high tumor grade for ccRCC in the TCGA cohort, the General cohort, and the CPTAC cohort, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (B, D, F) Comparations of the graderisk among patients with different tumor grades in the TCGA cohort, the General cohort, and the CPTAC cohort, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ROC, receiver operator characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC, area under the curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' TCGA, the Cancer Genome Atlas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CPTAC, Clinical Proteomic Tumor Analysis Consortium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CI, confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' A B < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0 TCGA cohort risk Grade 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0 AUC 95%CI TCGA: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='840 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='871) G1 G2 80 60 40 20 0 G3 G4 100 Specificity (%) c D < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 p 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00 General cohort 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='75 (%) Asue risk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='25 AUC 95%CI General: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='857 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='894) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00 100 80 60 40 20 0 G1 G2 G3 G4 Specificity (%) E F d <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 CPTAC cohort 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8 Grade risk 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='4 2 AUC 95%CI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0 CPTAC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='894 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='933) 80 60 40 / 100 20 0 G1 G2 G3 G4 Specificity (%)22 Figure S4 Prediction of the 5-year OS status for patients with clear cell renal cell carcinoma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (A, D, G) ROC curves for the prediction of the 5-year OS status for ccRCC in the TCGA cohort, the General cohort, and the CPTAC cohort, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (B, E, H) Comparations of the OSrisk among patients with different tumor grades in the TCGA cohort, the General cohort, and the CPTAC cohort, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' (C, F, I) Comparations of the OSrisk among patients with different tumor stages in the TCGA cohort, the General cohort, and the CPTAC cohort, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' OS, overall survival;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' ROC, receiver operator characteristics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' AUC, area under the curve;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' TCGA, the Cancer Genome Atlas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CPTAC, Clinical Proteomic Tumor Analysis Consortium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' CI, confidence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' A B C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 TCGA cohort 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='75 os 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='25 2 AUC 95%CI TCGA: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='784 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='746-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='819) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00 T 100 08 60 40 20 0 G1 G2 G3 G4 Stagei Stage ii Stage ili Stage iv Specificity (%) D E F 8 <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 p <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 I cohort 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8 Sensitivity ( risk General so 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0 AUC 95%C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0 General:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='774(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='723-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='820 T 100 80 60 40 20 0 G1 G2 G3 G4 Stagei Stage ii Stage ili Specificity (%) G H <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 d <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='75 CPTAC cohort 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='50 8 risk risk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='50 SO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='25 AUC 95%CI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='00- CPTAC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='702 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='632-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='765) 100 80 60 40 20 G1 0 G2 G3 G4 Stagei Stage ii Stage ili Stage iv Specificity (%)23 Supplemental Table Table S1 Basic clinical characteristics of patients recruited for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content=' General Cohort (401) TCGA Cohort (820) CPTAC Cohort (195) Age(years) ≥65 139(34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='7%) 263(32.' metadata={'source': 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138(70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8%) Female 113(28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2%) 272(33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2%) 57(29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2%) Stage i 362(90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='3%) 434(52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9%) 100(51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='3%) ii 25(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2%) 105(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8%) 20(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='3%) iii 14(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='5%) 182(22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='2%) 54(27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='7%) iv 0 99(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='1%) 21(10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='7%) WSI 401 847 195 Subtype ChRCC 44(11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='0%) 65(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='9%) / pRCC 51(12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='7%) 244(29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/C9E4T4oBgHgl3EQfFwy_/content/2301.04889v1.pdf'} +page_content='8%) / ccRCC 306(76.' metadata={'source': 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Cuozzo2,3, Nick E. Teslich1, Keith G. Ray1, Zurong Dai1, +Tian T. Li1, George F. Chapline1, Jonathan L. DuBois1, and Enrico Rossi3 +1Lawrence Livermore National Laboratory, Livermore, CA 94550, USA +2Sandia National Laboratories, Livermore, CA 94551, USA +3Department of Physics, William & Mary, Williamsburg, VA 23187, USA +January 3, 2023 +Superconducting topological systems formed by a strong 3D topological insulator (TI) +in proximity to a conventional s-wave superconductor (SC) have been intensely studied +as they may host Majorana zero modes. However, there are limited experimental realiza- +tions of TI-SC systems in which robust superconducting pairing is induced on the surface +states of the TI and a topological superconducting state is established. Here, we fabricate +a novel TI-SC system by depositing, via focused ion beam, tungsten (W) nanoscale clus- +ters on the surface of TI Bi0.91Sb0.09. We find that the resulting heterostructure supports +phase-slip lines that act as effective Josephson junctions. We probe the response of the +system to microwave radiation. We find that for some ac frequencies, and powers, the +resulting Shapiro steps’ structure of the voltage-current characteristic exhibits a missing +first step and an unexpectedly wide second Shapiro step. The theoretical analysis of the +measurements shows that the unusual Shapiro response arises from the interplay between +a static Josephson junction and a dynamic one, and allows us to identify the conditions +under which the missing first step can be attributed to the topological nature of the +Josephson junctions formed by the phase-slip lines. Our results suggest a new approach +to induce superconductivity in a TI, a novel route to realizing highly-transparent topo- +logical Josephson junctions, and show how the response of superconducting systems to +microwave radiation can be used to infer the dynamics of phase-slip lines. +Introduction +Hybrid structures formed by a strong topological insulator (TI) and a superconductor (SC) have +been theoretically predicted as a promising platform for realizing topological superconductivity [1– +6]. +Soon after the theoretical proposals, experiments showed that superconducting pairing can be +induced on the surface states of three dimensional (3D) TIs [7–10]. Experimental studies of Josephson +junctions (JJs) based on 2D or 3D TI-SC heterostructures then showed signatures in the current +voltage characteristic (I–V ) under microwave radiation consistent with the presence of a topological +superconducting state [11–13]. Over the past few years, a growing number of JJs with 3D TI weak links +have been realized and displayed signs suggesting the establishment of a topological superconducting +state [14–17]. Recently, several studies have provided further insight into the behavior of JJs based +on topological materials [17–20], and, in particular, have shown that signatures in the I–V properties +often associated with the topological character of the superconducting state can also be observed in +non-topological JJs [17,19,20]. +The main challenges to realize a robust topological JJ based on heterostructures formed by a 3D +TI and a SC are: (i) realization of an almost ideal TI-SC interface; (ii) suppression of disorder; (iii) +fabrication of short and very narrow JJs. In this work, to overcome these challenges we follow a very +different approach from previous ones: to create the TI-SC heterostructure we deposit tungsten (W) +clusters on TI Bi0.91Sb0.09 using the focused ion-beam technique (FIOB), and to form the JJ we rely on +the natural formation of phase-slip lines (PSLs), lines across which the phase of the superconducting +1 +arXiv:2301.00086v1 [cond-mat.supr-con] 31 Dec 2022 + +order parameter increases at different rates. Forming the TI-SC hybrid system by deposing W clusters +has two advantages: the W clusters, being separated and randomly placed, do not significantly modify +the electronic structure of the TI, and yet, can induce via the proximity effect pairing correlations +in the TI’s surface states at low temperature, given that the inter-cluster distance is comparable to +the normal-metal coherence length of Bi0.91Sb0.09; it minimizes the exposure of the TI’s surface to air +and it removes the need to perform any annealing, both of which can strongly affect the TI’s surface +properties and doping. By relying on the natural formation of a PSL we can realize an effective JJ +with a length of just few nanometers and a width controlled by the W coverage of the TI. Given that +W is deposited via FIOB the JJ width can be as small as few 10s nm. +We find that the W clusters induce on Bi0.91 Sb0.09’s surface a superconducting state with a critical +temperature Tc that is slightly below the Tc of W nanoclusters. Transport measurements in the dc +regime reveal that the system undergoes a Berenziskii-Kosterlitz-Thouless (BKT) transition. Jumps +in the voltage-current (V –I) characteristic can be associated to the presence of phase-slip lines which +form effective JJs. +To probe the properties of such JJs we measure the V –I characteristic under +microwave radiation for different ac frequencies and powers. We find that at intermediate frequencies +and powers the first Shapiro step is missing, and that at low frequencies and powers, in addition to the +first Shapiro step being missing, the second step can be very wide. We develop the theory to explain +such unusual features and find that for intermediate frequencies and powers the missing step can be +explained by the presence of Landau-Zener transitions (LZTs), and that for low frequencies and powers +the structure of the Shapiro steps can be understood considering the presence of two JJs, formed by +PSLs, one of which has its effective width dynamically driven by the biasing current. The results have +important implications for achieving proximity-induced superconductivity in a TI, understanding how +seemingly 4π-periodic Andreev bound states (ABSs) might arise in Josephson junctions formed by +PSLs, and understanding how signatures of the ac response can be used to infer the dynamics of PSLs +and the effect on such dynamics of the biasing currents. +Results +We present results for devices in which W leads are grown using the focused-ion-beam technique +on Bi0.91Sb0.09 flakes with a thickness of 2–5 µm. Due to the halo effect [21, 22], self-assembled W +islands with a thickness of 10–50 nm form around the deposited W. Details about the fabrication and +characterization of the devices can be found in the Methods section and Supplementary Information +(SI). We have studied the sample with the geometry shown in Figs. 1 (a) and (b), in which a bow- +tie-like strip of W islands was deposited within a 1-µm-wide region from the edge of the Bi0.91Sb0.09 +flake to produce a microbridge. The inset of Fig. 1 (c) shows a scanning-electron-microscopy (SEM) +image of the W islands. We find that the island diameter is typically in the range of 50–60 nm, and +edge-to-edge spacing between islands is 20 nm. The island size and inter-island spacing depend on +the ion dose and gradually decrease with increasing distance from the deposition region. +We first perform dc measurements to characterize the superconducting state of the W-TI het- +erostructure. +The inset of Fig. 1 (c) shows the contacts’ configuration used to measure the I–V +characteristic. Figure 1 (c) shows the resistance R versus temperature T profiles under a perpendicu- +lar magnetic field, H, stepping from 0 to 4 Tesla. The normal-state resistance displays an upturn at +low temperatures for all magnetic fields. This behavior arises from the current redistribution related +to sample non-homogeneity together with an out-of-line contact arrangement [23]. For H = 0, at +T ∼ 4 K, the system undergoes a broad superconducting transition, signaled by a sharp reduction of +the resistance, while inter-island phase coherence develops [24]. On further decreasing T below 1.6 +K, the resistance vanishes completely and the global phase coherence is reached. Increasing H de- +creases the temperature at which coherent superconducting states are established. Figure 1 (d) shows +the value of the upper critical field Hc2(T) as a function of temperature. A linear fit of this data +allows us to estimate the in-plane Ginzburg–Landau (GL) coherence length at zero temperature to be +ξGL(0) = 7.6 ± 1 nm. This value agrees with tungsten’s superconducting coherence length, ξW . +Figure 1 (e) shows, on a logarithmic scale, the dc V –I characteristic for H = 0 and different values +of T < 4 K. We see that when the current is larger than threshold values, that depend on T, V +grows with I following a power law, V ∝ Iα(T ), with a T-dependent α. This indicates the presence of +dissipation due to the motion of vortices and antivortices in the superconductor. As T grows the 2D +superconductor undergoes a BKT transition at the BKT transition temperature, TBKT. For T = TBKT +vortex-antivortex pairs break and α(TBKT) = 3 [25–28]. The black dashed line in Fig. 1 (e) shows the +2 + +slope, on the log-log scale, corresponding to α = 3. Figure 1(f) shows the evolution of α with T. We +determine TBKT = 2.96 K from where α = 3 interpolates. +The results presented in Fig. 1 show that our W-TI heterostructure is a proximity-coupled super- +conducting system [24, 29]. By examining the V -I characteristic at higher currents we observe the +presence of additional voltage jumps for I > 0.25 mA for all temperatures, Fig. 2 (a). We find that +the slopes of the V -I characteristic before and after each additional jump approximately extrapolate at +V = 0 to the same current value, the so called excess current Ie, as shown in Fig. 2 (b). The features +of the dc V –I characteristic at high currents are consistent with the formation of PSLs, resistive states +arising in thin superconducting films when the current is larger than a threshold value, It [30–36]. A +PSL has width ∼ ξ, the superconducting coherence length. In our case ξ = ξW given that Bi0.91Sb0.09’s +superconducting correlations are only induced by W via the proximity effect. Across the PSL a voltage +V = RP SL(I − ¯Is) is established, where I is the biasing current, RP SL is the effective resistance of +the PSL, and ¯Is the average supercurrent across the PSL. ¯Is can be identified with the excess current +Ie, i.e., the current that crosses the PSL even when V = 0. As a consequence a PSL can be described +effectively as a biased JJ, of length ξ, with critical current Ic = Ie. The dependence of dV/dI on the +perpendicular field B⊥ and dc bias current shows signatures of a Fraunhofer pattern consistent with a +JJ of length L ≈ ξW . Using an induced gap on Bi0.91Sb0.09 equal to W’s superconducting gap, for all +the Fermi pockets of Bi0.91Sb0.09’s surface states, we obtain a coherence length that is at least a few +times larger than ξW . As a result, for Bi0.91Sb0.09’s surface states a PSL in W-TI hybrid can be well +approximated as a short JJ. +For a superconducting TI, the effective JJ associated with a PSL can be expected to have a +topological character. In the presence of microwave radiation the V –I characteristic of a JJ exhibits +Shapiro steps [37] for V = nhf/2e, where f is the frequency of the radiation and n is an integer. For +a topological JJ the current-phase relation (CPR) is 4π-periodic [38, 39] and this results in missing +Shapiro steps for odd n [12,40–43]. However, in highly transparent JJs, Landau-Zener processes can +cause the odd Shapiro steps to be missing even when the junction is not topological [20]. +Figures 3 (a), (c), (e), and (g) show the color maps for dV /dI versus the ac power P and the bias +dc current I at microwave frequencies f = 2.3, 2.0, 1.6, and 1.4 GHz, respectively. The corresponding +V (I) dependence, obtained from the integration of the dV /dI curve over the peak area, is shown in +Figs. 3 (b), (d), (f), and (h), respectively. At high frequency, f = 2.3 GHz, we observe the usual +structure for the Shapiro steps consistent with a conventional 2π-periodic CPR. As f is decreased, +f = 2.0 GHz, we observe the appearance of additional peaks in the dV/dI at low bias currents +that result in regular Shapiro steps. As the f is decreased further, f = 1.6 GHz, we observe the +disappearance of the first, odd, Shapiro step indicating that the CPR of the JJ formed by the PSL has +a non-negligible 4π-periodic component either due to its topological character [12,14,41,44] or due to +Landau-Zener processes [20]. Because no hysteresis is observed in our devices the missing steps cannot +be attributed to hysteretic effects. At even lower frequencies, f = 1.4 GHz, the peaks in the dV/dI at +low bias currents result in a Shapiro steps’ structure in which the first step is absent, and the second +one is unusually long. For the steps at low bias currents shown in Fig. 3 (h) we also notice that the +in-gap critical current in the presence of an ac bias, Ic,ac appears to increase with power, rather than +decreasing, as in conventional JJs. This suggests that in our system some properties, such as the width +of the effective JJs created by PSLs, might be affected by the biasing current and ac power. +Theoretical analysis +To understand the anomalous structure of the Shapiro steps shown in Fig. 3, we developed and studied +an effective model to describe the JJs created by the PSLs. A calculation of the Shapiro steps from +a microscopic model is computationally prohibitive for the size of our devices [45], and so we describe +the dynamics of the JJs using a resistively and capacitively shunted junction (RCSJ) model. Within +the RCSJ model, for a current-drive junction the dynamics of the phase φ across the junction is given +by: +d2φ +dt2 + σ dφ +dt + Is(φ) +Ic += Idc +Ic ++ Iac +Ic +sin(ωt) +(1) +where t = +� +2eIc +ℏC t′ is a dimensionless time variable, σ = +� +ℏ +2eIcR2 +nC is the Stewart-McCumber param- +eter, Is(φ) is the supercurrent across the JJ, and Idc, Iac are the dc and ac bias currents, respectively. +3 + +For σ ≫ 1 the JJ is overdamped and we can neglect the first term on the left hand side of Eq. (1) and +simplify the model to a resistively shunted junction (RSJ) model. From the dc transport measure- +ments, Fig. 2, we extract RN ≈ 8.4 Ω, and from experimental results like the ones shown in Fig. 3 (a) +we extract Ic ∼ 0.1 mA. Assuming C ≈ 1 fF, the expected value for a JJ with a geometry similar to +the JJ formed by a PSL in our devices, we obtain σ ≈ 20 (see SI). This implies that to understand the +results shown in Figs. 3 (a), (b), and Figs. 3 (e), (f), to good approximation, we can treat the JJs as +overdamped. +In general, for JJs based on superconducting TIs, we have that Is has both a 2π-periodic, I2π, +component and 4π-periodic one, I4π. Because the topological nature of the JJ only guarantees one +crossing in the ABS’s spectrum at φ = π, it only contributes one 4π mode to the total supercurrent +across the JJ. The maximum supercurrent I(i) +c +carried by a single conducting mode is given by I(i) +c += +e∆/2ℏ. From the value of Tc for W, Tc = 4.4 K, we obtan ∆ = 1.76kBTc = 668 µeV and therefore +I(i) +c +≈ 81 nA. A junction with an I4π component exhibits missing odd Shapiro steps for frequencies +smaller than f4π = 2eRNI4π/h [46]. As a consequence, if there is only one mode contributing to I4π, +we obtain f4π < 0.5 GHz. Given that we observe missing odd steps for f > 1 GHz we conclude that +to explain dV/dI profiles like the one shown in Fig. 3 (e) (no in-gap steps) we need to have more +than a single mode contributing to I4π. Given the large width, W > ξ, of the bow-tie-like strip of +tungsten islands, and therefore of the JJs formed by PSLs located away from the center of the bow-tie, +we can have Andreev mid-gap states with small gaps at φ = π, and sizable detachment gaps from the +continuum at φ = 0 [20]. Such modes can contribute to the 4π-periodic component of the supercurrent +Is(φ) given that they have a large probability, PLZT,˜τ, to undergo a Landau-Zener transition (LZT) +at φ = π, and a negligible probability to undergo transitions at φ = 0 mod 2π into the continuum. To +good approximation we have [47]: +PLZT,˜τ(t = tnπ) = exp +� +−π ∆(1 − ˜τ) +e|V (tnπ)| +� +, +(2) +where tnπ is the time when φ → (2n+1)π (n ∈ N), ˜τ is the average transparency of high transparency +modes which also have a sizable detachment gap [20], and V (tnπ) = (ℏ/2e)(dφ/dt)|t=tnπ. +dV/dI +profiles like the one shown in Fig. 3 (e) can be understood considering an effective RSJ model in which +the supercurrent Is(φ) has two channels [20]: one low-transparency channel with a purely 2π-periodic +CPR, Is,2π = I2π sin(φ), and for which no LZTs can take place, and a high-transparency channel with +Is,˜τ = I˜τ sin(φ)/[1 − ˜τ sin2(φ/2)]1/2. To obtain the dynamics of the JJ we integrate Eq. (1), neglecting +the first term on the left hand side, setting Is(φ) = I2π sin(φ) + Is,˜τ(φ), evaluating PLZT,˜τ at times +t = tnπ and switching the sign in front of Is,˜τ for t = tnπ if a randomly generated number 0 < r < 1 +is smaller than PLZT,˜τ(tnπ). +Figure 4 (a) shows the dependence of time-averaged voltage, V , on the dc current for different values +of the ac power when ˜τ = 0.999, I˜τ/I2π = 2.0%, EJ ≡ 2eIcRN = 364.5 µeV , and hf = 0.026EJ. This +corresponds to a relatively high frequency regime compared to f4π, and we find that, for the powers +considered, the Shapiro steps’ structure does not exhibit missing steps, analogous to the experimental +V –I shown in Fig. 3 (b). Figure 4 (b) shows the results for the case when hf = 0.018EJ, all the other +parameters being the same as in Fig. 4 (a). For this lower value of the frequency the contribution to +the supercurrent from the high transparency channels qualitatively affects the structure of the Shapiro +steps: at low powers the odd steps are missing, as seen in the experimental results shown in Fig. 3 (f). +In the dV/dI profile showed in Fig. 3 (g) we have two sets of peaks: the “standard” peaks outside +the region where dV/dI is mostly zero, and isolated “in-gap” peaks inside this region, present only +when -9 dBm ≲ P ≲ -6 dBm and |I| ≲ 0.15 mA. To explain the presence of two sets of peaks in dV/dI +profiles like the one shown in Fig. 3 (g) it is natural to assume that two PSLs in series are present. +One JJ, JJ1, with a large Ic is responsible for the standard peaks, and one, JJ2, with a smaller Ic, +is responsible for the in-gap peaks. The resulting effective circuit describing the dynamics of the two +junctions is shown in Fig. 4 (d). +The V –I characteristic associated to the in-gap peaks, see Fig. 3 (h), has two very unique qualitative +features: (i) the critical current in the presence of ac bias (Ic,ac) increases with the microwave power +rather than decreasing, as expected for JJs; (ii) the width of the second step is very large, larger +than Ic,ac and of the width of the conventional steps seen at higher powers. The first feature strongly +suggests that the critical current of the JJ responsible for the in-gap dV/dI peaks might grow with the +ac power. This can be understood by considering that a weak link created by a PSL can be affected by +4 + +the biasing current: as the biasing current increases, if possible, the PSL will change to allow a larger +supercurrent across the JJ. In our setup we can expect that, as the biasing current increases a PSL, +initially at a point close to the center of the “bow-tie”, might move away from the center and become +wider, see Fig. 4 (c), causing JJ2 to have a larger Ic. +From the smallest value of Ic,ac we estimate the minimum width of JJ2 to be approximately 50 nm. +For such a small width we have that RN can be sufficiently large that even just one 4π-periodic +supercurrent channel can be sufficient to have f4π ≳ 1 GHz. The fact that in the V –I characteristic +corresponding to the in-gap peaks shown in Fig. 3 (h) the absence of the first Shapiro step is very +robust supports the hypothesis that its absence, at least for the smallest values of power and Idc, might +be due to the topological nature of JJ2. As discussed above, however, we cannot exclude contributions +to the 4π-periodic supercurrent arising from LZTs of highly transparent modes. For JJ2, a 4π-periodic +supercurrent channel appears to be sufficiently strong to determine the structure of the junction’s +Shapiro steps, and so for JJ2 we include only such a supercurrent channel. We describe JJ1 via an +RSJ model in which both a 2π- and 4π-periodic supercurrent channels are present. JJ2 is expected to +form close to the middle of the bow-tie, a region where W is expected to be thinner and so Ic smaller. +This suggests that for JJ2 σ might not be very large and therefore that for JJ2 the capacitive term in +Eq. (1) might not be negligible. Indeed, we find good agreement with the experimental results if for +JJ2 we set σ ∼ 6 − 7 and keep the capacitive term, resulting in the effective circuit model shown in +Fig. 4 (d). For the critical current of JJ2 we assume Ic,2 = I(0) +c,2 + αIac if Idc is smaller than Ionset and +Ic,2 ≈ I(0) +c,2 + αIac + (Idc − Ionset) if Idc > Ionset, with α > 0. The extension of the width of the second +Shapiro step in the V -I characteristic of Fig. 3 (h) allows us to fix the values of I(0) +c,2 , Ionset, and α (see +SI). Notice that given that we assume Ic,iRN,i = π∆/e = const., we have that for JJ2, as Ic,2 increases +RN,2 decreases, which is reasonable if we attribute the increase of Ic,2 to an increase of the PSL’s width. +Similarly, we keep the value of σ fixed, implying that as Ic,2 increases the capacitance also increase, +consistent with the idea that the PSL moves to regions of the bow-tie with larger cross-sectional areas. +Fig. 4 (e) shows the results for the V –I characteristics, for different microwave powers, obtained +integrating the RCJS model corresponding to the circuit diagram shown in Fig. 4 (d). We see that we +recover the main qualitative features observed experimentally at low frequencies and powers, Fig. 3 (h). +Figure 4 (f) shows how the V –I characteristic for JJ1 and JJ2 evolve as the microwave power is +increased: we see that Ic,ac for the two junctions approach each other as P increases. Given that the +two JJs are in series, the full V –I characteristic is given by the sum of the characteristics for JJ1 and +JJ2. +Discussion +In this work, by placing tungsten nanoislands on TI Bi0.91Sb0.09 using the focus ion beam technique, +we demonstrated a new approach to realize an air-stable heterostructure in which superconductivity +is induced at the surface of a 3D TI. By studying the transport properties in the dc limit we have +shown that the system undergoes a Berezinskii-Koasterlitz-Thouless transition at T = TBKT ≈ 3 K. +We have shown that when the biasing current is larger than a threshold value, PSLs are formed, which +can be described effectively as Josephson junctions. We have estimated the length of the PSLs to be +about 7 nm, and their width to be as small as 50 nm, making the geometry of the effective Josephson +junction to be at the limit of current fabrication techniques. At low frequencies, the V –I characteristic +of PSL-formed JJ exhibits missing odd Shapiro steps. Our theoretical analysis suggests that for wide +PSLs (of width of the order of a µm) the absence of odd Shapiro steps is due to the presence of +Andreev bound states with a large probability to undergo a Landau-Zener transition when the phase +difference across the PSL is close to π. +For PSLs of width ∼ 50 nm we estimate the topological +nature of the resulting Josephson junction might be sufficient to explain the observed absence of odd +Shapiro steps. We showed how, by analyzing the response of the system to microwave radiation it is +possible to infer the presence of multiple PSLs and how the microwave power and dc current can affect +their properties, in particular their width, and therefore the critical current of the effective Josepshon +junction formed by the PSL. Our results suggest that the width of a PSL can be controlled in a +superconductor-TI microbridge with a bow-tie geometry by tuning the biasing current, a result that +complements approaches in which the PSL’s nucleation site is controlled by other means, for instance +the application of localized mechanical stress [48]. +The unique properties of the phase-slip lines in heterostructures like W-Bi0.91Sb0.09, and the possi- +bility of engineering their width, make these structures a new platform to realize topological Josephson +5 + +junctions with geometries that stretch current fabrication techniques to the limit. The topological na- +ture of such junctions could be further probed by measuring via tunneling contacts the unique trans- +port properties [49, 50] of the associated Majorana modes. Replacing BixSb1−x with other TIs, e.g., +(BixSb1−x)2Te3, is also a promising step towards reducing the total number of conducting channels. +1 +Acknowledgement +We would like to thank Y. Rosen for helpful discussions, and K. Huang and A. A. Baker for assistance +in performing the experiments. This work was performed under the auspices of the US Department +of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. +The project was supported by the Laboratory Directed Research and Development (LDRD) programs +of LLNL (19-LW-040). J. J. Cuozzo and E. Rossi acknowledge support from DOE, Grant No DE- +SC0022245. +Sandia National Laboratories is a multimission laboratory managed and operated by +National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell +International Inc. for the U.S. DOE’s National Nuclear Security Administration under contract DE- +NA0003525. This paper describes objective technical results and analysis. Any subjective views or +opinions that might be expressed in the paper do not necessarily represent the views of the U.S. DOE +or the United States Government. +Methods +Bi0.91Sb0.09 single crystals were synthesized by the modified Bridgman method with high purity (5N) +Bi and Sb in a sealed quartz tube. The tube was heated up to 600 ◦C for 1–2 days and shaken to +homogenize the mixture. Then the tube was slowly cooled to 270 ◦C over a period of 3.5 months. +Finally the samples were annealed at 270 ◦C for 3 days. Our devices are fabricated by pressing single +crystal flakes onto a SiO2/Si substrate with pre-fabricated Au electrodes. +A micromanipulator is +used to pick up the flake with a flat surface and move it to the center of the Au electrode pattern. +Superconducting W-based focused-ion-beam technique was employed to perform the W deposition and +tungsten hexacarbonyl W(CO)6 gas was used as a precursor material. First, we deposited W leads +with a thickness of 200–500 nm by FIOB with a Ga+ ion-beam current of 0.92 nA. Then, we deposited +W pads to bridge the W leads to the pre-patterned Au electrodes. We iterated the W deposition +process in combination with the transport measurements four times until realizing the zero-resistance +state between the bottom W leads. +Our transport measurements are carried out with a four-probe configuration to eliminate the contact +resistance between W/Pt electrodes and Bi1−xSbx. To attenuate electronic noise, π filters are installed +between the shielded cryostat and the measurement apparatus. For the Shapiro step measurements, +microwave radiation is applied through a coaxial cable with a stripped end that is placed 1–2 mm above +the sample surface. All measurements are performed in a Helium-3 cryostat with a base temperature +of 0.54 K. +6 + +10 µm +0 +50 +100 +(a) +(b) +V +I +0 +1 +2 +3 +4 +5 +6 +0 +2 +4 +6 +8 +0.01 +0.1 +0.01 +0.1 +2.5 +3.0 +3.5 +4.0 +0 +3 +6 +0 +1 +2 +3 +4 +0 +2 +4 +6 +T (K) +4 T +3 T +2 T +1 T + +R (W) +0 T +(c) +I (mA) +(e) +V (mV) + 0.56 K + 0.92 + 1.64 + 1.95 + 2.29 + 2.70 + 2.81 + 2.91 + 3.05 + 3.26 + 3.47 + 3.74 +(f) + + +a +T (K) +TBKT = 2.96 K +(d) +µ0HC2(T) +T (K) +Figure 1: a, Scanning-electron-microscopy (SEM) image of the sample, where superconducting W pads +are fabricated on the Bi0.09Sb0.91 flake with a distance of L ∼ 30 µm apart. Scale bar = 10 µm. b, +The corresponding false-color energy-dispersive X-ray spectroscopy (EDS) elemental map shows the +distribution of elemental W. The W clusters spread out around the W leads, forming a bow-tie shaped +∼1 µm by 30 µm microbridge. Scale bar = 1 µm. c, Resistance R as a function of temperature T for the +2.6-µm-thick sample measured using the probe configuration I (see bottom inset). The magnetic field +is applied perpendicular to the sample surface and the bias current is 10 µA. Top inset: SEM image of +W islands on the Bi0.91Sb0.09 substrate, taken at a distance of 2.8 µm from a 200-nm-thick W deposit +(scale bar = 200 nm). d, Temperature dependence of the upper critical field Hc2, which follows the +GL theory for a 2D superconductor: Hc2 = +Φ0 +2πξGL(0)2 (1 − T +Tc ), where Φ0 is the flux quantum. e, V (I) +curves on a logarithmic scale. The long dashed line corresponds to V ∼I3 dependence. f, Temperature +dependence of the power-law exponent α. The data α is extracted from the fits to the V (I) curves +shown in e. +7 + +HV +mag □/t +tilt +HFW +WD +det +2 μm +5.00 kV| +50 000 x|0 |5.62 μm +15.0mm +TLD +Device1100 +95 +92 +89 +85 +82 +79 +76 +73 +69 +99 +9 +09 +56 +53 +50 +46 +43 +40 +36 +33 +30 +27 +24 +20 +17 +14 +11 +8 +5 +00.0 +0.1 +0.2 +0.3 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.15 +0.20 +0.25 +0.30 +0.35 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +V (mV) +I (mA) + 0.55 K + 1.46 + 1.88 + 2.27 + 2.47 + 2.77 +(a) +(b) +V (mV) +I (mA) + T = 1.46 K +Ie +Ic +Figure 2: +a, Temperature dependence of V –I characteristic obtained with configuration I. The black +arrows indicate the second voltage jump at a higher current. b, Voltage–current characteristic obtained +with configuration I at T = 1.46 K. The red and green lines are extrapolated linear V –I segments from +the first and second resistive branches, respectively. These two resistive branches exhibit approximately +the same excess current Ie, determined by the intersection of the red or green lines with the current +axis. This behavior is consistent with the signatures of phase-slip lines previously observed in quasi- +two-dimensional superconducting strips. +8 + +-0.1 +0.0 +0.1 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +f = 1.6 GHz + + +I (mA) +-8 dBm + + +V (hf/2e) +-4.5 dBm + + + + + + + + + + + + +(f) +-0.2 +-0.1 +0.0 +0.1 +0.2 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +V (hf/2e) +I (mA) +-2 dBm +f = 2.3 GHz +-5.5 dBm +(b) +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +-0.1 +0.0 +0.1 +-9.5 dBm +-8 dBm +f = 1.4 GHz +V (hf/2e) +I (mA) +(h) +-0.2 +-0.1 +0.0 +0.1 +0.2 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +f = 2 GHz +-9.5 dBm +V (hf/2e) +I (mA) +(d) +(a) +f = 2.3 GHz +f = 1.4 GHz +(g) +(c) +f = 2 GHz +f = 1.6 GHz +(e) +Figure 3: The ac Josephson effect measured using probe configuration I. a, c, e, g, color maps of the +differential resistance dV /dI as a function of the rf power P and dc bias current I for rf frequencies +f = 2.3, 2, 1.6, and 1.4 GHz at T = 0.56 K. The white arrows in c, e indicate the in-gap Shapiro +response. b, d, f, h, Shapiro steps at different irradiation powers. The voltage is scaled in the unit of +Shapiro voltage ∆V = hf/2e. +9 + +dV/d/ (2) +dV/d/ (2) +dV/d/ (2) +dV/d/ (2) +14 +９ +-5 +12 +12 +3 +8 +10 +10 +7 +-8 +P (dBm) +P (dBm) +-6 +6 +P (dBm) +P (dBm) +5 +6 +-9 +-5 +-14 +12 +2 +-12 +-8 +17 +-15 +-15 +-0.2-0.1 +0.1 +0.2 +-0.2-0.1 +0.1 +0.2 +-0.2-0.1 +0.1 +0.2 +-0.2-0.1 +0.1 +0.2 +0 +0(a) +(b) +hf = 0.026 EJ +(c) +(d) +hf = 0.018 EJ +JJ 2 +JJ 1 +C = 1e − 15 F; +f = 20 GHz +(e) +(f) +JJ 1 +JJ 2 +Figure 4: Shapiro steps calculated using the RCSJ model with LZTs using a hf/EJ = 0.026 and b +hf/EJ = 0.018. The effective transparency for the modes undergoing LZTs was taken to be τLZT = +0.999 and I˜τ/I2π = 2.0%. c, Schematic of two PSLs in series (denoted JJ1 and JJ2) where JJ1 is +fixed and JJ2 changes with an applied current bias. d, Circuit diagram for the dynamic two-junction +model. e, Shapiro steps calculated using a two-junction model describing PSL motion. Here σ = 6.7, +Ic,2 = Ic,1/8, α = 7 for JJ2 and hf = 0.09EJ in both junctions. f, Individual contributions of JJ1 and +JJ2 to panel e. +10 + +I(2元) +bias +I(4元) +T(4元) +1 +SReferences +[1] L. Fu and C. L. Kane, Superconducting proximity effect and Majorana fermions at the surface of +a topological insulator, Phys. Rev. Lett. 100, 096407 (2008). +[2] M. Sato and S. Fujimoto, Topological phases of noncentrosymmetric superconductors: edge states, +Majorana fermions, and non-Abelian statistics, Phys. Rev. B 79, 094504 (2009). +[3] M. Z. Hasan and C. L. Kane, Colloquium: topological insulators, Rev. Mod. 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Huang et al., Proximity-induced surface superconductivity in Dirac semimetal Cd3As2, Nat. +Commu. 10, 2217 (2019). +13 + diff --git a/EdAyT4oBgHgl3EQfSfdI/content/tmp_files/load_file.txt b/EdAyT4oBgHgl3EQfSfdI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b04cebf2fe9a49c76fb2aef3dfdd52cdee3275a1 --- /dev/null +++ b/EdAyT4oBgHgl3EQfSfdI/content/tmp_files/load_file.txt @@ -0,0 +1,830 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf,len=829 +page_content='Phase-Slip Lines and Anomalous Josephson Effects in a Tungsten Clusters-Topological Insulator Microbridge Dong-Xia Qu1, Joseph J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Cuozzo2,3, Nick E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Teslich1, Keith G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Ray1, Zurong Dai1, Tian T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Li1, George F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Chapline1, Jonathan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' DuBois1, and Enrico Rossi3 1Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 2Sandia National Laboratories, Livermore, CA 94551, USA 3Department of Physics, William & Mary, Williamsburg, VA 23187, USA January 3, 2023 Superconducting topological systems formed by a strong 3D topological insulator (TI) in proximity to a conventional s-wave superconductor (SC) have been intensely studied as they may host Majorana zero modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' However, there are limited experimental realiza- tions of TI-SC systems in which robust superconducting pairing is induced on the surface states of the TI and a topological superconducting state is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Here, we fabricate a novel TI-SC system by depositing, via focused ion beam, tungsten (W) nanoscale clus- ters on the surface of TI Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We find that the resulting heterostructure supports phase-slip lines that act as effective Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We probe the response of the system to microwave radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We find that for some ac frequencies, and powers, the resulting Shapiro steps’ structure of the voltage-current characteristic exhibits a missing first step and an unexpectedly wide second Shapiro step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The theoretical analysis of the measurements shows that the unusual Shapiro response arises from the interplay between a static Josephson junction and a dynamic one, and allows us to identify the conditions under which the missing first step can be attributed to the topological nature of the Josephson junctions formed by the phase-slip lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Our results suggest a new approach to induce superconductivity in a TI, a novel route to realizing highly-transparent topo- logical Josephson junctions, and show how the response of superconducting systems to microwave radiation can be used to infer the dynamics of phase-slip lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Introduction Hybrid structures formed by a strong topological insulator (TI) and a superconductor (SC) have been theoretically predicted as a promising platform for realizing topological superconductivity [1– 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Soon after the theoretical proposals, experiments showed that superconducting pairing can be induced on the surface states of three dimensional (3D) TIs [7–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Experimental studies of Josephson junctions (JJs) based on 2D or 3D TI-SC heterostructures then showed signatures in the current voltage characteristic (I–V ) under microwave radiation consistent with the presence of a topological superconducting state [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Over the past few years, a growing number of JJs with 3D TI weak links have been realized and displayed signs suggesting the establishment of a topological superconducting state [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Recently, several studies have provided further insight into the behavior of JJs based on topological materials [17–20], and, in particular, have shown that signatures in the I–V properties often associated with the topological character of the superconducting state can also be observed in non-topological JJs [17,19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The main challenges to realize a robust topological JJ based on heterostructures formed by a 3D TI and a SC are: (i) realization of an almost ideal TI-SC interface;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' (ii) suppression of disorder;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' (iii) fabrication of short and very narrow JJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' In this work, to overcome these challenges we follow a very different approach from previous ones: to create the TI-SC heterostructure we deposit tungsten (W) clusters on TI Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09 using the focused ion-beam technique (FIOB), and to form the JJ we rely on the natural formation of phase-slip lines (PSLs), lines across which the phase of the superconducting 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='00086v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='supr-con] 31 Dec 2022 order parameter increases at different rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Forming the TI-SC hybrid system by deposing W clusters has two advantages: the W clusters, being separated and randomly placed, do not significantly modify the electronic structure of the TI, and yet, can induce via the proximity effect pairing correlations in the TI’s surface states at low temperature, given that the inter-cluster distance is comparable to the normal-metal coherence length of Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' it minimizes the exposure of the TI’s surface to air and it removes the need to perform any annealing, both of which can strongly affect the TI’s surface properties and doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' By relying on the natural formation of a PSL we can realize an effective JJ with a length of just few nanometers and a width controlled by the W coverage of the TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Given that W is deposited via FIOB the JJ width can be as small as few 10s nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We find that the W clusters induce on Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91 Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09’s surface a superconducting state with a critical temperature Tc that is slightly below the Tc of W nanoclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Transport measurements in the dc regime reveal that the system undergoes a Berenziskii-Kosterlitz-Thouless (BKT) transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Jumps in the voltage-current (V –I) characteristic can be associated to the presence of phase-slip lines which form effective JJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' To probe the properties of such JJs we measure the V –I characteristic under microwave radiation for different ac frequencies and powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We find that at intermediate frequencies and powers the first Shapiro step is missing, and that at low frequencies and powers, in addition to the first Shapiro step being missing, the second step can be very wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We develop the theory to explain such unusual features and find that for intermediate frequencies and powers the missing step can be explained by the presence of Landau-Zener transitions (LZTs), and that for low frequencies and powers the structure of the Shapiro steps can be understood considering the presence of two JJs, formed by PSLs, one of which has its effective width dynamically driven by the biasing current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The results have important implications for achieving proximity-induced superconductivity in a TI, understanding how seemingly 4π-periodic Andreev bound states (ABSs) might arise in Josephson junctions formed by PSLs, and understanding how signatures of the ac response can be used to infer the dynamics of PSLs and the effect on such dynamics of the biasing currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Results We present results for devices in which W leads are grown using the focused-ion-beam technique on Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09 flakes with a thickness of 2–5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Due to the halo effect [21, 22], self-assembled W islands with a thickness of 10–50 nm form around the deposited W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Details about the fabrication and characterization of the devices can be found in the Methods section and Supplementary Information (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We have studied the sample with the geometry shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 1 (a) and (b), in which a bow- tie-like strip of W islands was deposited within a 1-µm-wide region from the edge of the Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09 flake to produce a microbridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 1 (c) shows a scanning-electron-microscopy (SEM) image of the W islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We find that the island diameter is typically in the range of 50–60 nm, and edge-to-edge spacing between islands is 20 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The island size and inter-island spacing depend on the ion dose and gradually decrease with increasing distance from the deposition region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We first perform dc measurements to characterize the superconducting state of the W-TI het- erostructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 1 (c) shows the contacts’ configuration used to measure the I–V characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figure 1 (c) shows the resistance R versus temperature T profiles under a perpendicu- lar magnetic field, H, stepping from 0 to 4 Tesla.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The normal-state resistance displays an upturn at low temperatures for all magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This behavior arises from the current redistribution related to sample non-homogeneity together with an out-of-line contact arrangement [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For H = 0, at T ∼ 4 K, the system undergoes a broad superconducting transition, signaled by a sharp reduction of the resistance, while inter-island phase coherence develops [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' On further decreasing T below 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6 K, the resistance vanishes completely and the global phase coherence is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Increasing H de- creases the temperature at which coherent superconducting states are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figure 1 (d) shows the value of the upper critical field Hc2(T) as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' A linear fit of this data allows us to estimate the in-plane Ginzburg–Landau (GL) coherence length at zero temperature to be ξGL(0) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6 ± 1 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This value agrees with tungsten’s superconducting coherence length, ξW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figure 1 (e) shows, on a logarithmic scale, the dc V –I characteristic for H = 0 and different values of T < 4 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We see that when the current is larger than threshold values, that depend on T, V grows with I following a power law, V ∝ Iα(T ), with a T-dependent α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This indicates the presence of dissipation due to the motion of vortices and antivortices in the superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' As T grows the 2D superconductor undergoes a BKT transition at the BKT transition temperature, TBKT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For T = TBKT vortex-antivortex pairs break and α(TBKT) = 3 [25–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The black dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 1 (e) shows the 2 slope, on the log-log scale, corresponding to α = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figure 1(f) shows the evolution of α with T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We determine TBKT = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='96 K from where α = 3 interpolates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 1 show that our W-TI heterostructure is a proximity-coupled super- conducting system [24, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' By examining the V -I characteristic at higher currents we observe the presence of additional voltage jumps for I > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='25 mA for all temperatures, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We find that the slopes of the V -I characteristic before and after each additional jump approximately extrapolate at V = 0 to the same current value, the so called excess current Ie, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 2 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The features of the dc V –I characteristic at high currents are consistent with the formation of PSLs, resistive states arising in thin superconducting films when the current is larger than a threshold value, It [30–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' A PSL has width ∼ ξ, the superconducting coherence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' In our case ξ = ξW given that Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09’s superconducting correlations are only induced by W via the proximity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Across the PSL a voltage V = RP SL(I − ¯Is) is established, where I is the biasing current, RP SL is the effective resistance of the PSL, and ¯Is the average supercurrent across the PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' ¯Is can be identified with the excess current Ie, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=', the current that crosses the PSL even when V = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' As a consequence a PSL can be described effectively as a biased JJ, of length ξ, with critical current Ic = Ie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The dependence of dV/dI on the perpendicular field B⊥ and dc bias current shows signatures of a Fraunhofer pattern consistent with a JJ of length L ≈ ξW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Using an induced gap on Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09 equal to W’s superconducting gap, for all the Fermi pockets of Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09’s surface states, we obtain a coherence length that is at least a few times larger than ξW .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' As a result, for Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09’s surface states a PSL in W-TI hybrid can be well approximated as a short JJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For a superconducting TI, the effective JJ associated with a PSL can be expected to have a topological character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' In the presence of microwave radiation the V –I characteristic of a JJ exhibits Shapiro steps [37] for V = nhf/2e, where f is the frequency of the radiation and n is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For a topological JJ the current-phase relation (CPR) is 4π-periodic [38, 39] and this results in missing Shapiro steps for odd n [12,40–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' However, in highly transparent JJs, Landau-Zener processes can cause the odd Shapiro steps to be missing even when the junction is not topological [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figures 3 (a), (c), (e), and (g) show the color maps for dV /dI versus the ac power P and the bias dc current I at microwave frequencies f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='4 GHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The corresponding V (I) dependence, obtained from the integration of the dV /dI curve over the peak area, is shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (b), (d), (f), and (h), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' At high frequency, f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='3 GHz, we observe the usual structure for the Shapiro steps consistent with a conventional 2π-periodic CPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' As f is decreased, f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 GHz, we observe the appearance of additional peaks in the dV/dI at low bias currents that result in regular Shapiro steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' As the f is decreased further, f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6 GHz, we observe the disappearance of the first, odd, Shapiro step indicating that the CPR of the JJ formed by the PSL has a non-negligible 4π-periodic component either due to its topological character [12,14,41,44] or due to Landau-Zener processes [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Because no hysteresis is observed in our devices the missing steps cannot be attributed to hysteretic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' At even lower frequencies, f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='4 GHz, the peaks in the dV/dI at low bias currents result in a Shapiro steps’ structure in which the first step is absent, and the second one is unusually long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For the steps at low bias currents shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (h) we also notice that the in-gap critical current in the presence of an ac bias, Ic,ac appears to increase with power, rather than decreasing, as in conventional JJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This suggests that in our system some properties, such as the width of the effective JJs created by PSLs, might be affected by the biasing current and ac power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Theoretical analysis To understand the anomalous structure of the Shapiro steps shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3, we developed and studied an effective model to describe the JJs created by the PSLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' A calculation of the Shapiro steps from a microscopic model is computationally prohibitive for the size of our devices [45], and so we describe the dynamics of the JJs using a resistively and capacitively shunted junction (RCSJ) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Within the RCSJ model, for a current-drive junction the dynamics of the phase φ across the junction is given by: d2φ dt2 + σ dφ dt + Is(φ) Ic = Idc Ic + Iac Ic sin(ωt) (1) where t = � 2eIc ℏC t′ is a dimensionless time variable, σ = � ℏ 2eIcR2 nC is the Stewart-McCumber param- eter, Is(φ) is the supercurrent across the JJ, and Idc, Iac are the dc and ac bias currents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 For σ ≫ 1 the JJ is overdamped and we can neglect the first term on the left hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' (1) and simplify the model to a resistively shunted junction (RSJ) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' From the dc transport measure- ments, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 2, we extract RN ≈ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='4 Ω, and from experimental results like the ones shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (a) we extract Ic ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Assuming C ≈ 1 fF, the expected value for a JJ with a geometry similar to the JJ formed by a PSL in our devices, we obtain σ ≈ 20 (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This implies that to understand the results shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (a), (b), and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (e), (f), to good approximation, we can treat the JJs as overdamped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' In general, for JJs based on superconducting TIs, we have that Is has both a 2π-periodic, I2π, component and 4π-periodic one, I4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Because the topological nature of the JJ only guarantees one crossing in the ABS’s spectrum at φ = π, it only contributes one 4π mode to the total supercurrent across the JJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The maximum supercurrent I(i) c carried by a single conducting mode is given by I(i) c = e∆/2ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' From the value of Tc for W, Tc = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='4 K, we obtan ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='76kBTc = 668 µeV and therefore I(i) c ≈ 81 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' A junction with an I4π component exhibits missing odd Shapiro steps for frequencies smaller than f4π = 2eRNI4π/h [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' As a consequence, if there is only one mode contributing to I4π, we obtain f4π < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Given that we observe missing odd steps for f > 1 GHz we conclude that to explain dV/dI profiles like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (e) (no in-gap steps) we need to have more than a single mode contributing to I4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Given the large width, W > ξ, of the bow-tie-like strip of tungsten islands, and therefore of the JJs formed by PSLs located away from the center of the bow-tie, we can have Andreev mid-gap states with small gaps at φ = π, and sizable detachment gaps from the continuum at φ = 0 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Such modes can contribute to the 4π-periodic component of the supercurrent Is(φ) given that they have a large probability, PLZT,˜τ, to undergo a Landau-Zener transition (LZT) at φ = π, and a negligible probability to undergo transitions at φ = 0 mod 2π into the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' To good approximation we have [47]: PLZT,˜τ(t = tnπ) = exp � −π ∆(1 − ˜τ) e|V (tnπ)| � , (2) where tnπ is the time when φ → (2n+1)π (n ∈ N), ˜τ is the average transparency of high transparency modes which also have a sizable detachment gap [20], and V (tnπ) = (ℏ/2e)(dφ/dt)|t=tnπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' dV/dI profiles like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (e) can be understood considering an effective RSJ model in which the supercurrent Is(φ) has two channels [20]: one low-transparency channel with a purely 2π-periodic CPR, Is,2π = I2π sin(φ), and for which no LZTs can take place, and a high-transparency channel with Is,˜τ = I˜τ sin(φ)/[1 − ˜τ sin2(φ/2)]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' To obtain the dynamics of the JJ we integrate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' (1), neglecting the first term on the left hand side, setting Is(φ) = I2π sin(φ) + Is,˜τ(φ), evaluating PLZT,˜τ at times t = tnπ and switching the sign in front of Is,˜τ for t = tnπ if a randomly generated number 0 < r < 1 is smaller than PLZT,˜τ(tnπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figure 4 (a) shows the dependence of time-averaged voltage, V , on the dc current for different values of the ac power when ˜τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='999, I˜τ/I2π = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0%, EJ ≡ 2eIcRN = 364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 µeV , and hf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='026EJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This corresponds to a relatively high frequency regime compared to f4π, and we find that, for the powers considered, the Shapiro steps’ structure does not exhibit missing steps, analogous to the experimental V –I shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figure 4 (b) shows the results for the case when hf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='018EJ, all the other parameters being the same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For this lower value of the frequency the contribution to the supercurrent from the high transparency channels qualitatively affects the structure of the Shapiro steps: at low powers the odd steps are missing, as seen in the experimental results shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' In the dV/dI profile showed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (g) we have two sets of peaks: the “standard” peaks outside the region where dV/dI is mostly zero, and isolated “in-gap” peaks inside this region, present only when -9 dBm ≲ P ≲ -6 dBm and |I| ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='15 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' To explain the presence of two sets of peaks in dV/dI profiles like the one shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (g) it is natural to assume that two PSLs in series are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' One JJ, JJ1, with a large Ic is responsible for the standard peaks, and one, JJ2, with a smaller Ic, is responsible for the in-gap peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The resulting effective circuit describing the dynamics of the two junctions is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 4 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The V –I characteristic associated to the in-gap peaks, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (h), has two very unique qualitative features: (i) the critical current in the presence of ac bias (Ic,ac) increases with the microwave power rather than decreasing, as expected for JJs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' (ii) the width of the second step is very large, larger than Ic,ac and of the width of the conventional steps seen at higher powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The first feature strongly suggests that the critical current of the JJ responsible for the in-gap dV/dI peaks might grow with the ac power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This can be understood by considering that a weak link created by a PSL can be affected by 4 the biasing current: as the biasing current increases, if possible, the PSL will change to allow a larger supercurrent across the JJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' In our setup we can expect that, as the biasing current increases a PSL, initially at a point close to the center of the “bow-tie”, might move away from the center and become wider, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 4 (c), causing JJ2 to have a larger Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' From the smallest value of Ic,ac we estimate the minimum width of JJ2 to be approximately 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For such a small width we have that RN can be sufficiently large that even just one 4π-periodic supercurrent channel can be sufficient to have f4π ≳ 1 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The fact that in the V –I characteristic corresponding to the in-gap peaks shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (h) the absence of the first Shapiro step is very robust supports the hypothesis that its absence, at least for the smallest values of power and Idc, might be due to the topological nature of JJ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' As discussed above, however, we cannot exclude contributions to the 4π-periodic supercurrent arising from LZTs of highly transparent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For JJ2, a 4π-periodic supercurrent channel appears to be sufficiently strong to determine the structure of the junction’s Shapiro steps, and so for JJ2 we include only such a supercurrent channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We describe JJ1 via an RSJ model in which both a 2π- and 4π-periodic supercurrent channels are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' JJ2 is expected to form close to the middle of the bow-tie, a region where W is expected to be thinner and so Ic smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This suggests that for JJ2 σ might not be very large and therefore that for JJ2 the capacitive term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' (1) might not be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Indeed, we find good agreement with the experimental results if for JJ2 we set σ ∼ 6 − 7 and keep the capacitive term, resulting in the effective circuit model shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 4 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For the critical current of JJ2 we assume Ic,2 = I(0) c,2 + αIac if Idc is smaller than Ionset and Ic,2 ≈ I(0) c,2 + αIac + (Idc − Ionset) if Idc > Ionset, with α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The extension of the width of the second Shapiro step in the V -I characteristic of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (h) allows us to fix the values of I(0) c,2 , Ionset, and α (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Notice that given that we assume Ic,iRN,i = π∆/e = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=', we have that for JJ2, as Ic,2 increases RN,2 decreases, which is reasonable if we attribute the increase of Ic,2 to an increase of the PSL’s width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Similarly, we keep the value of σ fixed, implying that as Ic,2 increases the capacitance also increase, consistent with the idea that the PSL moves to regions of the bow-tie with larger cross-sectional areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 4 (e) shows the results for the V –I characteristics, for different microwave powers, obtained integrating the RCJS model corresponding to the circuit diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 4 (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We see that we recover the main qualitative features observed experimentally at low frequencies and powers, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 3 (h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Figure 4 (f) shows how the V –I characteristic for JJ1 and JJ2 evolve as the microwave power is increased: we see that Ic,ac for the two junctions approach each other as P increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Given that the two JJs are in series, the full V –I characteristic is given by the sum of the characteristics for JJ1 and JJ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Discussion In this work, by placing tungsten nanoislands on TI Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09 using the focus ion beam technique, we demonstrated a new approach to realize an air-stable heterostructure in which superconductivity is induced at the surface of a 3D TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' By studying the transport properties in the dc limit we have shown that the system undergoes a Berezinskii-Koasterlitz-Thouless transition at T = TBKT ≈ 3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We have shown that when the biasing current is larger than a threshold value, PSLs are formed, which can be described effectively as Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We have estimated the length of the PSLs to be about 7 nm, and their width to be as small as 50 nm, making the geometry of the effective Josephson junction to be at the limit of current fabrication techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' At low frequencies, the V –I characteristic of PSL-formed JJ exhibits missing odd Shapiro steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Our theoretical analysis suggests that for wide PSLs (of width of the order of a µm) the absence of odd Shapiro steps is due to the presence of Andreev bound states with a large probability to undergo a Landau-Zener transition when the phase difference across the PSL is close to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For PSLs of width ∼ 50 nm we estimate the topological nature of the resulting Josephson junction might be sufficient to explain the observed absence of odd Shapiro steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We showed how, by analyzing the response of the system to microwave radiation it is possible to infer the presence of multiple PSLs and how the microwave power and dc current can affect their properties, in particular their width, and therefore the critical current of the effective Josepshon junction formed by the PSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Our results suggest that the width of a PSL can be controlled in a superconductor-TI microbridge with a bow-tie geometry by tuning the biasing current, a result that complements approaches in which the PSL’s nucleation site is controlled by other means, for instance the application of localized mechanical stress [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The unique properties of the phase-slip lines in heterostructures like W-Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09, and the possi- bility of engineering their width, make these structures a new platform to realize topological Josephson 5 junctions with geometries that stretch current fabrication techniques to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The topological na- ture of such junctions could be further probed by measuring via tunneling contacts the unique trans- port properties [49, 50] of the associated Majorana modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Replacing BixSb1−x with other TIs, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=', (BixSb1−x)2Te3, is also a promising step towards reducing the total number of conducting channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 1 Acknowledgement We would like to thank Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Rosen for helpful discussions, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Huang and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Baker for assistance in performing the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This work was performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' DE-AC52-07NA27344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The project was supported by the Laboratory Directed Research and Development (LDRD) programs of LLNL (19-LW-040).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Cuozzo and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Rossi acknowledge support from DOE, Grant No DE- SC0022245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' for the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' DOE’s National Nuclear Security Administration under contract DE- NA0003525.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This paper describes objective technical results and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' DOE or the United States Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Methods Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09 single crystals were synthesized by the modified Bridgman method with high purity (5N) Bi and Sb in a sealed quartz tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The tube was heated up to 600 ◦C for 1–2 days and shaken to homogenize the mixture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Then the tube was slowly cooled to 270 ◦C over a period of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Finally the samples were annealed at 270 ◦C for 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Our devices are fabricated by pressing single crystal flakes onto a SiO2/Si substrate with pre-fabricated Au electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' A micromanipulator is used to pick up the flake with a flat surface and move it to the center of the Au electrode pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Superconducting W-based focused-ion-beam technique was employed to perform the W deposition and tungsten hexacarbonyl W(CO)6 gas was used as a precursor material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' First, we deposited W leads with a thickness of 200–500 nm by FIOB with a Ga+ ion-beam current of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='92 nA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Then, we deposited W pads to bridge the W leads to the pre-patterned Au electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' We iterated the W deposition process in combination with the transport measurements four times until realizing the zero-resistance state between the bottom W leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Our transport measurements are carried out with a four-probe configuration to eliminate the contact resistance between W/Pt electrodes and Bi1−xSbx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' To attenuate electronic noise, π filters are installed between the shielded cryostat and the measurement apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' For the Shapiro step measurements, microwave radiation is applied through a coaxial cable with a stripped end that is placed 1–2 mm above the sample surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' All measurements are performed in a Helium-3 cryostat with a base temperature of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='54 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 6 10 µm 0 50 100 (a) (b) V I 0 1 2 3 4 5 6 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 0 3 6 0 1 2 3 4 0 2 4 6 T (K) 4 T 3 T 2 T 1 T R (W) 0 T (c) I (mA) (e) V (mV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='56 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='92 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='64 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='70 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='74 (f) a T (K) TBKT = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='96 K (d) µ0HC2(T) T (K) Figure 1: a, Scanning-electron-microscopy (SEM) image of the sample, where superconducting W pads are fabricated on the Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91 flake with a distance of L ∼ 30 µm apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Scale bar = 10 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' b, The corresponding false-color energy-dispersive X-ray spectroscopy (EDS) elemental map shows the distribution of elemental W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The W clusters spread out around the W leads, forming a bow-tie shaped ∼1 µm by 30 µm microbridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Scale bar = 1 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' c, Resistance R as a function of temperature T for the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6-µm-thick sample measured using the probe configuration I (see bottom inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The magnetic field is applied perpendicular to the sample surface and the bias current is 10 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Top inset: SEM image of W islands on the Bi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='91Sb0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09 substrate, taken at a distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='8 µm from a 200-nm-thick W deposit (scale bar = 200 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' d, Temperature dependence of the upper critical field Hc2, which follows the GL theory for a 2D superconductor: Hc2 = Φ0 2πξGL(0)2 (1 − T Tc ), where Φ0 is the flux quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' e, V (I) curves on a logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The long dashed line corresponds to V ∼I3 dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' f, Temperature dependence of the power-law exponent α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The data α is extracted from the fits to the V (I) curves shown in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 7 HV mag □/t tilt HFW WD det 2 μm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='00 kV| 50 000 x|0 |5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='62 μm 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0mm TLD Device1100 95 92 89 85 82 79 76 73 69 99 9 09 56 53 50 46 43 40 36 33 30 27 24 20 17 14 11 8 5 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='3 0.' metadata={'source': 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Voltage–current characteristic obtained with configuration I at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='46 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The red and green lines are extrapolated linear V –I segments from the first and second resistive branches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' These two resistive branches exhibit approximately the same excess current Ie, determined by the intersection of the red or green lines with the current axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' This behavior is consistent with the signatures of phase-slip lines previously observed in quasi- two-dimensional superconducting strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 8 6 4 2 0 2 4 6 8 f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6 GHz I (mA) 8 dBm V (hf/2e) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 dBm (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 8 6 4 2 0 2 4 6 8 V (hf/2e) I (mA) 2 dBm f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='3 GHz 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 dBm (b) 8 6 4 2 0 2 4 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 dBm 8 dBm f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='4 GHz V (hf/2e) I (mA) (h) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 8 6 4 2 0 2 4 6 8 f = 2 GHz 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='5 dBm V (hf/2e) I (mA) (d) (a) f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='3 GHz f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='4 GHz (g) (c) f = 2 GHz f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6 GHz (e) Figure 3: The ac Josephson effect measured using probe configuration I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' a, c, e, g, color maps of the differential resistance dV /dI as a function of the rf power P and dc bias current I for rf frequencies f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='3, 2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='6, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='4 GHz at T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='56 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The white arrows in c, e indicate the in-gap Shapiro response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' b, d, f, h, Shapiro steps at different irradiation powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The voltage is scaled in the unit of Shapiro voltage ∆V = hf/2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 9 dV/d/ (2) dV/d/ (2) dV/d/ (2) dV/d/ (2) 14 ９ 5 12 12 3 8 10 10 7 8 P (dBm) P (dBm) 6 6 P (dBm) P (dBm) 5 6 9 5 14 12 2 12 8 17 15 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='2 0 0(a) (b) hf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='026 EJ (c) (d) hf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='018 EJ JJ 2 JJ 1 C = 1e − 15 F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' f = 20 GHz (e) (f) JJ 1 JJ 2 Figure 4: Shapiro steps calculated using the RCSJ model with LZTs using a hf/EJ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='026 and b hf/EJ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' The effective transparency for the modes undergoing LZTs was taken to be τLZT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='999 and I˜τ/I2π = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='0%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' c, Schematic of two PSLs in series (denoted JJ1 and JJ2) where JJ1 is fixed and JJ2 changes with an applied current bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' d, Circuit diagram for the dynamic two-junction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' e, Shapiro steps calculated using a two-junction model describing PSL motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Here σ = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='7, Ic,2 = Ic,1/8, α = 7 for JJ2 and hf = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content='09EJ in both junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' f, Individual contributions of JJ1 and JJ2 to panel e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 10 I(2元) bias I(4元) T(4元) 1 SReferences [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Fu and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Kane, Superconducting proximity effect and Majorana fermions at the surface of a topological insulator, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' 100, 096407 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Sato and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Fujimoto, Topological phases of noncentrosymmetric superconductors: edge states, Majorana fermions, and non-Abelian statistics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdAyT4oBgHgl3EQfSfdI/content/2301.00086v1.pdf'} +page_content=' Rev.' metadata={'source': 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0000000000000000000000000000000000000000..8ffa79f3a0063a0fc1a09de4a9de0e7aad4a02df --- /dev/null +++ b/FtAzT4oBgHgl3EQfHPtI/content/tmp_files/2301.01041v1.pdf.txt @@ -0,0 +1,1491 @@ +On the Numerical Integration of Singular Initial and +Boundary Value Problems for Generalised +Lane–Emden and Thomas–Fermi Equations +Werner M. Seilera, Matthias Seißa +aInstitut f¨ur Mathematik, Universit¨at Kassel, 34132 Kassel, Germany +Abstract +We propose a geometric approach for the numerical integration of singular initial +value problems for (systems of) quasi-linear differential equations. It transforms +the original problem into the problem of computing the unstable manifold at a +stationary point of an associated vector field and thus into one which can be +solved in an efficient and robust manner. Using the shooting method, our ap- +proach also works well for boundary value problems. As examples, we treat +some (generalised) Lane–Emden equations and the Thomas–Fermi equation. +Keywords: singular initial value problems, singular boundary value problems, +Vessiot distribution, unstable manifold, numerical integration, Lane–Emden +equation, Thomas–Fermi equation, Majorana transformation +2010 MSC: 34A09, 34A26, 34B16, 65L05 +1. Introduction +The Lane–Emden equation was originally derived in astrophysics [1, p. 40] +and represents a dimensionless form of Poisson’s equation for the gravitational +potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid +(see [2–4] and references therein for a more detailed discussion): +u′′ + 2 +xu′ = −un +(1) +Email addresses: seiler@mathematik.uni-kassel.de (Werner M. Seiler), +mseiss@mathematik.uni-kassel.de (Matthias Seiß) +URL: http://www.mathematik.uni-kassel.de/~seiler (Werner M. Seiler) +Preprint submitted to Elsevier +January 4, 2023 +arXiv:2301.01041v1 [math.NA] 3 Jan 2023 + +with n the polytropic index. Astrophysicists want to solve the initial value prob- +lem u(0) = 1 and u′(0) = 0. Eq. (1) is prototypical for ordinary differential +equations arising in the construction of radially symmetric steady state solutions +of reaction-diffusion equations, as the left hand side of (1) represents the Laplace +operator in spherical coordinates. In an N-dimensional space, the numerator 2 +has to be replaced by N − 1. This leads to generalised Lane–Emden equations +u′′ + N − 1 +x +u′ = h(x, u) , +(2) +where h represents the reaction term. Besides the classical form from astro- +physics, we will later consider examples arising in chemical engineering (biocat- +alysts) and in physiology (oxygen uptake of cells). There, one needs the solution +of boundary value problems with u′(0) = 0 and αu(1) + βu′(1) = γ. +Thomas [5] and Fermi [6] derived independently of each other in a statistical +model of atoms treating electrons as a gas of particles a Lane–Emden equation +(1) with polytropic index n = 3/2 for the electrostatic potential V(x), however +with the “initial condition” that V(x) behaves like 1/x for x → 0. Writing V(x) = +u(x)/x, one obtains the Thomas–Fermi equation +u′′ = +� +u3/x +(3) +together with the initial condition u(0) = 1 (see [7–9] for more physical and +historical details and [10, 11] for a mathematical analysis). In addition, one +imposes one of the following three types of boundary conditions: +bu′(b) − u(b) = 0 , +(4a) +lim +x→∞ u(x) = 0 , +(4b) +u(a) = 0 +(4c) +with 0 < a, b < ∞ given positions. The infinite case (4b) occurs only for a crit- +ical value ω ≈ −1.588 . . . of the initial slope u′(0) and represents physically an +isolated neutral atom. For larger initial slopes, one can prescribe the boundary +condition (4a) and obtains solutions going through a minimum and then growing +rapidly. Physically, such solutions are relevant for certain crystals. The bound- +ary condition (4c) leads to solutions with a smaller initial slope and represent +physically ions with radius a. +Numerical methods from textbooks cannot be directly applied here, as all +considered equations are singular at x = 0 and at least one initial/boundary con- +dition is imposed there. In the vast literature on the numerical integration of +2 + +Lane–Emden or Thomas–Fermi equations, three different types of approaches +prevail. Astrophysicists apply for initial value problems a very simple approach: +they use for the first step a series expansion of the solution to get away from the +singularity and then use some standard integrator [3, Sect. 7.7.2] (see also [12]). +For boundary value problems, collocation methods are popular, as they are easily +adapted to the singularity, see e. g. [13]. Finally, various kinds of semi-analytic +expansions like Adomian decomposition have been adapted to the singularity +(see the references given below and references therein). +We propose here a new and rather different alternative. In the geometric +theory of differential equations [14, 15], one associates with any implicit ordi- +nary differential equation a vector field on a higher-dimensional space such that +the graphs of prolonged solutions of the implicit equation are integral curves of +this vector field. Most of the literature on singularity theory is concerned with +fully implicit equations. However, in applications quasi-linear equations like +the Lane–Emden equations prevail. In [16, 17], we showed that such equations +possess a special geometry allowing us to work in a lower order. Singulari- +ties, now called impasse points, are typically stationary points of the associated +vector field. If there is a unique solution, its prolonged solution graph is the one- +dimensional unstable manifold of this stationary point. Such an unstable man- +ifold can numerically be computed very robustly. In [18], we already sketched +this possibility to exploit ideas from singularity theory for numerical analysis. +Here, we want to demonstrate for concrete problems of practical relevance that +it is easy to apply and efficiently provides accurate results. +The paper is structured as follows. In the next section, we recall the neces- +sary elements of the geometric theory of differential equations and how one can +translate an implicit problem into an explicit one. Section 3 is then devoted to +the application of these ideas to (generalised) Lane–Emden equations and to the +numerical solution of some concrete problems from the literature. In Section 4 +we discuss the Thomas–Fermi equation by first reducing it via a transformation +introduced by Majorana and then applying the geometric theory. We compare +the obtained numerical results with some high precision calculations from the +literature. Finally, some conclusions are given. +2. Geometric Theory of Ordinary Differential Equations +We use a differential geometric approach to differential equations. It is be- +yond the scope of this article to provide deeper explanations of it; for this we +refer to [19] and references therein. For notational simplicity, we concentrate +on the scalar case; the extension to systems will be briefly discussed at the end. +3 + +Similarly, we restrict here to second-order equations, but equations of arbitrary +order can be treated in an analogous manner. +We consider a fully implicit differential equation of the form +F(x, u, u′, u′′) = 0 . +(5) +In the second-order jet bundle J2 (intuitively expressed, this is simply a four- +dimensional affine space with coordinates called x, u, u′, u′′), this equation de- +fines a hypersurface R2 ⊂ J2 which represents our geometric model of the dif- +ferential equation. We will assume throughout that R2 is actually a submanifold. +Given a function ψ(x), we may consider its graph as a curve in the jet bundle +J0 of order zero, i. e. the x-u space, given by the map x �→ �x, ψ(x)). Assuming +that ψ is at least twice differentiable, we can prolong this curve to a curve in J2 +defined by the map x �→ �x, ψ(x), ψ′(x), ψ′′(x)�. The function ψ is a solution of +(5), if and only if this curve lies completely in the hypersurface R2. +In an initial value problem for the implicit equation (5), one prescribes a +point ρ = (y, u0, u1, u2) ∈ R2 and asks for solutions such that ρ lies on their +prolonged graphs. Note that opposed to explicit problems, we must also specify +the value u2, as the algebraic equation F(y, u0, u1, u′′) = 0 may have several +(possibly infinitely many) solutions and thus may not uniquely determine u2. +A key ingredient of the geometry of jet bundles is the contact structure. In +the case of J2, the contact distribution C(2) is spanned by the two vector fields +Ctrans = ∂x + u′∂u + u′′∂u′ , +Cvert = ∂u′′ . +(6) +A curve x �→ �x, ψ0(x), ψ1(x), ψ2(x)� in J2 is a prolonged graph (i. e. ψ1 = ψ′ +0 and +ψ2 = ψ′′ +0 ), if and only if all its tangent vectors lie in the contact distribution. +The Vessiot distribution V[R2] of (5) is that part of the tangent space of R2 +which also lies in the contact distribution C(2). Writing X = aCtrans + bCvert for a +general vector in the contact distribution, X lies in the Vessiot distribution, if and +only if its coefficients a, b satisfy the linear equation +�Fx + u′Fu + u′′Fu′�a + Fu′′b = 0 . +(7) +A singularity is a point ρ = (y, u0, u1, u2) ∈ R2 such that Fu′′(ρ) = 0. One speaks +of a regular singularity, if the coefficient of a in (7) does not vanish at ρ, and of +an irregular singularity, if it does. Outside of irregular singularities, the Vessiot +distribution is one-dimensional and locally spanned by the vector field +X = Fu′′Ctrans − �Fx + u′Fu + u′′Fu′�Cvert +(8) +4 + +(note that X is defined only on the submanifold R2 ⊂ J2). The prolonged graph +of any solution of (5) must be integral curves of this vector field. The converse +is not necessarily true in the presence of singularities. +At regular singularities, the vector field X becomes vertical. Generically, only +one-sided solutions exist at such points and if two-sided solutions exist, then their +third derivative will blow up [20, Thm. 4.1]. At irregular singularities, typically +several (possibly infinitely many) solutions exist. In [21] it is shown how for +arbitrary systems of ordinary or partial differential equations with polynomial +nonlinearities all singularities can be automatically detected. +Irregular singularities are stationary points of X. Prolonged solution graphs +through them are one-dimensional invariant manifolds. Any one-dimensional +(un)stable or centre manifold (with transversal tangent vectors) at such a station- +ary point defines a solution. For higher-dimensional invariant manifolds, one +must study the induced dynamics on them to identify solutions. In any case, we +note that the numerical determination of invariant manifolds at stationary points +is a well-studied topic – see e. g. [22, 23]. +In general, the direct numerical integration of (5) faces some problems, if +it is not possible to solve (uniquely) for u′′, and typically breaks down, if one +gets too close to a singularity. The geometric theory offers here as alternative +the numerical integration of the dynamical system defined by the vector field X. +Thus an implicit problem is transformed into an explicit one! The price one +has to pay is an increase of the dimension: while (5) is a scalar equation (but +second-order), the vector field X lives on the three-dimensional manifold R2 in +the four-dimensional jet bundle J2 (more generally, a scalar equation of order q +leads to a vector field on a (q − 1)-dimensional manifold). +The key difference is, however, that we obtain a parametric solution repre- +sentation. We work now with the explicit autonomous system1 +dx +ds = Fu′′ , +du +ds = u′Fu′′ , +du′ +ds = u′′Fu′′ , +du′′ +ds = −Fx − u′Fu − u′′Fu′ , +(9) +where s is some auxiliary variable used to parametrise the integral curves of X. +A solution of it will thus be a curve s �→ �x(s), u(s), u′(s), u′′(s)� on R2 ⊂ J2. A +numerical integration will provide a discrete approximation of this curve. +1Strictly speaking, we are dealing here with a three-dimensional system, as X lives on the +three-dimensional manifold R2. As we do not know a parametrisation of R2, we must work with +all four coordinates of J2. One could augment (9) by its first integral F(x, u, u′, u′′) = 0 and +enforce it during a numerical integration, but in our experience this is not necessary. +5 + +In applications, quasi-linear equations prevail. We restrict here even to semi- +linear differential equations of the form +F(x, u, u′, u′′) = g(x)u′′ − f(x, u, u′) = 0 , +(10) +with smooth functions f, g, as both the Lane–Emden and the Thomas–Fermi +equation can be brought into this form. A point (y, u0, u1, u2) ∈ R2 is then a +singularity, if and only if g(y) = 0. +As first shown in [16] and later discussed in more details in [17], quasi-linear +equations possess their own special geometry, as it is possible to project the +Vessiot distribution to the jet bundle of one order less, i. e. in our case to the +first-order jet bundle J1 with coordinates (x, u, u′). Projecting the vector field X +defined by (8) with F as in (10) to J1 yields the vector field +Y = g(x)∂x + g(x)u′∂u + f(x, u, u′)∂u′ . +(11) +It is only defined on the canonical projection of R2 to J1 which may be a proper +subset. Assuming that f, g are defined everywhere on J1, we analytically extend +Y to all of J1 and replace (9) by the three-dimensional system +dx +ds = g(x) , +du +ds = g(x)u′ , +du′ +ds = f(x, u, u′) . +(12) +The first equation is decoupled and can be interpreted as describing a change of +the independent variable, but we will not pursue this point of view. +A point ρ = (y, u0, u1) ∈ J1 is an impasse point for (10), if the vector field Y +is not transversal at ρ, i. e. if its x-component vanishes. Here, this is equivalent to +g(y) = 0. We call ρ a proper impasse point, if R2 contains points which project on +ρ; otherwise, ρ is improper. Here, proper impasse points are obviously stationary +points of Y or (12), respectively. Prolonged graphs of solutions of (10) are one- +dimensional invariant manifolds of Y (or (12), resp.) and again the converse is +not necessarily true. In [17], we proved geometrically the following result (a +classical analytic proof for the special case g(x) = x can be found in [24]). +Theorem 1. Consider (10) for f, g smooth together with the initial conditions +u(y) = u0 and u′(y) = u1 where g(y) = 0 and f(y, u0, u1) = 0. If δ = g′(y) and +γ = fu′(y, u0, u1) are both non zero and of opposite sign, then the initial value +problem possesses a unique smooth solution. +Under the made assumptions, the initial point ρ = (y, u0, u1) is a proper im- +passe point of (10). One readily verifies that the Jacobian J of Y at ρ has the +eigenvalues δ, 0 and γ and thus we find three one-dimensional invariant man- +ifolds at ρ: the stable, the unstable and the centre manifold.2 Without loss of +2The centre manifold is here unique, as there exists a whole curve of stationary points [25]. +6 + +generality, we assume that δ > 0 (otherwise we multiply (10) by −1). It is then +shown in [17] that the prolonged graph of the unique solution is the unstable +manifold and thus at ρ it is tangent to the eigenvector of J for δ. +Remark 2. The extension to implicit systems F(x, u, u′, u′′) = 0 is straightfor- +ward. Assuming that the unknown function u is vector valued, u: I ⊆ R → Rn, +the jet bundle J2 is (3n + 1)-dimensional and the contact distribution C(2) is gen- +erated by the n + 1 vector fields Ctrans = ∂x + u′ · ∂u + u′′ · ∂u′ and Cvert = ∂u′′, +where the dot denotes the standard scalar product. Again the Vessiot distribution +is generically one-dimensional and the coefficients of a vector field X spanning +it are readily determined by solving a linear system of equations. Numerical +integration of X allows us to approximate solutions of the given system. +We restrict to semi-linear first-order systems of the form g(x)u′ = f(x, u) +with g still a scalar functions. For initial conditions u(y) = u0 with g(y) = 0 +and f(y, u0) = 0, we introduce δ = g′(y) (assuming δ > 0) and the Jacobian +Γ = fu(y, u0). In [26], it is shown that if all eigenvalues of Γ have a negative real +part, then the initial value problem has a unique smooth solution. A classical an- +alytical proof was given by Vainikko by first studying extensively the linear case +[27] and then extending to the nonlinear one [28]. In the geometric approach, one +sees again that the graph of the solution is a one-dimensional unstable manifold +of the vector field Y spanning the projected Vessiot distribution. +3. (Generalised) Lane–Emden Equations +3.1. Geometric Treatment +If we consider the generalised Lane–Emden equation (2), then one obtains +after multiplication by x the special case of (10) given by +g(x) = x , +f(x, u, u′) = xh(x, u) − (N − 1)u′ , +(13) +where we always assume N > 1. For arbitrary initial conditions u(0) = u0 and +u′(0) = u1, we find that δ = 1 and γ = −(N −1) are nonzero and of opposite sign. +The initial point ρ = (0, u0, u1) is a proper impasse point, if and only if u1 = 0. +In this case, Theorem 1 asserts the existence of a unique smooth solution. +The projected Vessiot distribution is spanned by the vector field +Y = x∂x + xu′∂u + �xh(x, u) − (N − 1)u′�∂u′ . +(14) +For u1 � 0, no solution can exist. Indeed, the vector field Y has then no stationary +point and the unique trajectory through the initial point ρ = (0, u0, u1) is the +vertical line s �→ (0, u0, u1 + s) which does not define a prolonged graph. +7 + +We thus assume u1 = 0, which unsurprisingly is the case in all applications of +(2) in the literature. Independent of the value of u0, the initial point ρ = (0, u0, 0) +is a stationary point of the vector field Y. The Jacobian of Y at ρ is +J = +���������� +1 +0 +0 +0 +0 +0 +h(0, u0) +0 +−(N − 1) +���������� . +(15) +Its eigenvalues are 1, 0 and −(N − 1). Relevant for us is only the eigenvector +to the eigenvalue 1, as it is tangential to the unstable manifold. It is given by +v = �N, 0, h(0, u0)�T. +For the numerical solution of our given initial value problem, we integrate the +vector field Y for the initial data �x(0), u(0), u′(0)�T = �0, u0, 0�T + ϵv with some +small ϵ > 0. The concrete value of ϵ is not very relevant. As the exact solution +corresponds to the unstable manifold, any error is automatically damped by the +dynamics of Y. In our experiments, we typically used ϵ = 10−3 or ϵ = 10−4. +We can easily extend this approach to coupled systems of the form +u′′ + N − 1 +x +u′ = h(x, u) , +(16) +where u is a vector valued function and the coupling occurs solely through the +reaction terms. If u is a d-dimensional vector, then the dimension of the first- +order jet bundle J1 is 2d + 1. Thus (12) becomes a system of this dimension: +dx +ds = x , +du +ds = xu′ , +du′ +ds = xh(x, u) − (N − 1)u′ . +(17) +By the same arguments as in the scalar case, we restrict to the initial condition +u′(0) = 0 so that the initial point ρ = (0, u0, 0) is again a proper impasse point. +The Jacobian at ρ is a block form of (15): +J = +���������� +1 +0T +0T +0 +0d +0d +h(0, u0) +0d +−(N − 1)Ed +���������� , +(18) +where 0d and Ed denote the d × d zero and unit matrix, resp. We still have 1 as +a simple eigenvalue, whereas the eigenvalues 0 and −(N − 1) have both the al- +gebraic multiplicity d. The d-dimensional stable and centre manifolds are again +vertical and irrelevant for a solution theory. But we still find a one-dimensional +unstable manifold corresponding to the prolonged graph of the unique solution. +It is tangential to the vector v = �N, 0T, h(0, u0)T�T and as in the scalar case we +use as initial data for its determination the point �0, uT +0 , 0T�T + ϵv. +8 + +3.2. Numerical Results +As our main goal consists of showing how easy the numerical integration +of singular problems becomes with our geometric approach, we did not de- +velop any sophisticated production code. We performed all our computations +with the built-in numerical capabilities of Maple. We used most of the time +the dsolve/numeric command with its standard settings, i. e. a Runge–Kutta– +Fehlberg pair of order 4/5 is applied with a tolerance of 10−6 for the relative error +and 10−7 for the absolute error. +Our geometric ansatz does not determine approximations un ≈ u(xn) of +the solution u(x) on a discrete mesh (xn), but approximations xn = x(sn) and +un = u(sn) for a parametric representation �x(s), u(s)� of the graph of the solu- +tion. Hence, for computing an approximated solution value u(¯x), one must first +determine a parameter value ¯s such that x(¯s) ≈ ¯x. This can easily be accom- +plished either with a nonlinear solver or with a numerical integrator with event +handling. We used the latter option in most of our experiments. +For boundary value problems, we applied the shooting method which worked +very well. As Maple provides no built-in command for it, we wrote our own sim- +ple version. In scalar problems, we solved the arising nonlinear equation most +of the time with the Steffensen method (with Aitken ∆2 acceleration). As our +equations are dimensionfree, suitable starting values were easy to find: typically, +u(x) varied between 0 and 1 and we chose 0.5 as starting point. +We encountered difficulties only in the simulation of a biocatalyst. For some +parameter values, the correct initial value was very close to zero and the Stef- +fensen iterations produced sometimes intermediate approximations which were +negative and for which the numerical integration became meaningless. Here we +resorted to a simple bisection method. +For Lane–Emden systems, we used the Newton method for the arising non- +linear systems. The Jacobian was determined via the variational equation of the +differential system. Thus for an n-dimensional differential system where k < n +initial conditions have to be determined via shooting, we had to solve an addi- +tional kn-dimensional linear differential system with variable coefficients. +3.2.1. Scalar Lane–Emden Equations +We consider scalar Lane–Emden equations of the generalised form +u′′ + m +x u′ = f(x, u) +(19a) +together with either the initial conditions +u(0) = u0 , +u′(0) = 0 +(19b) +9 + +or the boundary conditions +u′(0) = 0 , +αu(1) + βu′(1) = γ . +(19c) +Chawla and Shivakumar [29] proved for boundary value problems with α = 1 +and β = 0 an existence and uniqueness theorem under the following assumption +on the right hand side f(x, u): the supremum M of the negative partial derivative +− fu(x, u) on [0, 1] × R must be less than the first positive root t1 of the Bessel +function J(m−1)/2( √t) (in the frequent case m = 2, we thus need M < π2). +The numerical integration of (19a) has been studied by many authors using +many different approaches; we refer to [30] for an overview of many works be- +fore 2010. We will discuss three different situations: (i) initial value problems in +astrophysics, (ii) Dirichlet boundary value problem in chemical engineering and +(iii) mixed boundary value problems in physiology. +Initial Value Problems from Astrophysics. In the classical Lane–Emden equa- +tions, one has m = N − 1 with N the space dimension and f(x, u) = −un. The +solutions for u0 = 1 are known as polytropes. Physically meaningful is the range +0 ≤ n < 5 (with n not necessarily an integer). For three polytropic indices, +namely n = 0, 1, 5, exact solutions are known [4, Sect. 2.3]. Of physical rele- +vance are in particular the first zero ξ1 of u (corresponding to the scaled radius +of the sphere) and the value of u′(ξ1) (e. g. the ratio of the central density to the +mean density is given by r = −ξ1/3u′(ξ1)). +Figure 1: Logarithmic plot of absolute deviation from exact solution for some polytropes. +We numerically solved the Lane–Emden equations by integrating the dynam- +ical system (12) with f, g given by (13). As concrete test cases, we used some +10 + +10 +-6 +10 +.7 +10 +n=0 N=2 +err +n=1 N=2 +8 +n=0 N=3 +10 +n=1 N=3 +n=5 N=3 +9 +10 +.10 +10 +0 +2 +3 +4 +xpolytropic cylinders and spheres where the exact solutions are known. Figure 1 +shows the observed errors in logarithmic scale. Obviously, the results are within +the expected range for the default settings of Maple’s numerical integrator. +N, n +ξ1 +r +2, 0 +3.2 · 10−6 +4.0 · 10−7 +2, 2 +4.2 · 10−7 +5.7 · 10−6 +3, 0 +4.3 · 10−7 +2.2 · 10−10 +3, 1 +1.5 · 10−7 +9.3 · 10−7 +Table 1: Relative errors for first zero ξ1 and +density ratio r for the cases with ξ1 < ∞. +Our approach also determines approx- +imations u′ +n = u′(sn) for the first deriva- +tives of the solution, as the integral curves +of the vector field Y define a parametrisa- +tion �x(s), u(s), u′(s)� of the solution and +its first derivative. We use this to approx- +imate also the quantities ξ1 and r. +Ta- +ble 1 exhibits their relative errors com- +pared with the exact solution for those +cases where ξ1 is finite. Again, the ob- +served accuracy corresponds well to the settings of the numerical integrator. +Boundary Value Problems for (Bio)Catalysts. In chemical engineering, the Lane– +Emden equation arises in the analysis of diffusive transport and chemical reac- +tions of species inside a porous catalyst pellet [31, §6.4] with boundary condi- +tions of the form (19c) with α = γ = 1 and β = 0. Flockerzi and Sundmacher +[32] considered the case m = 2 and f(x, u) = φ2un for a single species obeying +Fick’s law with constant diffusivity and power-law kinetics (the constant φ2 is +the Thiele modulus describing the ratio of surface reaction rate to diffusion rate). +As this corresponds up to a sign exactly to the above considered polytropes, we +omit concrete calculations and only note that [32] also provides a nice geomet- +ric proof of the existence of a unique solution of this particular boundary value +problem which, unfortunately, seems not be extendable to other functions f. +Using a Michaelis–Menten kinetics for a biocatalyst, one obtains right hand +sides like f(x, u) = 9φ2 +u +1+Ku, where φ is again the Thiele modulus and K the +dimensionless Michaelis–Menten constant (see [33] for some further variants). +This model was analysed by a homotopy perturbation method in [34]. A quantity +relevant for engineers is the effectiveness factor which is here given by η = +K+1 +3φ2 u′(1). A numerical study of the dependency of η on φ2 and K leads to the +surface shown in Fig. 2 based on a 17 × 17 grid, i. e. on the numerical solution of +289 boundary value problems with different combinations of parameter values. +As indicated above, we had to use here a bisection method for locating the right +initial value. Bisecting until an interval length of 10−5 was reached, the whole +computation required only 2–3sec on a laptop (equipped with eight Intel Core +i7-11370H (11th generation) working with 3.3GHz and 16GB of RAM running +Maple 2022 under Windows 11). +11 + +Figure 2: Dependency of the effectiveness +factor η on Thiele modulus φ2 and dimen- +sionless Michaelis–Menten constant K. +Matlab’s solvers bvp4c and bvp5c +are finite difference methods based on a +three- and four-stage, resp., Lobatto IIIa +collocation formulae and provide a special +option for the type of singularity appear- +ing in Lane–Emden equations [35, 36]. +However, it turned out to be nontrivial to +determine a plot like Fig. 2 with them, +as for some parameter values they re- +act rather sensitive to the required ini- +tial guess. Using a simple constant func- +tion lead sometimes either to completely +wrong solutions or the collocation equa- +tions could not be solved. We then com- +puted one solution with “harmless” pa- +rameter values and used it as initial guess for all other parameter values. But the +computations required with 5–6sec about twice as much time as our approach. +An alternative approach consists in transforming the problem into a reaction- +diffusion equation by adding a time derivative. +The desired solution of our +boundary value problem arises then as asymptotic for long times. Matlab pro- +vides here with pdepe a specialised solver admitting again our type of singu- +larity. It employs a method for parabolic partial differential equations proposed +by Skeel and Brezins [37] using a spatial discretisation derived with a Galerkin +approach. Here, one does not need an initial guess and it turns out that a steady +state is reached very rapidly (already t = 1 is sufficient). But one needs an ad- +ditional interpolation with pdeval to determine derivative values. Furthermore, +the computation time for a plot like Fig. 2 increases significantly to about 17sec.3 +Mixed Boundary Conditions for a Physiological Model. The same differential +equation is used to model the steady state oxygen diffusion in a spherical cell +with Michaelis-Menten uptake kinetics [38, 39], m = 2 and f(x, u) = +au +u+K, but +with mixed boundary conditions (19c) where α = γ, β = 1. Hiltmann and Lory +[40] proved explicitly the existence and uniqueness of a solution of this problem. +In the first two references above, concrete, physiologically meaningful values +3This approach was also used by the authors of [34] to compute reference solutions. However, +the plots presented there do not agree with our results. As they provided a listing of their Matlab +code, we could repeat their numerical experiments and obtained the same results as with our +method and not what they show in their paper. +12 + +0.8- +-9'0 +n +0.4- +0.2- +人 +0 +15 +5 +10 +2 +10 +15 +5 +Kfor the parameters are determined and numerical results are presented which are, +however, contradictory. We used for our experiments four different parameter +sets proposed by McElwain [39] and which can be found in Table 2. +a +K +α +1 +0.38065 +0.03119 +5 +2 +0.38065 +0.03119 +0.5 +3 +0.76129 +0.03119 +5 +4 +0.38065 +0.31187 +5 +Table 2: Parameter values for the oxygen +uptake model following McElwain [40]. +In particular for the third parame- +ter set, several authors performed similar +computations starting with Hiltmann und +Lory [40]. Khuri and Sayfy [41, Ex. 3] +combined a decomposition method in the +vicinity of the singularity with a colloca- +tion method in the rest of the integration +interval. They provided – like Hiltmann +and Lory – approximations of u(xi) for +xi = i/10 with i = 0, . . . , 10 [41, Tbl. 5] +and compared with results of C¸ a˘glar et al. [42]. It turned out that for the first six +digits all three approaches and our method yield exactly the same result – a quite +remarkable agreement. Fig. 3 provides plots of the oxygen concentration u(x) +and of its rate of change v(x) = u′(x) for all four different sets of parameters as +obtained by our method. The concentration plot agrees well with the one given +by McElwain [39, Fig. 1] (and confirmed by Hiltmann und Lory [40]). +Figure 3: Numerical solutions of the boundary value problem for the oxygen uptake model for +four different sets of parameters given in Table 2. Left: oxygen concentration u(x). Right: rate +of change of oxygen concentration u′(x). +Hiltmann and Lory [40] report that they used a sophisticated implementation +of a multiple shooting procedure based on four different integrators for initial +value problems together with a special treatment of the singularity using both a +technique of de Hoog and Weiss [43] and a Taylor series method (no further de- +tails are given). They prescribed a tolerance of 10−8 for their Newton solver and +10−10 for the integrator. By contrast, we used a simple shooting method with the +13 + +1.0 +0.9 +8'0 +1 +11 +4 +0.7 +90 +- +0 +02 +0.4 +9'0 +80 +1 +x0.3 +0.2 +1 +4 +ro +0.2 +0.4 +90 +0.8 +1 +0 +1 +xMaple built-in Runge–Kutta–Fehlberg integrator and a hand-coded Steffensen +method for the nonlinear system with a tolerance of 10−7. This comparison again +demonstrates how much simplicity and robustness one gains by using the asso- +ciated vector field for the numerical integration in singular situations. +3.2.2. Lane–Emden Systems +Our approach works for systems in the same manner as for scalar equations, +as one still finds a one-dimensional unstable manifold corresponding to pro- +longed graph of the solution. Thus we restrict to just one example of dimension +d = 3. We now have to integrate the system (17) of dimension n = 2d +1 = 7 for +the above given initial data. We used a Newton method for solving the nonlinear +system arising in the shooting method. Since we had to determine d = 3 initial +conditions via shooting, we had to augment (17) by a linear matrix differential +equation with variable coefficients of dimension 7 × 3. +Campesi et al. [44] proposed a system of coupled Lane–Emden equations as +model for the combustion of ethanol and ethyl acetate over an MnCu catalyst us- +ing a Langmuir–Hinshelwood–Hougen–Watson kinetics. In dimensionless form, +the system is given by (see [45]) +u′′ + 2 +xu′ = +µuu +1 + λuu + λvv + λww , +v′′ + 2 +xv′ = +µvv − µuu +1 + λuu + λvv + λww , +w′′ + 2 +xw′ = +µww +1 + λuu + λvv + λww , +(20) +where u, v, w represent (dimensionless) molar concentrations of ethanol, ac- +etaldehyde and ethyl acetate, respectively. The boundary conditions require that +at x = 0 all first derivatives vanish and that at x = 1 all concentrations are 1. +The authors of [44] used for numerically integrating (20) an approach devel- +oped by essentially the same group [46] based on an integral formulation and +an h-adaptive mesh procedure. Unfortunately, [44] does not provide all the pa- +rameters used in the computations so that it is not possible to compare with their +results. We used instead for our experiments data given in [45] (employing a +modified Adomian decomposition method). However, the plots given there are +not correct, as apparently wrong differential equations were used – at least in the +Matlab code presented in the appendix. We compared with analogous Matlab +computations using the right differential equations and again pdepe as a numeri- +cal solver and obtained an excellent agreement. Figure 4 presents solution curves +for the values µu = 30, µv/w = 0.01, λu = 3 and λv/w = 0.1 used in [45]. +14 + +Figure 4: Numerical solutions of the boundary value problem for the dimensionless model of +the MnCu catalyst. Left: concentrations of ethanol, acetaldehyde and ethyl acetate, respectively. +Right: corresponding rates of change. +4. Thomas–Fermi Equation +4.1. Majorana Transformation +The Thomas–Fermi equation (3) belongs also to the class (10), but with +g(x) = √x , +f(x, u, u′) = +√ +u3 . +(21) +The initial condition u(0) = 1 leads to a rather different situation as for the Lane– +Emden equation: the implicit form of the Thomas–Fermi equation entails that the +only points on R2 which project on x = 0 are of the form ρ = (0, 0, u1, u2) with +arbitrary values u1, u2. Hence no solution satisfying the above initial condition +can be twice differentiable at x = 0. Solutions with a higher regularity exist only +for the initial condition u(0) = 0 which has no physical relevance. +Any point of the form ρ = (0, 1, u1) is an improper impasse point. The vector +field Y defined by (11) does not vanish at such points but takes the form ∂u′ and +it is not Lipschitz continuous there. While Peano’s theorem still asserts the ex- +istence of solutions, we cannot apply the Picard–Lindel¨of theorem to guarantee +uniqueness. We could rescale Y by some function like x which does not change +its trajectories for obtaining an everywhere differentiable vector field ˜Y = xY. +Now all points of the above form are stationary points. But the Jacobian of ˜Y has +0 as a triple eigenvalue at them making it hard to analyse the local phase portrait. +We use therefore a different approach. As Esposito [47] reported only in +2002, Majorana proposed already in 1928 a differential transformation to a new +independent variable t and a new dependent variable v of the form +t = 144−1/6x1/2u1/6 , +v = −(16/3)1/3u−4/3u′ . +(22) +15 + +2 +1.5 +u,V,w +0.5 +0 +0.2 +0.4 +0.6 +0.8 +1 +x2 +u',v',w' +0 +u +1 +-2- +0 +0.2 +0.4 +0.6 +0.8 +1 +xThis at first sight rather miraculous transformation stems from a particular kind +of scaling symmetry [48]. A computation detailed in [47] shows that if it is +applied to any solution of the Thomas–Fermi equation (3), then the transformed +variables satisfy the reduced equation +(1 − t2v)dv +dt = 8(tv2 − 1) . +(23) +The boundary condition (4b), i. e. limx→∞ u(x) = 0, translates into the condition +v(1) = 1.4 We will see below that the thus defined singular initial value problem +for (23) possesses two solutions. Only one of them is also defined for t = 0 and +thus is the physically relevant one. It follows from (22) that the initial slope u′(0) +for the Thomas–Fermi equation is obtained from a solution of (23) by +u′(0) = −(3/16)1/3v(0) . +(24) +The reduced equation (23) is quasi-linear and of first order. Opposed to the +Lane–Emden equations, it is not semi-linear. Thus singular behaviour does not +simply occur at specific t-values. Instead it appears whenever a solution graph +contains a point (t, v) with t2v = 1. Nevertheless, one can apply the same kind of +approach. One first computes a vector field X living on the hypersurface R1 ⊂ J1 +defined by (23) and spanning there the Vessiot distribution. Then one projects X +to the jet bundle J0 and obtains there the vector field +Yred = (t2v − 1)∂t + 8(1 − tv2)∂v . +(25) +As we are now on J0, one-dimensional invariant manifolds of Yred which are +transversal can be directly identified with the graphs of solutions of (23). Our +initial point (1, 1) is a proper impasse point where Yred vanishes. +Fig. 5 shows the phase portrait of the vector field Yred. It has (1, 1) as its only +stationary point. The plot shows in blue some integral curves. Most, but not +all of them can be considered as the graphs of solutions of (23). The plot also +contains in red the t-nullcline given by v = 1/t2 – which is simultaneously the +singular locus of (23) – and in green the v-nullcline given by v = ±1/ √t. The +integral curves that cross the t-nullcline show at the intersection a turning point +behaviour, as the t-component of Yred changes its sign there. If (ti, vi) is such an +4The Majorana transformation is not bijective. A well-known solution of the Thomas–Fermi +equation already given by Thomas [5] is us(x) = 144x−3. It does not satisfy the left boundary +condition, as it is not even defined for x = 0, but the asymptotic condition at infinity. One easily +verifies that any point of the form �x, us(x), u′ +s(x)� is mapped into the point (1, 1). +16 + +intersection point, then it splits the corresponding integral curve into two solution +graphs where both solutions are defined only for t < ti, as they both loose their +differentiability at t = ti. With traditional numerical methods applied to (23), it +would be difficult to determine these solutions; as integral curves of Yred they are +trivial to obtain numerically. +Figure 5: Phase portrait of the vector field +associated to the reduced system (23). The +unstable manifold is shown in cyan, the sta- +ble manifold in magenta. +The Jacobian of Yred at the stationary +point (1, 1) is the matrix J = � −2 −1 +8 +16 +� with +eigenvalues −7± +√ +73 ≈ (1.544, −15.544). +Thus we are dealing with a saddle point. +The unstable and the stable manifold +shown in Fig. 5 in cyan and magenta, +resp., correspond to the above mentioned +two solutions of the initial value problem +with v(1) = 1. There cannot exist any ad- +ditional solutions, as there are no further +invariant manifolds entering or leaving the +saddle point. One sees that in the positive +quadrant the stable manifold cannot cross +the nullclines outside of the saddle point +and hence can never reach the v-axis. +Thus we may conclude that the part of +the unstable manifold between the v-axis +and the stationary point corresponds to the +unique solution u∞ of the boundary value problem with the condition (4b). The +abscissa of the intersection of the unstable manifold with the v-axis determines +via (24) the critical initial slope ω (see below for numerical values). The ex- +istence of such a unique solution for this specific boundary value problem was +proven in 1929 by Mambriani [49] (see also the discussion in [11]). +It will turn out that the integral curves to the right of the stable manifold have +no relevance for our problem. The integral curves to the left of it and above the +unstable manifold correspond to solutions of the boundary value problem with +the condition (4c), i. e. solutions with a zero, whereas the integral curves below +the stable manifold lead to solutions for (4a). This can be deduced from their +intersections with the v-axis and (24). +Much of the literature on numerically solving the Thomas–Fermi equation +is concerned with the solution u∞ of (4b) defined on the semi-infinite interval +[0, ∞) and concentrates on the determination of the critical slope ω. Most so- +lutions reported in the literature are either shown only on rather small intervals +17 + +1 +个 +个 +个 +个 +←↑ +← +-→ +→ +个 +→ +→ +→ +个 +个 +3 +→ +→ +V +→ +1 +2 +→ +→ +→ +→ +→ +→ +↑ +1 +1 +→ +↑ +↑ +→ +T +T +T +T +1[0, x0] with typically x0 < 10 or clearly deteriorate for larger x. One reason +for this effect is surely that many approaches are based on some kind of series +expansion. Another, more intrinsic reason becomes apparent from the phase por- +trait in Figure 5. As the sought solution corresponds to a branch of the unstable +manifold of the saddle point (1, 1), even small errors close to the saddle point +(corresponding to points with large x coordinates) are amplified by the dynamics +and the numerical solutions tend to diverge from a finite limit. +By contrast, our approach to determine u∞ leads to the standard problem +of determining a branch of the unstable manifold of a stationary point – a task +which can be performed numerically very robustly and efficiently. As the posi- +tive eigenvalue has about the tenfold magnitude of the negative one, trajectories +approach the unstable manifold very fast which ensures a high accuracy. +Following Majorana, Esposito [47] (and subsequent authors) determines a +series solution of the initial value problem v(1) = 1 for (23). In the first step, +one obtains a quadratic equation with two solutions. Esposito then argues that +one should take the smaller solution, as this was a perturbation calculation which +is not a convincing argument. The reduced initial value problem has two solu- +tions. As one can see in Figure 5, the second solution corresponding to the stable +manifold grows very rapidly. Therefore it is not surprising that several authors +suspected that the second solution of the quadratic equation leads to a divergent +power series and thus could be discarded. However, a second solution to the +initial value problem does exist, although it seems that it cannot be determined +with a power series ansatz. But as already discussed above, u∞ is nevertheless +unique and corresponds to the unstable manifold. +For the series solution, one expands around t = 1 and makes the ansatz v(t) = +�∞ +i=0 ai(1 − t)i. The initial condition yields a0 = 1 and for the arising quadratic +equation for a1 one chooses the root5 a1 = 9 − +√ +73 ≈ 0.456. After lengthy +computations sketched in [47], one obtains the following recursive expression +for the remaining coefficients with i > 1: +ai = +1 +2(i + 8) − (i − 1)a1 +� +(i + 6)a1ai−2 + +� +(i + 7) − 2(i + 3)a1 +� +ai−1 + +i−2 +� +j=1 +� +(j + 1)aj+1 − 2( j + 4)aj + ( j + 7)aj−1 +� +ai− j +� +. +(26) +5This value is related to the spectrum of the Jacobian of the vector field Yred: −a1 is the slope +of the tangent space of the unstable manifold at the saddle point. This is not surprising, as the +tangent space is the linear approximation of the solution. +18 + +Setting t = 0 yields for the critical slope the series representation +ω = − +� 3 +16 +�1/3 +∞ +� +i=0 +ai , +(27) +which can be evaluated to arbitrary precision. +To obtain whole solutions u(x), one must be able to transform back from the +variables (t, v) to the original variables (x, u). Esposito [47] exhibited a conve- +nient method for this. We express the solution in parametric form using t as +parameter: x = x(t) and u = u(t). Then we make the ansatz +u(t) = exp +�� t +0 +w(τ)dτ +� +(28) +with w a yet to be determined function. Assuming x(t = 0) = 0, this ansatz au- +tomatically satisfies the initial condition u(x = 0) = 1. Using the transformation +(22), one can show that w(t) = +6tv(t) +t2v(t)−1 and that x(t) can be expressed via w(t) as +x(t) = 1441/3t2 exp +� +−1 +3 +� t +0 +w(τ)dτ +� +(29) +(which shows that indeed x(0) = 0). Esposito [47] proposed to enter the above +determined series solution for v(t) into these formulae and to compute this way +a series expansion of u∞. This requires essentially one quadrature. +4.2. Numerical Results +We refrain from citing the many papers written on computing u∞ and in par- +ticular ω and instead refer only to [50, 51] both listing a large number of ap- +proaches with references. We emphasise again that our main point is to show +that the geometric theory allows us – here in combination with the Majorana +transformation – to translate a singular problem into basic tasks from the theory +of dynamical systems which can be easily solved by standard methods. +4.2.1. The “Critical” Solution u∞ and the Critical Slope ω +We consider first the problem of only determining the initial slope ω belong- +ing to the solution u∞ for (4b). With classical approaches, this is a non-trivial +problem and in the literature one often finds values with a very low number of +correct digits. Using our geometric approach, we can determine ω to (almost) +19 + +any desired precision in about 10 lines of Maple code. We write the dynamical +system corresponding to the vector field Yred defined by (25) as +dt +ds = t2v − 1 , +dv +ds = 8(1 − tv2) , +(30) +i. e. we determine integral curves of Yred in parametric form �t(s), v(s)�. As dis- +cussed above, the sought trajectory corresponds to the unstable manifold of the +saddle point (1, 1). An eigenvector for the positive eigenvalue λ = −7 + +√ +73 is +given by e = �1, −9 + +√ +73�T and we denote by ˆe = (e1, e2)T the corresponding +normalised vector. Then we choose as initial point for a numerical integration +t(0) = 1 + ϵe1 and v(0) = 1 + ϵe2 with ϵ > 0 some small number (we typically +used 10−3 or 10−4, but this had no effect on the obtained slope) and integrated +until t(s) = 0 for s = s0. Finally, we obtain ω from v(s0) via (24). +We control the precision with an integer parameter N specifying that the +numerical integration of (30) should take place with an absolute and relative +error of 10−N and that for this purpose Maple should compute with N + 5 digits. +In a recent work, Fern´andez and Garcia [51] determined ω based on the first +5000 terms of the Majorana series (27) to a precision of several hundred digits. +This is by far the best approximation available and our reference solution. +tolerance +rel. error +time +10−5 +3.2 · 10−6 +0.6 +10−10 +7.3 · 10−12 +0.6 +10−15 +5.5 · 10−17 +2.7 +10−20 +5.3 · 10−22 +22.7 +10−25 +5.5 · 10−27 +231.5 +Table 3: Relative error and computation +time in seconds for different tolerances. +Our numerical results are summarised +in Table 3. Our relative error is always +smaller than the prescribed tolerance. For +smaller tolerances, the computational ef- +fort is rapidly increasing and on a laptop +we needed for 25 digits less than 4 min- +utes. We made no effort to optimise the +computations. For example, we are using +the default integration method of Maple +(a Runge–Kutta–Fehlberg method of or- +der 4/5 with a degree four interpolant), al- +though a higher order scheme would probably be more efficient (Maple offers +such schemes – but not in combination with the automated root finding used in +our code). Nevertheless, one may conclude that for practically relevant preci- +sions, our geometric approach combined with the Majorana transformation pro- +vides very accurate results fast and almost effortless. +Fern´andez and Garcia [51] analyse also the convergence rate of the Majorana +series (27) and consider it as fast (see also the comments by Esposito [47]). We +compared for a relative small accuracy, Maple hardware floats with 10 digits, the +value for the initial slope obtained with our approach with the approximations +20 + +terms +10 +20 +30 +40 +50 +rel. err. +5.8 · 10−2 +6.7 · 10−3 +8.3 · 10−4 +1.1 · 10−4 +1.4 · 10−5 +terms +60 +70 +80 +90 +100 +rel. err. +1.9 · 10−6 +2.7 · 10−7 +3.7 · 10−8 +4.4 · 10−9 +0 +Table 4: Relative error for different truncation degrees of the Majorana series. +delivered by various truncations of the series. Somewhat surprisingly, our ap- +proach gets all 10 digits right, despite the considerably higher tolerances (10−6) +used by the integrator. Table 4 contains the approximations obtained by evalu- +ating the first N terms of the Majorana series (27). One needs 100 terms for a +similarly accurate result. On average, one needs 10 more terms for one additional +digit corresponding to a linear convergence as already theoretically predicted in +[47, 51]. This observation also roughly agrees with the fact that Fern´andez and +Garcia used 5000 terms for obtaining about 500 digits [51]. +For determining the whole solution u∞(x) instead of only the critical slope +ω = u′ +∞(0), we have to perform a transformation back from the variables (t, v) to +(x, u). We described above Esposito’s approach for this. For a purely numerical +computation instead of series expansions, we modify it in a way which fits nicely +into our approach. We introduce as Esposito [47] the function +I(t) = +� t +0 +τv(τ) +1 − τ2v(τ)dτ . +(31) +We then express I(t) as a function of the parameter s which we use to parametrise +solution curves. If s0 is the (first) parameter value satisfying t(s0) = 0, then an +elementary application of the substitution rule yields +I(s) = − +� s +s0 +t(σ)v(σ)dσ , +(32) +which immediately implies that I satisfies the differential equation dI +ds = −tv +by which we augment the system (30). We thus obtain a free boundary value +problem for the augmented system, as the function I(s) satisfies the condition +I(s0) = 0 with the a priori unknown value s0. As usual, we consider s0 as +an additional unknown function and introduce the rescaled independent variable +σ = s/s0. Then we finally obtain the following two-point boundary value prob- +21 + +lem with non-separated boundary conditions +dt +dσ = s0(t2v − 1) , +t(0) = 1 + ϵe1 , +t(1) = 0 , +dv +dσ = 8s0(1 − tv2) , +v(0) = 1 + ϵe2 +dI +dσ = −s0tv , +I(1) = 0 , +ds0 +dσ = 0 . +(33) +Once this boundary value problem is solved, (28) and (29) imply that parametri- +sations of the graph of u∞(x) are given by +x(σ) = 1441/3t(σ)2 exp �2I(σ)� , +u(σ) = exp �−6I(σ)� . +(34) +Figure 6: Comparison of values obtained +via (34) and Majorana’s series for different +numbers N of terms. +We implemented this approach in +Maple using the built-in solver for bound- +ary value problems which could handle +(33) without problems. We compared the +results with solutions obtained via Majo- +rana’s series, i. e. following Esposito [47], +we entered a given number N of terms into +the integral defining I and performed a +numerical integration. Fig. 6 shows on a +logarithmic scale the absolute difference +between our curve �x(σ), u(σ)� and the +curves computed via the series for differ- +ent values of N. Obviously, our results are +in an excellent agreement with the series +solutions. The fact that all error curves +have their maximum close to x = 0 is easy +to explain. As the expansion point of the +series corresponds to x = ∞ (i. e. t = 1), the series solutions become less accu- +rate the closer one gets to x = 0; at x = 0 of course no error occurs, as this value +is fixed by an initial condition. We did not make an extensive comparison of +computation times. But plotting the series solution for N = 10 over the interval +[0, 5] required more than 10 times as much computation time than solving above +boundary value problem demonstrating again the efficiency of our approach. +22 + +10-5 +err +10-6 +10~7 +10°8 +0 +m +x +N=10 +N=20 +N=30We mentioned already above that in the literature results are often presented +only for rather small values of x, although the solution is defined for all non- +negative real numbers. One exception is Amore et al. [52, Tbl. 3/4] who used +a Pad´e–Hankel method and asymptotic expansions to present highly accurate +values of the solution u∞(x) and its first derivative u′ +∞(x) up to x = 400. +x +u∞(x) +u′ +∞(x) +0 +1 +−1.58807101687867 +10 +0.0243142929534589 +−0.00460288186903816 +50 +0.000632254782228818 +−0.0000324989019998445 +100 +0.000100242568239745 +−2.73935106365787 · 10−6 +150 +0.0000326339644201454 +−6.09139947257267 · 10−7 +200 +0.0000145018034835377 +−2.05753231409599 · 10−7 +250 +7.67729076668264 · 10−6 +−8.78946798702223 · 10−8 +300 +4.54857195240339 · 10−6 +−4.36594961733055 · 10−8 +350 +2.91510210708972 · 10−6 +−2.40920109677041 · 10−8 +400 +1.97973262954641 · 10−6 +−1.43668230750324 · 10−8 +Table 5: Solution values u∞(x) and derivative values u′ +∞(x) for large x. +Table 5 contains similar values obtained with our approach. For determining +the values of u′ +∞(x), we must augment (34) by an equation for u′(σ), i. e. we +must extend the parametrisation to the prolonged graph. By a straightforward +application of the chain rule, one obtains +u′(σ) = −3 · 144−1/3v(σ) exp �−8I(σ)� . +(35) +To compile such a table, one must then determine for each x the corresponding +value of the parameter σ via the solution of a nonlinear equation. Nevertheless, +the complete computation of the values at the ten points contained in the table +required only about 0.1 seconds. Amore et al. [52] claim that in their tables all +digits are correct. Assuming that this is indeed the case, we can conclude that +we obtained with minimal effort for each value of x at least eight correct digits +for u∞(x) and seven correct digits for u′ +∞(x). Given the settings for the tolerances +of our integrator and the use of hardware floats with only 10 digits, these results +demonstrate again a very remarkable precision and efficiency of our approach. +As large values of x correspond to small values of σ and thus to values of t close +to 1, one may have to choose a smaller value of ϵ for very large values of x. The +largest value appearing in above table, x = 400, corresponds to σ ≈ 0.35 and +t ≈ 0.9789. We chose for our numerical calculation the value ϵ = 10−3 and thus +23 + +used as right end of the approximated unstable manifold instead of the saddle +point (1, 1) the point (t1, v1) ≈ (0.9978, 1.001). For x = 400, one may say that +we are still sufficiently far away from this point, but for larger values of x one +should probably start working with a smaller value of ϵ which will increase the +computation time, as the dynamics is very slow so close to a stationary point. +4.2.2. Other Solutions +So far, we only considered the particular solution u∞ (which has attracted the +most attention in the literature). In Fig. 5 we presented the phase portrait for the +Majorana transformed Thomas–Fermi equation. Using a slight modification (and +simplification) of the above described backtransformation via the solution of an +extended differential system, we can also obtain a “phase portrait” of the original +Thomas–Fermi equation, i. e. we compute solutions for different values of the +initial slope u′(0) keeping the initial condition u(0) = 1. While the Majorana +transformation itself is valid for any solution of the Thomas–Fermi equation, our +ansatz for the back transformation has encoded this second initial condition (one +could easily adapt to a different value u(0) = c by multiplying (28) with the +constant c). According to (24), each value of u′(0) corresponds uniquely to a +value of v(0). We now take the vector field −Yred and use a parametrisation such +that s = 0 corresponds to t = 0 (and thus also x = 0). This leads to the following +augmented initial value problem: +dt +ds = 1 − t2v , +dv +ds = 8(tv2 − 1) , +dI +ds = tv , +t(0) = 0 , +v(0) = v0 , +I(0) = 0 . +(36) +Its solutions are then transformed into x- and u-coordinates via (34). +Fig. 7 shows that the solution u∞ vanishing at infinity acts as a kind of “sep- +aratrix”. The solutions above it, i. e. with an initial slope higher than ω, pass +through a minimum and then grow faster than exponentially (note the logarith- +mic scale). The solutions below it approach rapidly zero, reaching it at a finite +value of x (recall that the separatrix reaches zero at infinity). It turns out that +around the critical value ω, the trajectories are rather sensitive with respect to +the initial slope. For some of the curves shown in Fig. 7, u′(0) differs only in +the fifth or sixth digit. For the curves approaching zero, it is also non-trivial +to determine the exact location of the zero, as here v goes towards infinity. In +our computations, we actually integrated only until some threshold like 10−8. +Probably a “hybrid” approach using (36) only to get away from the singularity +at x = 0 and applying afterwards a standard integrator to the Thomas–Fermi +equation would be a good alternative. +24 + +Figure 7: Solutions of the Thomas–Fermi +equation with u(0) = 1 and different u′(0) +using a logarithmic scale for u. The curve +in magenta shows u∞. +For solving concrete boundary value +problems with boundary conditions of the +form (4a) or (4c), resp., for given values of +a or b, resp., one can use an adapted ver- +sion of a shooting method. Starting with +an initial guess v0 for the unknown value +of v(0) for the sought solution, one inte- +grates the initial value problem (36) un- +til a condition of the desired form is sat- +isfied. However, in general, the condition +will be satisfied at a wrong position a∗ or +b∗, resp. Using a bisection, one modifies +v0 until one is sufficiently close to the ac- +tually prescribed values. As in both cases, +Fig. 7 shows that there is a monotone re- +lation between v0 and a∗ or b∗, resp., it is +always clear in which direction one has to +change v0. But for larger values of a or b, one gets again into areas where very +small changes in v0 lead to significant changes in a∗ or b∗, resp. Despite this +sensitivity, the approach worked in tests very well for a ≤ 27 and b ≤ 30. +5. Conclusions +The Lane–Emden and the Thomas–Fermi equation are prototypical exam- +ples for ordinary differential equations with singularities. Their singularities are +determined by a specific value of the independent variable: x = 0. Any initial or +boundary value problem with conditions prescribed at x = 0 cannot be tackled +by standard methods and this concerns both theoretical and numerical studies. +The Lane–Emden equations fit into the framework of so-called Fuchsian +equations (see e. g. [53]), i. e. equations of the form Lu = f(x, u) where L is +a linear differential operator of Fuchsian type and where only the right hand side +may contain nonlinear terms. For the theoretical treatment of such equations, +some form of quasilinearisation is often fruitful, as it allows to use the far devel- +oped theory of the linear counterpart Lu = ˜f(x). For example, the existence and +uniqueness proof for boundary value problems for (generalised) Lane–Emden +equations given in [29] follows such strategy. For the numerical integration, [54] +presents methods for first- and second-order systems of this particular form. +A key consequence of this special structure is the above mentioned loca- +tion of the singularities depending only on x which facilitates the design of spe- +25 + +10 +100 +102 +10-4 +10-6. +10-8 +10 +20 +30 +xcialised numerical methods. Therefore it is not surprising that so many different +techniques have been proposed in the literature. Our approach is independent +of such a special form, as one can see from our treatment of the Thomas–Fermi +equation based on the reduced equation (23). The location of its singularities +depends on t and v making an integration with standard numerical methods more +difficult. By contrast, our approach can handle all forms of quasilinear problems. +In some computations related to the Thomas–Fermi equations, we encoun- +tered problems, for example when computing u∞(x) for very large values of x or +when u(x) approaches zero. In the first case, the reason lies in an often highly +nonlinear relationship between the variable t used in the reduced system and the +variable x where “microscopic” changes in t may correspond to huge differences +in x. In the second case, u can approach 0 only when v tends towards infinity. +In both cases, one could probably extend the applicability of our method by a +rescaling of the reduced equation. For computing u∞ for large x, an alternative, +semianalytic approach would consist of determining a higher order approxima- +tion of the unstable manifold close to the saddle point (1, 1) – in fact, the Majo- +rana series is nothing else than such an approximation. This could lead to very +accurate values even for extremely large values of x. +One may wonder why we used in the case of the Lane–Emden equations the +shooting method for boundary value problems and not also a formulation as free +boundary value problem as for the Thomas–Fermi equation. In both cases, one +faces the problem that at one boundary one has to deal with a two-dimensional +plane of stationary points and that the boundary conditions enforces that one end +point of the solution trajectory lies on this plane. In the case of the Thomas– +Fermi equation, we resolved this problem by moving a bit in the direction of +the unstable eigenspace. This was possible, as this direction is the same for all +points on the plane. In the case of the Lane–Emden equations, the direction of +the unstable eigenspace depends on the value u(0) and thus differs for different +points on the plane. Probably one could adapt typical approaches to boundary +value problems like collocation methods to this dependency. But as our emphasis +in this paper lies on the use of standard methods, we refrained from studying this +possibility in more details. 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Comp. 232 (2014) 929–943. +[53] S. Kichenassamy, Fuchsian Reduction, Progress in Nonlinear Differential Equations and +Their Applications 71, Birkh¨auser, Boston, 2007. +[54] O. Koch, P. Kofler, E. Weinm¨uller, Initial value problems for systems of ordinary first and +second order differential equations with a singularity of the first kind, Analysis 21 (2001) +373–389. +29 + diff --git a/FtAzT4oBgHgl3EQfHPtI/content/tmp_files/load_file.txt b/FtAzT4oBgHgl3EQfHPtI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..42744a211bc1b1ecdf0d9c26b2949f1106678cd7 --- /dev/null +++ b/FtAzT4oBgHgl3EQfHPtI/content/tmp_files/load_file.txt @@ -0,0 +1,964 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf,len=963 +page_content='On the Numerical Integration of Singular Initial and Boundary Value Problems for Generalised Lane–Emden and Thomas–Fermi Equations Werner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Seilera, Matthias Seißa aInstitut f¨ur Mathematik, Universit¨at Kassel, 34132 Kassel, Germany Abstract We propose a geometric approach for the numerical integration of singular initial value problems for (systems of) quasi-linear differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It transforms the original problem into the problem of computing the unstable manifold at a stationary point of an associated vector field and thus into one which can be solved in an efficient and robust manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Using the shooting method, our ap- proach also works well for boundary value problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As examples, we treat some (generalised) Lane–Emden equations and the Thomas–Fermi equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Keywords: singular initial value problems, singular boundary value problems, Vessiot distribution, unstable manifold, numerical integration, Lane–Emden equation, Thomas–Fermi equation, Majorana transformation 2010 MSC: 34A09, 34A26, 34B16, 65L05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Introduction The Lane–Emden equation was originally derived in astrophysics [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 40] and represents a dimensionless form of Poisson’s equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid (see [2–4] and references therein for a more detailed discussion): u′′ + 2 xu′ = −un (1) Email addresses: seiler@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='uni-kassel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='de (Werner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Seiler), mseiss@mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='uni-kassel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='de (Matthias Seiß) URL: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='mathematik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='uni-kassel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='de/~seiler (Werner M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Seiler) Preprint submitted to Elsevier January 4, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='01041v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='NA] 3 Jan 2023 with n the polytropic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Astrophysicists want to solve the initial value prob- lem u(0) = 1 and u′(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (1) is prototypical for ordinary differential equations arising in the construction of radially symmetric steady state solutions of reaction-diffusion equations, as the left hand side of (1) represents the Laplace operator in spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In an N-dimensional space, the numerator 2 has to be replaced by N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This leads to generalised Lane–Emden equations u′′ + N − 1 x u′ = h(x, u) , (2) where h represents the reaction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Besides the classical form from astro- physics, we will later consider examples arising in chemical engineering (biocat- alysts) and in physiology (oxygen uptake of cells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' There, one needs the solution of boundary value problems with u′(0) = 0 and αu(1) + βu′(1) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thomas [5] and Fermi [6] derived independently of each other in a statistical model of atoms treating electrons as a gas of particles a Lane–Emden equation (1) with polytropic index n = 3/2 for the electrostatic potential V(x), however with the “initial condition” that V(x) behaves like 1/x for x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Writing V(x) = u(x)/x, one obtains the Thomas–Fermi equation u′′ = � u3/x (3) together with the initial condition u(0) = 1 (see [7–9] for more physical and historical details and [10, 11] for a mathematical analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In addition, one imposes one of the following three types of boundary conditions: bu′(b) − u(b) = 0 , (4a) lim x→∞ u(x) = 0 , (4b) u(a) = 0 (4c) with 0 < a, b < ∞ given positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The infinite case (4b) occurs only for a crit- ical value ω ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='588 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' of the initial slope u′(0) and represents physically an isolated neutral atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For larger initial slopes, one can prescribe the boundary condition (4a) and obtains solutions going through a minimum and then growing rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Physically, such solutions are relevant for certain crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The bound- ary condition (4c) leads to solutions with a smaller initial slope and represent physically ions with radius a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Numerical methods from textbooks cannot be directly applied here, as all considered equations are singular at x = 0 and at least one initial/boundary con- dition is imposed there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the vast literature on the numerical integration of 2 Lane–Emden or Thomas–Fermi equations, three different types of approaches prevail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Astrophysicists apply for initial value problems a very simple approach: they use for the first step a series expansion of the solution to get away from the singularity and then use some standard integrator [3, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2] (see also [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For boundary value problems, collocation methods are popular, as they are easily adapted to the singularity, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Finally, various kinds of semi-analytic expansions like Adomian decomposition have been adapted to the singularity (see the references given below and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We propose here a new and rather different alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the geometric theory of differential equations [14, 15], one associates with any implicit ordi- nary differential equation a vector field on a higher-dimensional space such that the graphs of prolonged solutions of the implicit equation are integral curves of this vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Most of the literature on singularity theory is concerned with fully implicit equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' However, in applications quasi-linear equations like the Lane–Emden equations prevail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In [16, 17], we showed that such equations possess a special geometry allowing us to work in a lower order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Singulari- ties, now called impasse points, are typically stationary points of the associated vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' If there is a unique solution, its prolonged solution graph is the one- dimensional unstable manifold of this stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Such an unstable man- ifold can numerically be computed very robustly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In [18], we already sketched this possibility to exploit ideas from singularity theory for numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Here, we want to demonstrate for concrete problems of practical relevance that it is easy to apply and efficiently provides accurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the next section, we recall the neces- sary elements of the geometric theory of differential equations and how one can translate an implicit problem into an explicit one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Section 3 is then devoted to the application of these ideas to (generalised) Lane–Emden equations and to the numerical solution of some concrete problems from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In Section 4 we discuss the Thomas–Fermi equation by first reducing it via a transformation introduced by Majorana and then applying the geometric theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We compare the obtained numerical results with some high precision calculations from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Finally, some conclusions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Geometric Theory of Ordinary Differential Equations We use a differential geometric approach to differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It is be- yond the scope of this article to provide deeper explanations of it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' for this we refer to [19] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For notational simplicity, we concentrate on the scalar case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' the extension to systems will be briefly discussed at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 3 Similarly, we restrict here to second-order equations, but equations of arbitrary order can be treated in an analogous manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We consider a fully implicit differential equation of the form F(x, u, u′, u′′) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (5) In the second-order jet bundle J2 (intuitively expressed, this is simply a four- dimensional affine space with coordinates called x, u, u′, u′′), this equation de- fines a hypersurface R2 ⊂ J2 which represents our geometric model of the dif- ferential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We will assume throughout that R2 is actually a submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Given a function ψ(x), we may consider its graph as a curve in the jet bundle J0 of order zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' the x-u space, given by the map x �→ �x, ψ(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Assuming that ψ is at least twice differentiable, we can prolong this curve to a curve in J2 defined by the map x �→ �x, ψ(x), ψ′(x), ψ′′(x)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The function ψ is a solution of (5), if and only if this curve lies completely in the hypersurface R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In an initial value problem for the implicit equation (5), one prescribes a point ρ = (y, u0, u1, u2) ∈ R2 and asks for solutions such that ρ lies on their prolonged graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Note that opposed to explicit problems, we must also specify the value u2, as the algebraic equation F(y, u0, u1, u′′) = 0 may have several (possibly infinitely many) solutions and thus may not uniquely determine u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A key ingredient of the geometry of jet bundles is the contact structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the case of J2, the contact distribution C(2) is spanned by the two vector fields Ctrans = ∂x + u′∂u + u′′∂u′ , Cvert = ∂u′′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (6) A curve x �→ �x, ψ0(x), ψ1(x), ψ2(x)� in J2 is a prolonged graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' ψ1 = ψ′ 0 and ψ2 = ψ′′ 0 ), if and only if all its tangent vectors lie in the contact distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The Vessiot distribution V[R2] of (5) is that part of the tangent space of R2 which also lies in the contact distribution C(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Writing X = aCtrans + bCvert for a general vector in the contact distribution, X lies in the Vessiot distribution, if and only if its coefficients a, b satisfy the linear equation �Fx + u′Fu + u′′Fu′�a + Fu′′b = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (7) A singularity is a point ρ = (y, u0, u1, u2) ∈ R2 such that Fu′′(ρ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One speaks of a regular singularity, if the coefficient of a in (7) does not vanish at ρ, and of an irregular singularity, if it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Outside of irregular singularities, the Vessiot distribution is one-dimensional and locally spanned by the vector field X = Fu′′Ctrans − �Fx + u′Fu + u′′Fu′�Cvert (8) 4 (note that X is defined only on the submanifold R2 ⊂ J2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The prolonged graph of any solution of (5) must be integral curves of this vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The converse is not necessarily true in the presence of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' At regular singularities, the vector field X becomes vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Generically, only one-sided solutions exist at such points and if two-sided solutions exist, then their third derivative will blow up [20, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' At irregular singularities, typically several (possibly infinitely many) solutions exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In [21] it is shown how for arbitrary systems of ordinary or partial differential equations with polynomial nonlinearities all singularities can be automatically detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Irregular singularities are stationary points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Prolonged solution graphs through them are one-dimensional invariant manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Any one-dimensional (un)stable or centre manifold (with transversal tangent vectors) at such a station- ary point defines a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For higher-dimensional invariant manifolds, one must study the induced dynamics on them to identify solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In any case, we note that the numerical determination of invariant manifolds at stationary points is a well-studied topic – see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In general, the direct numerical integration of (5) faces some problems, if it is not possible to solve (uniquely) for u′′, and typically breaks down, if one gets too close to a singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The geometric theory offers here as alternative the numerical integration of the dynamical system defined by the vector field X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thus an implicit problem is transformed into an explicit one!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The price one has to pay is an increase of the dimension: while (5) is a scalar equation (but second-order), the vector field X lives on the three-dimensional manifold R2 in the four-dimensional jet bundle J2 (more generally, a scalar equation of order q leads to a vector field on a (q − 1)-dimensional manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The key difference is, however, that we obtain a parametric solution repre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We work now with the explicit autonomous system1 dx ds = Fu′′ , du ds = u′Fu′′ , du′ ds = u′′Fu′′ , du′′ ds = −Fx − u′Fu − u′′Fu′ , (9) where s is some auxiliary variable used to parametrise the integral curves of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A solution of it will thus be a curve s �→ �x(s), u(s), u′(s), u′′(s)� on R2 ⊂ J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A numerical integration will provide a discrete approximation of this curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 1Strictly speaking, we are dealing here with a three-dimensional system, as X lives on the three-dimensional manifold R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As we do not know a parametrisation of R2, we must work with all four coordinates of J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One could augment (9) by its first integral F(x, u, u′, u′′) = 0 and enforce it during a numerical integration, but in our experience this is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 5 In applications, quasi-linear equations prevail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We restrict here even to semi- linear differential equations of the form F(x, u, u′, u′′) = g(x)u′′ − f(x, u, u′) = 0 , (10) with smooth functions f, g, as both the Lane–Emden and the Thomas–Fermi equation can be brought into this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A point (y, u0, u1, u2) ∈ R2 is then a singularity, if and only if g(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As first shown in [16] and later discussed in more details in [17], quasi-linear equations possess their own special geometry, as it is possible to project the Vessiot distribution to the jet bundle of one order less, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' in our case to the first-order jet bundle J1 with coordinates (x, u, u′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Projecting the vector field X defined by (8) with F as in (10) to J1 yields the vector field Y = g(x)∂x + g(x)u′∂u + f(x, u, u′)∂u′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (11) It is only defined on the canonical projection of R2 to J1 which may be a proper subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Assuming that f, g are defined everywhere on J1, we analytically extend Y to all of J1 and replace (9) by the three-dimensional system dx ds = g(x) , du ds = g(x)u′ , du′ ds = f(x, u, u′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (12) The first equation is decoupled and can be interpreted as describing a change of the independent variable, but we will not pursue this point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A point ρ = (y, u0, u1) ∈ J1 is an impasse point for (10), if the vector field Y is not transversal at ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' if its x-component vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Here, this is equivalent to g(y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We call ρ a proper impasse point, if R2 contains points which project on ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' otherwise, ρ is improper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Here, proper impasse points are obviously stationary points of Y or (12), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Prolonged graphs of solutions of (10) are one- dimensional invariant manifolds of Y (or (12), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=') and again the converse is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In [17], we proved geometrically the following result (a classical analytic proof for the special case g(x) = x can be found in [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Consider (10) for f, g smooth together with the initial conditions u(y) = u0 and u′(y) = u1 where g(y) = 0 and f(y, u0, u1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' If δ = g′(y) and γ = fu′(y, u0, u1) are both non zero and of opposite sign, then the initial value problem possesses a unique smooth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Under the made assumptions, the initial point ρ = (y, u0, u1) is a proper im- passe point of (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One readily verifies that the Jacobian J of Y at ρ has the eigenvalues δ, 0 and γ and thus we find three one-dimensional invariant man- ifolds at ρ: the stable, the unstable and the centre manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 Without loss of 2The centre manifold is here unique, as there exists a whole curve of stationary points [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 6 generality, we assume that δ > 0 (otherwise we multiply (10) by −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It is then shown in [17] that the prolonged graph of the unique solution is the unstable manifold and thus at ρ it is tangent to the eigenvector of J for δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The extension to implicit systems F(x, u, u′, u′′) = 0 is straightfor- ward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Assuming that the unknown function u is vector valued, u: I ⊆ R → Rn, the jet bundle J2 is (3n + 1)-dimensional and the contact distribution C(2) is gen- erated by the n + 1 vector fields Ctrans = ∂x + u′ · ∂u + u′′ · ∂u′ and Cvert = ∂u′′, where the dot denotes the standard scalar product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Again the Vessiot distribution is generically one-dimensional and the coefficients of a vector field X spanning it are readily determined by solving a linear system of equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Numerical integration of X allows us to approximate solutions of the given system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We restrict to semi-linear first-order systems of the form g(x)u′ = f(x, u) with g still a scalar functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For initial conditions u(y) = u0 with g(y) = 0 and f(y, u0) = 0, we introduce δ = g′(y) (assuming δ > 0) and the Jacobian Γ = fu(y, u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In [26], it is shown that if all eigenvalues of Γ have a negative real part, then the initial value problem has a unique smooth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A classical an- alytical proof was given by Vainikko by first studying extensively the linear case [27] and then extending to the nonlinear one [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the geometric approach, one sees again that the graph of the solution is a one-dimensional unstable manifold of the vector field Y spanning the projected Vessiot distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (Generalised) Lane–Emden Equations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Geometric Treatment If we consider the generalised Lane–Emden equation (2), then one obtains after multiplication by x the special case of (10) given by g(x) = x , f(x, u, u′) = xh(x, u) − (N − 1)u′ , (13) where we always assume N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For arbitrary initial conditions u(0) = u0 and u′(0) = u1, we find that δ = 1 and γ = −(N −1) are nonzero and of opposite sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The initial point ρ = (0, u0, u1) is a proper impasse point, if and only if u1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In this case, Theorem 1 asserts the existence of a unique smooth solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The projected Vessiot distribution is spanned by the vector field Y = x∂x + xu′∂u + �xh(x, u) − (N − 1)u′�∂u′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (14) For u1 � 0, no solution can exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Indeed, the vector field Y has then no stationary point and the unique trajectory through the initial point ρ = (0, u0, u1) is the vertical line s �→ (0, u0, u1 + s) which does not define a prolonged graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 7 We thus assume u1 = 0, which unsurprisingly is the case in all applications of (2) in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Independent of the value of u0, the initial point ρ = (0, u0, 0) is a stationary point of the vector field Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The Jacobian of Y at ρ is J = ���������� 1 0 0 0 0 0 h(0, u0) 0 −(N − 1) ���������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (15) Its eigenvalues are 1, 0 and −(N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Relevant for us is only the eigenvector to the eigenvalue 1, as it is tangential to the unstable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It is given by v = �N, 0, h(0, u0)�T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For the numerical solution of our given initial value problem, we integrate the vector field Y for the initial data �x(0), u(0), u′(0)�T = �0, u0, 0�T + ϵv with some small ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The concrete value of ϵ is not very relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As the exact solution corresponds to the unstable manifold, any error is automatically damped by the dynamics of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In our experiments, we typically used ϵ = 10−3 or ϵ = 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We can easily extend this approach to coupled systems of the form u′′ + N − 1 x u′ = h(x, u) , (16) where u is a vector valued function and the coupling occurs solely through the reaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' If u is a d-dimensional vector, then the dimension of the first- order jet bundle J1 is 2d + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thus (12) becomes a system of this dimension: dx ds = x , du ds = xu′ , du′ ds = xh(x, u) − (N − 1)u′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (17) By the same arguments as in the scalar case, we restrict to the initial condition u′(0) = 0 so that the initial point ρ = (0, u0, 0) is again a proper impasse point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The Jacobian at ρ is a block form of (15): J = ���������� 1 0T 0T 0 0d 0d h(0, u0) 0d −(N − 1)Ed ���������� , (18) where 0d and Ed denote the d × d zero and unit matrix, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We still have 1 as a simple eigenvalue, whereas the eigenvalues 0 and −(N − 1) have both the al- gebraic multiplicity d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The d-dimensional stable and centre manifolds are again vertical and irrelevant for a solution theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But we still find a one-dimensional unstable manifold corresponding to the prolonged graph of the unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It is tangential to the vector v = �N, 0T, h(0, u0)T�T and as in the scalar case we use as initial data for its determination the point �0, uT 0 , 0T�T + ϵv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Numerical Results As our main goal consists of showing how easy the numerical integration of singular problems becomes with our geometric approach, we did not de- velop any sophisticated production code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We performed all our computations with the built-in numerical capabilities of Maple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We used most of the time the dsolve/numeric command with its standard settings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' a Runge–Kutta– Fehlberg pair of order 4/5 is applied with a tolerance of 10−6 for the relative error and 10−7 for the absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Our geometric ansatz does not determine approximations un ≈ u(xn) of the solution u(x) on a discrete mesh (xn), but approximations xn = x(sn) and un = u(sn) for a parametric representation �x(s), u(s)� of the graph of the solu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Hence, for computing an approximated solution value u(¯x), one must first determine a parameter value ¯s such that x(¯s) ≈ ¯x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This can easily be accom- plished either with a nonlinear solver or with a numerical integrator with event handling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We used the latter option in most of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For boundary value problems, we applied the shooting method which worked very well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As Maple provides no built-in command for it, we wrote our own sim- ple version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In scalar problems, we solved the arising nonlinear equation most of the time with the Steffensen method (with Aitken ∆2 acceleration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As our equations are dimensionfree, suitable starting values were easy to find: typically, u(x) varied between 0 and 1 and we chose 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 as starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We encountered difficulties only in the simulation of a biocatalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For some parameter values, the correct initial value was very close to zero and the Stef- fensen iterations produced sometimes intermediate approximations which were negative and for which the numerical integration became meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Here we resorted to a simple bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For Lane–Emden systems, we used the Newton method for the arising non- linear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The Jacobian was determined via the variational equation of the differential system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thus for an n-dimensional differential system where k < n initial conditions have to be determined via shooting, we had to solve an addi- tional kn-dimensional linear differential system with variable coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Scalar Lane–Emden Equations We consider scalar Lane–Emden equations of the generalised form u′′ + m x u′ = f(x, u) (19a) together with either the initial conditions u(0) = u0 , u′(0) = 0 (19b) 9 or the boundary conditions u′(0) = 0 , αu(1) + βu′(1) = γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (19c) Chawla and Shivakumar [29] proved for boundary value problems with α = 1 and β = 0 an existence and uniqueness theorem under the following assumption on the right hand side f(x, u): the supremum M of the negative partial derivative − fu(x, u) on [0, 1] × R must be less than the first positive root t1 of the Bessel function J(m−1)/2( √t) (in the frequent case m = 2, we thus need M < π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The numerical integration of (19a) has been studied by many authors using many different approaches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' we refer to [30] for an overview of many works be- fore 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We will discuss three different situations: (i) initial value problems in astrophysics, (ii) Dirichlet boundary value problem in chemical engineering and (iii) mixed boundary value problems in physiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Initial Value Problems from Astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the classical Lane–Emden equa- tions, one has m = N − 1 with N the space dimension and f(x, u) = −un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The solutions for u0 = 1 are known as polytropes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Physically meaningful is the range 0 ≤ n < 5 (with n not necessarily an integer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For three polytropic indices, namely n = 0, 1, 5, exact solutions are known [4, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Of physical rele- vance are in particular the first zero ξ1 of u (corresponding to the scaled radius of the sphere) and the value of u′(ξ1) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' the ratio of the central density to the mean density is given by r = −ξ1/3u′(ξ1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Figure 1: Logarithmic plot of absolute deviation from exact solution for some polytropes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We numerically solved the Lane–Emden equations by integrating the dynam- ical system (12) with f, g given by (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As concrete test cases, we used some 10 10 6 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 10 n=0 N=2 err n=1 N=2 8 n=0 N=3 10 n=1 N=3 n=5 N=3 9 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='10 10 0 2 3 4 xpolytropic cylinders and spheres where the exact solutions are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Figure 1 shows the observed errors in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Obviously, the results are within the expected range for the default settings of Maple’s numerical integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' N, n ξ1 r 2, 0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 · 10−6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='0 · 10−7 2, 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 · 10−7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 · 10−6 3, 0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3 · 10−7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 · 10−10 3, 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 · 10−7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3 · 10−7 Table 1: Relative errors for first zero ξ1 and density ratio r for the cases with ξ1 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Our approach also determines approx- imations u′ n = u′(sn) for the first deriva- tives of the solution, as the integral curves of the vector field Y define a parametrisa- tion �x(s), u(s), u′(s)� of the solution and its first derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We use this to approx- imate also the quantities ξ1 and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Ta- ble 1 exhibits their relative errors com- pared with the exact solution for those cases where ξ1 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Again, the ob- served accuracy corresponds well to the settings of the numerical integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Boundary Value Problems for (Bio)Catalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In chemical engineering, the Lane– Emden equation arises in the analysis of diffusive transport and chemical reac- tions of species inside a porous catalyst pellet [31, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4] with boundary condi- tions of the form (19c) with α = γ = 1 and β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Flockerzi and Sundmacher [32] considered the case m = 2 and f(x, u) = φ2un for a single species obeying Fick’s law with constant diffusivity and power-law kinetics (the constant φ2 is the Thiele modulus describing the ratio of surface reaction rate to diffusion rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As this corresponds up to a sign exactly to the above considered polytropes, we omit concrete calculations and only note that [32] also provides a nice geomet- ric proof of the existence of a unique solution of this particular boundary value problem which, unfortunately, seems not be extendable to other functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Using a Michaelis–Menten kinetics for a biocatalyst, one obtains right hand sides like f(x, u) = 9φ2 u 1+Ku, where φ is again the Thiele modulus and K the dimensionless Michaelis–Menten constant (see [33] for some further variants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This model was analysed by a homotopy perturbation method in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A quantity relevant for engineers is the effectiveness factor which is here given by η = K+1 3φ2 u′(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A numerical study of the dependency of η on φ2 and K leads to the surface shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 2 based on a 17 × 17 grid, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' on the numerical solution of 289 boundary value problems with different combinations of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As indicated above, we had to use here a bisection method for locating the right initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Bisecting until an interval length of 10−5 was reached, the whole computation required only 2–3sec on a laptop (equipped with eight Intel Core i7-11370H (11th generation) working with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3GHz and 16GB of RAM running Maple 2022 under Windows 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 11 Figure 2: Dependency of the effectiveness factor η on Thiele modulus φ2 and dimen- sionless Michaelis–Menten constant K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Matlab’s solvers bvp4c and bvp5c are finite difference methods based on a three- and four-stage, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=', Lobatto IIIa collocation formulae and provide a special option for the type of singularity appear- ing in Lane–Emden equations [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' However, it turned out to be nontrivial to determine a plot like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 2 with them, as for some parameter values they re- act rather sensitive to the required ini- tial guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Using a simple constant func- tion lead sometimes either to completely wrong solutions or the collocation equa- tions could not be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We then com- puted one solution with “harmless” pa- rameter values and used it as initial guess for all other parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But the computations required with 5–6sec about twice as much time as our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' An alternative approach consists in transforming the problem into a reaction- diffusion equation by adding a time derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The desired solution of our boundary value problem arises then as asymptotic for long times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Matlab pro- vides here with pdepe a specialised solver admitting again our type of singu- larity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It employs a method for parabolic partial differential equations proposed by Skeel and Brezins [37] using a spatial discretisation derived with a Galerkin approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Here, one does not need an initial guess and it turns out that a steady state is reached very rapidly (already t = 1 is sufficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But one needs an ad- ditional interpolation with pdeval to determine derivative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Furthermore, the computation time for a plot like Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 2 increases significantly to about 17sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3 Mixed Boundary Conditions for a Physiological Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The same differential equation is used to model the steady state oxygen diffusion in a spherical cell with Michaelis-Menten uptake kinetics [38, 39], m = 2 and f(x, u) = au u+K, but with mixed boundary conditions (19c) where α = γ, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Hiltmann and Lory [40] proved explicitly the existence and uniqueness of a solution of this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the first two references above, concrete, physiologically meaningful values 3This approach was also used by the authors of [34] to compute reference solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' However, the plots presented there do not agree with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As they provided a listing of their Matlab code, we could repeat their numerical experiments and obtained the same results as with our method and not what they show in their paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content="8- 9'0 n 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2- 人 0 15 5 10 2 10 15 5 Kfor the parameters are determined and numerical results are presented which are, however, contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We used for our experiments four different parameter sets proposed by McElwain [39] and which can be found in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' a K α 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='38065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='03119 5 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='38065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='03119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='76129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='03119 5 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='38065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='31187 5 Table 2: Parameter values for the oxygen uptake model following McElwain [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In particular for the third parame- ter set, several authors performed similar computations starting with Hiltmann und Lory [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Khuri and Sayfy [41, Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 3] combined a decomposition method in the vicinity of the singularity with a colloca- tion method in the rest of the integration interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' They provided – like Hiltmann and Lory – approximations of u(xi) for xi = i/10 with i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' , 10 [41, Tbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 5] and compared with results of C¸ a˘glar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It turned out that for the first six digits all three approaches and our method yield exactly the same result – a quite remarkable agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 3 provides plots of the oxygen concentration u(x) and of its rate of change v(x) = u′(x) for all four different sets of parameters as obtained by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The concentration plot agrees well with the one given by McElwain [39, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 1] (and confirmed by Hiltmann und Lory [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Figure 3: Numerical solutions of the boundary value problem for the oxygen uptake model for four different sets of parameters given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Left: oxygen concentration u(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Right: rate of change of oxygen concentration u′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Hiltmann and Lory [40] report that they used a sophisticated implementation of a multiple shooting procedure based on four different integrators for initial value problems together with a special treatment of the singularity using both a technique of de Hoog and Weiss [43] and a Taylor series method (no further de- tails are given).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' They prescribed a tolerance of 10−8 for their Newton solver and 10−10 for the integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' By contrast, we used a simple shooting method with the 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content="9 8'0 1 11 4 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 90 0 02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content="4 9'0 80 1 x0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 1 4 ro 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='8 1 0 1 xMaple built-in Runge–Kutta–Fehlberg integrator and a hand-coded Steffensen method for the nonlinear system with a tolerance of 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This comparison again demonstrates how much simplicity and robustness one gains by using the asso- ciated vector field for the numerical integration in singular situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Lane–Emden Systems Our approach works for systems in the same manner as for scalar equations, as one still finds a one-dimensional unstable manifold corresponding to pro- longed graph of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thus we restrict to just one example of dimension d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We now have to integrate the system (17) of dimension n = 2d +1 = 7 for the above given initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We used a Newton method for solving the nonlinear system arising in the shooting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Since we had to determine d = 3 initial conditions via shooting, we had to augment (17) by a linear matrix differential equation with variable coefficients of dimension 7 × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Campesi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' [44] proposed a system of coupled Lane–Emden equations as model for the combustion of ethanol and ethyl acetate over an MnCu catalyst us- ing a Langmuir–Hinshelwood–Hougen–Watson kinetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In dimensionless form, the system is given by (see [45]) u′′ + 2 xu′ = µuu 1 + λuu + λvv + λww , v′′ + 2 xv′ = µvv − µuu 1 + λuu + λvv + λww , w′′ + 2 xw′ = µww 1 + λuu + λvv + λww , (20) where u, v, w represent (dimensionless) molar concentrations of ethanol, ac- etaldehyde and ethyl acetate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The boundary conditions require that at x = 0 all first derivatives vanish and that at x = 1 all concentrations are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The authors of [44] used for numerically integrating (20) an approach devel- oped by essentially the same group [46] based on an integral formulation and an h-adaptive mesh procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Unfortunately, [44] does not provide all the pa- rameters used in the computations so that it is not possible to compare with their results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We used instead for our experiments data given in [45] (employing a modified Adomian decomposition method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' However, the plots given there are not correct, as apparently wrong differential equations were used – at least in the Matlab code presented in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We compared with analogous Matlab computations using the right differential equations and again pdepe as a numeri- cal solver and obtained an excellent agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Figure 4 presents solution curves for the values µu = 30, µv/w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='01, λu = 3 and λv/w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1 used in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 14 Figure 4: Numerical solutions of the boundary value problem for the dimensionless model of the MnCu catalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Left: concentrations of ethanol, acetaldehyde and ethyl acetate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Right: corresponding rates of change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thomas–Fermi Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Majorana Transformation The Thomas–Fermi equation (3) belongs also to the class (10), but with g(x) = √x , f(x, u, u′) = √ u3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (21) The initial condition u(0) = 1 leads to a rather different situation as for the Lane– Emden equation: the implicit form of the Thomas–Fermi equation entails that the only points on R2 which project on x = 0 are of the form ρ = (0, 0, u1, u2) with arbitrary values u1, u2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Hence no solution satisfying the above initial condition can be twice differentiable at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Solutions with a higher regularity exist only for the initial condition u(0) = 0 which has no physical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Any point of the form ρ = (0, 1, u1) is an improper impasse point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The vector field Y defined by (11) does not vanish at such points but takes the form ∂u′ and it is not Lipschitz continuous there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' While Peano’s theorem still asserts the ex- istence of solutions, we cannot apply the Picard–Lindel¨of theorem to guarantee uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We could rescale Y by some function like x which does not change its trajectories for obtaining an everywhere differentiable vector field ˜Y = xY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Now all points of the above form are stationary points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But the Jacobian of ˜Y has 0 as a triple eigenvalue at them making it hard to analyse the local phase portrait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We use therefore a different approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As Esposito [47] reported only in 2002, Majorana proposed already in 1928 a differential transformation to a new independent variable t and a new dependent variable v of the form t = 144−1/6x1/2u1/6 , v = −(16/3)1/3u−4/3u′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (22) 15 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 u,V,w 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content="8 1 x2 u',v',w' 0 u 1 2- 0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='8 1 xThis at first sight rather miraculous transformation stems from a particular kind of scaling symmetry [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A computation detailed in [47] shows that if it is applied to any solution of the Thomas–Fermi equation (3), then the transformed variables satisfy the reduced equation (1 − t2v)dv dt = 8(tv2 − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (23) The boundary condition (4b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' limx→∞ u(x) = 0, translates into the condition v(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4 We will see below that the thus defined singular initial value problem for (23) possesses two solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Only one of them is also defined for t = 0 and thus is the physically relevant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It follows from (22) that the initial slope u′(0) for the Thomas–Fermi equation is obtained from a solution of (23) by u′(0) = −(3/16)1/3v(0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (24) The reduced equation (23) is quasi-linear and of first order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Opposed to the Lane–Emden equations, it is not semi-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thus singular behaviour does not simply occur at specific t-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Instead it appears whenever a solution graph contains a point (t, v) with t2v = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Nevertheless, one can apply the same kind of approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One first computes a vector field X living on the hypersurface R1 ⊂ J1 defined by (23) and spanning there the Vessiot distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Then one projects X to the jet bundle J0 and obtains there the vector field Yred = (t2v − 1)∂t + 8(1 − tv2)∂v .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (25) As we are now on J0, one-dimensional invariant manifolds of Yred which are transversal can be directly identified with the graphs of solutions of (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Our initial point (1, 1) is a proper impasse point where Yred vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 5 shows the phase portrait of the vector field Yred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It has (1, 1) as its only stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The plot shows in blue some integral curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Most, but not all of them can be considered as the graphs of solutions of (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The plot also contains in red the t-nullcline given by v = 1/t2 – which is simultaneously the singular locus of (23) – and in green the v-nullcline given by v = ±1/ √t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The integral curves that cross the t-nullcline show at the intersection a turning point behaviour, as the t-component of Yred changes its sign there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' If (ti, vi) is such an 4The Majorana transformation is not bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A well-known solution of the Thomas–Fermi equation already given by Thomas [5] is us(x) = 144x−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It does not satisfy the left boundary condition, as it is not even defined for x = 0, but the asymptotic condition at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One easily verifies that any point of the form �x, us(x), u′ s(x)� is mapped into the point (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 16 intersection point, then it splits the corresponding integral curve into two solution graphs where both solutions are defined only for t < ti, as they both loose their differentiability at t = ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' With traditional numerical methods applied to (23), it would be difficult to determine these solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' as integral curves of Yred they are trivial to obtain numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Figure 5: Phase portrait of the vector field associated to the reduced system (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The unstable manifold is shown in cyan, the sta- ble manifold in magenta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The Jacobian of Yred at the stationary point (1, 1) is the matrix J = � −2 −1 8 16 � with eigenvalues −7± √ 73 ≈ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='544, −15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='544).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thus we are dealing with a saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The unstable and the stable manifold shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 5 in cyan and magenta, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=', correspond to the above mentioned two solutions of the initial value problem with v(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' There cannot exist any ad- ditional solutions, as there are no further invariant manifolds entering or leaving the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One sees that in the positive quadrant the stable manifold cannot cross the nullclines outside of the saddle point and hence can never reach the v-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Thus we may conclude that the part of the unstable manifold between the v-axis and the stationary point corresponds to the unique solution u∞ of the boundary value problem with the condition (4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The abscissa of the intersection of the unstable manifold with the v-axis determines via (24) the critical initial slope ω (see below for numerical values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The ex- istence of such a unique solution for this specific boundary value problem was proven in 1929 by Mambriani [49] (see also the discussion in [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It will turn out that the integral curves to the right of the stable manifold have no relevance for our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The integral curves to the left of it and above the unstable manifold correspond to solutions of the boundary value problem with the condition (4c), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' solutions with a zero, whereas the integral curves below the stable manifold lead to solutions for (4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This can be deduced from their intersections with the v-axis and (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Much of the literature on numerically solving the Thomas–Fermi equation is concerned with the solution u∞ of (4b) defined on the semi-infinite interval [0, ∞) and concentrates on the determination of the critical slope ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Most so- lutions reported in the literature are either shown only on rather small intervals 17 1 个 个 个 个 ←↑ ← → → 个 → → → 个 个 3 → → V → 1 2 → → → → → → ↑ 1 1 → ↑ ↑ → T T T T 1[0, x0] with typically x0 < 10 or clearly deteriorate for larger x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One reason for this effect is surely that many approaches are based on some kind of series expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Another, more intrinsic reason becomes apparent from the phase por- trait in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As the sought solution corresponds to a branch of the unstable manifold of the saddle point (1, 1), even small errors close to the saddle point (corresponding to points with large x coordinates) are amplified by the dynamics and the numerical solutions tend to diverge from a finite limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' By contrast, our approach to determine u∞ leads to the standard problem of determining a branch of the unstable manifold of a stationary point – a task which can be performed numerically very robustly and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As the posi- tive eigenvalue has about the tenfold magnitude of the negative one, trajectories approach the unstable manifold very fast which ensures a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Following Majorana, Esposito [47] (and subsequent authors) determines a series solution of the initial value problem v(1) = 1 for (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the first step, one obtains a quadratic equation with two solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Esposito then argues that one should take the smaller solution, as this was a perturbation calculation which is not a convincing argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The reduced initial value problem has two solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As one can see in Figure 5, the second solution corresponding to the stable manifold grows very rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Therefore it is not surprising that several authors suspected that the second solution of the quadratic equation leads to a divergent power series and thus could be discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' However, a second solution to the initial value problem does exist, although it seems that it cannot be determined with a power series ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But as already discussed above, u∞ is nevertheless unique and corresponds to the unstable manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For the series solution, one expands around t = 1 and makes the ansatz v(t) = �∞ i=0 ai(1 − t)i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The initial condition yields a0 = 1 and for the arising quadratic equation for a1 one chooses the root5 a1 = 9 − √ 73 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' After lengthy computations sketched in [47], one obtains the following recursive expression for the remaining coefficients with i > 1: ai = 1 2(i + 8) − (i − 1)a1 � (i + 6)a1ai−2 + � (i + 7) − 2(i + 3)a1 � ai−1 + i−2 � j=1 � (j + 1)aj+1 − 2( j + 4)aj + ( j + 7)aj−1 � ai− j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (26) 5This value is related to the spectrum of the Jacobian of the vector field Yred: −a1 is the slope of the tangent space of the unstable manifold at the saddle point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This is not surprising, as the tangent space is the linear approximation of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 18 Setting t = 0 yields for the critical slope the series representation ω = − � 3 16 �1/3 ∞ � i=0 ai , (27) which can be evaluated to arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' To obtain whole solutions u(x), one must be able to transform back from the variables (t, v) to the original variables (x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Esposito [47] exhibited a conve- nient method for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We express the solution in parametric form using t as parameter: x = x(t) and u = u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Then we make the ansatz u(t) = exp �� t 0 w(τ)dτ � (28) with w a yet to be determined function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Assuming x(t = 0) = 0, this ansatz au- tomatically satisfies the initial condition u(x = 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Using the transformation (22), one can show that w(t) = 6tv(t) t2v(t)−1 and that x(t) can be expressed via w(t) as x(t) = 1441/3t2 exp � −1 3 � t 0 w(τ)dτ � (29) (which shows that indeed x(0) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Esposito [47] proposed to enter the above determined series solution for v(t) into these formulae and to compute this way a series expansion of u∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This requires essentially one quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Numerical Results We refrain from citing the many papers written on computing u∞ and in par- ticular ω and instead refer only to [50, 51] both listing a large number of ap- proaches with references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We emphasise again that our main point is to show that the geometric theory allows us – here in combination with the Majorana transformation – to translate a singular problem into basic tasks from the theory of dynamical systems which can be easily solved by standard methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The “Critical” Solution u∞ and the Critical Slope ω We consider first the problem of only determining the initial slope ω belong- ing to the solution u∞ for (4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' With classical approaches, this is a non-trivial problem and in the literature one often finds values with a very low number of correct digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Using our geometric approach, we can determine ω to (almost) 19 any desired precision in about 10 lines of Maple code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We write the dynamical system corresponding to the vector field Yred defined by (25) as dt ds = t2v − 1 , dv ds = 8(1 − tv2) , (30) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' we determine integral curves of Yred in parametric form �t(s), v(s)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As dis- cussed above, the sought trajectory corresponds to the unstable manifold of the saddle point (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' An eigenvector for the positive eigenvalue λ = −7 + √ 73 is given by e = �1, −9 + √ 73�T and we denote by ˆe = (e1, e2)T the corresponding normalised vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Then we choose as initial point for a numerical integration t(0) = 1 + ϵe1 and v(0) = 1 + ϵe2 with ϵ > 0 some small number (we typically used 10−3 or 10−4, but this had no effect on the obtained slope) and integrated until t(s) = 0 for s = s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Finally, we obtain ω from v(s0) via (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We control the precision with an integer parameter N specifying that the numerical integration of (30) should take place with an absolute and relative error of 10−N and that for this purpose Maple should compute with N + 5 digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In a recent work, Fern´andez and Garcia [51] determined ω based on the first 5000 terms of the Majorana series (27) to a precision of several hundred digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This is by far the best approximation available and our reference solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' tolerance rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' error time 10−5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2 · 10−6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='6 10−10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3 · 10−12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='6 10−15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 · 10−17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 10−20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3 · 10−22 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 10−25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 · 10−27 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='5 Table 3: Relative error and computation time in seconds for different tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Our numerical results are summarised in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Our relative error is always smaller than the prescribed tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For smaller tolerances, the computational ef- fort is rapidly increasing and on a laptop we needed for 25 digits less than 4 min- utes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We made no effort to optimise the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For example, we are using the default integration method of Maple (a Runge–Kutta–Fehlberg method of or- der 4/5 with a degree four interpolant), al- though a higher order scheme would probably be more efficient (Maple offers such schemes – but not in combination with the automated root finding used in our code).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Nevertheless, one may conclude that for practically relevant preci- sions, our geometric approach combined with the Majorana transformation pro- vides very accurate results fast and almost effortless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Fern´andez and Garcia [51] analyse also the convergence rate of the Majorana series (27) and consider it as fast (see also the comments by Esposito [47]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We compared for a relative small accuracy, Maple hardware floats with 10 digits, the value for the initial slope obtained with our approach with the approximations 20 terms 10 20 30 40 50 rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='8 · 10−2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 · 10−3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='3 · 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1 · 10−4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4 · 10−5 terms 60 70 80 90 100 rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' err.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='9 · 10−6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 · 10−7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='7 · 10−8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='4 · 10−9 0 Table 4: Relative error for different truncation degrees of the Majorana series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' delivered by various truncations of the series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Somewhat surprisingly, our ap- proach gets all 10 digits right, despite the considerably higher tolerances (10−6) used by the integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Table 4 contains the approximations obtained by evalu- ating the first N terms of the Majorana series (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One needs 100 terms for a similarly accurate result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' On average, one needs 10 more terms for one additional digit corresponding to a linear convergence as already theoretically predicted in [47, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This observation also roughly agrees with the fact that Fern´andez and Garcia used 5000 terms for obtaining about 500 digits [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For determining the whole solution u∞(x) instead of only the critical slope ω = u′ ∞(0), we have to perform a transformation back from the variables (t, v) to (x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We described above Esposito’s approach for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For a purely numerical computation instead of series expansions, we modify it in a way which fits nicely into our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We introduce as Esposito [47] the function I(t) = � t 0 τv(τ) 1 − τ2v(τ)dτ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (31) We then express I(t) as a function of the parameter s which we use to parametrise solution curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' If s0 is the (first) parameter value satisfying t(s0) = 0, then an elementary application of the substitution rule yields I(s) = − � s s0 t(σ)v(σ)dσ , (32) which immediately implies that I satisfies the differential equation dI ds = −tv by which we augment the system (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We thus obtain a free boundary value problem for the augmented system, as the function I(s) satisfies the condition I(s0) = 0 with the a priori unknown value s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As usual, we consider s0 as an additional unknown function and introduce the rescaled independent variable σ = s/s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Then we finally obtain the following two-point boundary value prob- 21 lem with non-separated boundary conditions dt dσ = s0(t2v − 1) , t(0) = 1 + ϵe1 , t(1) = 0 , dv dσ = 8s0(1 − tv2) , v(0) = 1 + ϵe2 dI dσ = −s0tv , I(1) = 0 , ds0 dσ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (33) Once this boundary value problem is solved, (28) and (29) imply that parametri- sations of the graph of u∞(x) are given by x(σ) = 1441/3t(σ)2 exp �2I(σ)� , u(σ) = exp �−6I(σ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (34) Figure 6: Comparison of values obtained via (34) and Majorana’s series for different numbers N of terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We implemented this approach in Maple using the built-in solver for bound- ary value problems which could handle (33) without problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We compared the results with solutions obtained via Majo- rana’s series, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' following Esposito [47], we entered a given number N of terms into the integral defining I and performed a numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 6 shows on a logarithmic scale the absolute difference between our curve �x(σ), u(σ)� and the curves computed via the series for differ- ent values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Obviously, our results are in an excellent agreement with the series solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The fact that all error curves have their maximum close to x = 0 is easy to explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As the expansion point of the series corresponds to x = ∞ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' t = 1), the series solutions become less accu- rate the closer one gets to x = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' at x = 0 of course no error occurs, as this value is fixed by an initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We did not make an extensive comparison of computation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But plotting the series solution for N = 10 over the interval [0, 5] required more than 10 times as much computation time than solving above boundary value problem demonstrating again the efficiency of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 22 10-5 err 10-6 10~7 10°8 0 m x N=10 N=20 N=30We mentioned already above that in the literature results are often presented only for rather small values of x, although the solution is defined for all non- negative real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One exception is Amore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' [52, Tbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 3/4] who used a Pad´e–Hankel method and asymptotic expansions to present highly accurate values of the solution u∞(x) and its first derivative u′ ∞(x) up to x = 400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' x u∞(x) u′ ∞(x) 0 1 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='58807101687867 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='0243142929534589 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='00460288186903816 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='000632254782228818 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='0000324989019998445 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='000100242568239745 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='73935106365787 · 10−6 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='0000326339644201454 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='09139947257267 · 10−7 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='0000145018034835377 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='05753231409599 · 10−7 250 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='67729076668264 · 10−6 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='78946798702223 · 10−8 300 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='54857195240339 · 10−6 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='36594961733055 · 10−8 350 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='91510210708972 · 10−6 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='40920109677041 · 10−8 400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='97973262954641 · 10−6 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='43668230750324 · 10−8 Table 5: Solution values u∞(x) and derivative values u′ ∞(x) for large x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Table 5 contains similar values obtained with our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For determining the values of u′ ∞(x), we must augment (34) by an equation for u′(σ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' we must extend the parametrisation to the prolonged graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' By a straightforward application of the chain rule, one obtains u′(σ) = −3 · 144−1/3v(σ) exp �−8I(σ)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (35) To compile such a table, one must then determine for each x the corresponding value of the parameter σ via the solution of a nonlinear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Nevertheless, the complete computation of the values at the ten points contained in the table required only about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='1 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Amore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' [52] claim that in their tables all digits are correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Assuming that this is indeed the case, we can conclude that we obtained with minimal effort for each value of x at least eight correct digits for u∞(x) and seven correct digits for u′ ∞(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Given the settings for the tolerances of our integrator and the use of hardware floats with only 10 digits, these results demonstrate again a very remarkable precision and efficiency of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As large values of x correspond to small values of σ and thus to values of t close to 1, one may have to choose a smaller value of ϵ for very large values of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The largest value appearing in above table, x = 400, corresponds to σ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='35 and t ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='9789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We chose for our numerical calculation the value ϵ = 10−3 and thus 23 used as right end of the approximated unstable manifold instead of the saddle point (1, 1) the point (t1, v1) ≈ (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='9978, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For x = 400, one may say that we are still sufficiently far away from this point, but for larger values of x one should probably start working with a smaller value of ϵ which will increase the computation time, as the dynamics is very slow so close to a stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Other Solutions So far, we only considered the particular solution u∞ (which has attracted the most attention in the literature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 5 we presented the phase portrait for the Majorana transformed Thomas–Fermi equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Using a slight modification (and simplification) of the above described backtransformation via the solution of an extended differential system, we can also obtain a “phase portrait” of the original Thomas–Fermi equation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' we compute solutions for different values of the initial slope u′(0) keeping the initial condition u(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' While the Majorana transformation itself is valid for any solution of the Thomas–Fermi equation, our ansatz for the back transformation has encoded this second initial condition (one could easily adapt to a different value u(0) = c by multiplying (28) with the constant c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' According to (24), each value of u′(0) corresponds uniquely to a value of v(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' We now take the vector field −Yred and use a parametrisation such that s = 0 corresponds to t = 0 (and thus also x = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This leads to the following augmented initial value problem: dt ds = 1 − t2v , dv ds = 8(tv2 − 1) , dI ds = tv , t(0) = 0 , v(0) = v0 , I(0) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' (36) Its solutions are then transformed into x- and u-coordinates via (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 7 shows that the solution u∞ vanishing at infinity acts as a kind of “sep- aratrix”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The solutions above it, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' with an initial slope higher than ω, pass through a minimum and then grow faster than exponentially (note the logarith- mic scale).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The solutions below it approach rapidly zero, reaching it at a finite value of x (recall that the separatrix reaches zero at infinity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' It turns out that around the critical value ω, the trajectories are rather sensitive with respect to the initial slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For some of the curves shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 7, u′(0) differs only in the fifth or sixth digit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For the curves approaching zero, it is also non-trivial to determine the exact location of the zero, as here v goes towards infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In our computations, we actually integrated only until some threshold like 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Probably a “hybrid” approach using (36) only to get away from the singularity at x = 0 and applying afterwards a standard integrator to the Thomas–Fermi equation would be a good alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 24 Figure 7: Solutions of the Thomas–Fermi equation with u(0) = 1 and different u′(0) using a logarithmic scale for u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The curve in magenta shows u∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For solving concrete boundary value problems with boundary conditions of the form (4a) or (4c), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=', for given values of a or b, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=', one can use an adapted ver- sion of a shooting method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Starting with an initial guess v0 for the unknown value of v(0) for the sought solution, one inte- grates the initial value problem (36) un- til a condition of the desired form is sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' However, in general, the condition will be satisfied at a wrong position a∗ or b∗, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Using a bisection, one modifies v0 until one is sufficiently close to the ac- tually prescribed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' As in both cases, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 7 shows that there is a monotone re- lation between v0 and a∗ or b∗, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=', it is always clear in which direction one has to change v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But for larger values of a or b, one gets again into areas where very small changes in v0 lead to significant changes in a∗ or b∗, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Despite this sensitivity, the approach worked in tests very well for a ≤ 27 and b ≤ 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Conclusions The Lane–Emden and the Thomas–Fermi equation are prototypical exam- ples for ordinary differential equations with singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Their singularities are determined by a specific value of the independent variable: x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Any initial or boundary value problem with conditions prescribed at x = 0 cannot be tackled by standard methods and this concerns both theoretical and numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The Lane–Emden equations fit into the framework of so-called Fuchsian equations (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' [53]), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' equations of the form Lu = f(x, u) where L is a linear differential operator of Fuchsian type and where only the right hand side may contain nonlinear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For the theoretical treatment of such equations, some form of quasilinearisation is often fruitful, as it allows to use the far devel- oped theory of the linear counterpart Lu = ˜f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For example, the existence and uniqueness proof for boundary value problems for (generalised) Lane–Emden equations given in [29] follows such strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For the numerical integration, [54] presents methods for first- and second-order systems of this particular form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' A key consequence of this special structure is the above mentioned loca- tion of the singularities depending only on x which facilitates the design of spe- 25 10 100 102 10-4 10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 10-8 10 20 30 xcialised numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Therefore it is not surprising that so many different techniques have been proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Our approach is independent of such a special form, as one can see from our treatment of the Thomas–Fermi equation based on the reduced equation (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' The location of its singularities depends on t and v making an integration with standard numerical methods more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' By contrast, our approach can handle all forms of quasilinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In some computations related to the Thomas–Fermi equations, we encoun- tered problems, for example when computing u∞(x) for very large values of x or when u(x) approaches zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the first case, the reason lies in an often highly nonlinear relationship between the variable t used in the reduced system and the variable x where “microscopic” changes in t may correspond to huge differences in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the second case, u can approach 0 only when v tends towards infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In both cases, one could probably extend the applicability of our method by a rescaling of the reduced equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' For computing u∞ for large x, an alternative, semianalytic approach would consist of determining a higher order approxima- tion of the unstable manifold close to the saddle point (1, 1) – in fact, the Majo- rana series is nothing else than such an approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This could lead to very accurate values even for extremely large values of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' One may wonder why we used in the case of the Lane–Emden equations the shooting method for boundary value problems and not also a formulation as free boundary value problem as for the Thomas–Fermi equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In both cases, one faces the problem that at one boundary one has to deal with a two-dimensional plane of stationary points and that the boundary conditions enforces that one end point of the solution trajectory lies on this plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the case of the Thomas– Fermi equation, we resolved this problem by moving a bit in the direction of the unstable eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' This was possible, as this direction is the same for all points on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' In the case of the Lane–Emden equations, the direction of the unstable eigenspace depends on the value u(0) and thus differs for different points on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Probably one could adapt typical approaches to boundary value problems like collocation methods to this dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' But as our emphasis in this paper lies on the use of standard methods, we refrained from studying this possibility in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' Furthermore, the simple shooting method works very well and reliable for this class of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/FtAzT4oBgHgl3EQfHPtI/content/2301.01041v1.pdf'} +page_content=' 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[Phys. Med. Biol. 60 (2015) +4149–4168] +Hans Rabus 1 +1 Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany + +E-mail: hans.rabus@ptb.de + + +Abstract +In their article published in Phys. Med. Biol. 60 (2015) 4149–4168, Lin et al studied the +radiosensitizing effect of gold nanoparticles (GNPs) using radiation transport simulations and +a biological model for the survival of irradiated cells. This comment points out several +caveats to the methodlogy used by Lin et al. that may not be evident to readers and may +contribute to confusion in the literature about the radiation effects of gold nanoparticles. The +two main caveats are the high mass fraction of gold considered and a potential problem with +the modified local effect model used to predict cell survival. +Keywords: gold nanoparticle, radiotherapy, proton therapy, local effect, model + +1. Gold concentration +In the paper of Lin et al (2015), the main studied nanoparticle size and concentration of GNPs are 50 nm and 1 µM, +respectively. Assuming that the mass density of gold in the GNPs is that of bulk gold, namely Au = 19.32 g/cm3, a 50 nm GNP +contains + +(50×10-7 cm)3×π/6×19.32 g/cm3/(196.97 g/mol)×6.022×1023 mol-1 = 3.81×106 +(1) +gold atoms. Thus, a concentration of 1 µM GNPs corresponds to a concentration of gold atoms of about 3.8 mol/L. This implies +a mass density of gold in solution of 750 g/L, which corresponds to a mass fraction of gold of about 43%! +When irradiated with a 50 kVp photon spectrum, most photons have energies in the range where the mass-energy absorption +coefficients of gold and water differ by two orders of magnitude (Hubbell and Seltzer 2004). Therefore, a photon fluence that +produces an absorbed dose of 1 Gy in water in the absence of the GNPs results in an average dose of about 40 Gy when the +GNPs are present. So it is not a big surprise that negligibly small survival rates are predicted for the 50 kVp spectrum! +For these low-energy photons, Lin et al. (2015) also investigated the dependence on GNP concentration in the range between +10 nM and 1 μM. From the argument presented above, a GNP concentration of 10 nM corresponds to a mass fraction of gold +of about 0.75%, which is still high but closer to the range of realistic values. For the linac spectrum and protons, on the other +hand, the increase in average absorbed dose is much smaller. Here, Lin et al. (2015) studied concentrations between 100 nM +and 10 μM, corresponding to mass densities of gold in solution between 75 g/L and 7.5 kg/L and mass fractions between 7% +and 88%! These are definitely unrealistically high values. + + + +2 + + +2. Inconsistencies in the description of the simulation setup +Apart from the issue of high GNP concentration, the data in the “Materials and Methods” section of the paper appear +contradictory. The paper states, “A concentration of 1 µM using 50 nm diameter GNPs results in 1.4×105 GNPs for the Nucleus, +CellHomo and Cytoplasm geometries (based on a cylindrical volume of 13.5 µm diameter and 2 µm thickness).” The three +geometries refer to the cases where the GNPs are located only in the cell nucleus, uniformly distributed throughout the cell, +and only in the cytoplasm. It is obviously impossible for the same number of GNPs to correspond to the same concentration in +all three cases. For a given concentration, the number of GNPs must be different for the cell and for the cell nucleus, simply +because the cell has a larger volume. +A cylinder with a diameter of 13.5 μm and a height of 2 µm has a volume Vc of + +Vc = (13.5 µm)2×π/4×2 µm = 2.86×102 µm3 = 2.86×10-13 L +(2) +At a concentration cGNP of nanoparticles of 1 µM, the number NGNP,c of GNPs in the cell is given by +NGNP,c = cGNP ×Vc×NA = 1×10-6 mol/L × 2.86×10-13 L × 6.02×1023 mol-1 = 1.72×105. +Conversely, if the number of GNPs in the nucleus, NGNP,n, is 1.4×105 and cGNP = 1 µM, then the volume Vn of the nucleus is + +Vn = NGNP,n / (cGNP × NA) = 1.4×105 / (1×10-6 mol/L × 6.022×1023 mol-1) = 2.33×10-13 L = 233 µm3 +(3) +An 8 µm diameter circle has an area of (8 µm)2/4 = 50.3 µm2, so a cylindrical cell nucleus of volume Vn = 233 µm3 has a +height of 4.64 µm, which exceeds the cell’s assumed thickness of 2 µm. If the nucleus is assumed to be spherical with a diameter +of 8 µm, its volume Vn is + +Vn = (8 µm)3×π/6 = 2.68×102 µm3 = 2.68×10-13 L +(4) +and NGNP,n = 1.4×105 corresponds to a GNP concentration of + +cGNP = NGNP,n/Vn/NA = 1.4×105 / (2.68×10-13 L × 6.022×1023 mol-1) = 0.87 µM. +(5) +It should be noted that a sphere with a diameter of 8 µm will not fit into a cylinder 2 µm high, and that the volumes given in +Eqs. 2 and 4 are similar but not identical. It therefore remains unclear what geometry and concentration of GNPs was actually +used. +3. Local effect model +Section 2.3 of (Lin et al 2015) describes a variant of the local effect model (LEM), called GNP-LEM, which uses a dose +distribution composed of the dose contribution from interactions in water and the localized additional dose contribution around +GNPs. The paper states that the latter dose contribution is obtained “by multiplying the dose from a single ionizing event by +the number of GNPs, the interaction probability per Gray and the prescribed dose” and that “The GNP-LEM developed in this +study was implemented in 2D, where the volume integration is reduced to an area integration over the cell nucleus.” +It is not clear what these two statements actually mean. The first statement suggests that the spatial arrangement of the GNPs +was not taken into account. The second statement suggests that GNPs are treated in analogy to ion beams in the original LEM, +where the dose distribution has a cylindrical symmetry around the ion trajectory. If one then performs the integral over a plane +perpendicular to this trajectory, one obtains the number of lesions produced per pathlength of the ion. For ions with low energy +loss in the nucleus and a nucleus with cylindrical shape irradiated along the cylinder axis, the total number of lesions is obtained +by multiplying the cylinder height with the number of lesions produced per pathlength. +How this can be applied to GNPs is not clear. In this context, it should be mentioned that the formula given in the article of +Lin et al (2015) for the total number of lethal lesions (second formula on page 4149) is incorrect because the logarithm of the +survival probability (appearing in the first formula on page 4149) is missing. The correct formula is + +������� = � +� �� ������,�,��� +� +�� +� + +(6) +Since the procedure used calculate the integral is not described in sufficient detail, it is not possible to assess whether or not +“area integration over the cell nucleus” gives a correct evaluation of the total number of induced lesions. In conjunction with +the first unclear statement, there is a possibility that Lin et al (2015) implicitly assumed (as did Jones et al (2010)) that a two- +dimensional projection of the dose distributions around GNPs onto a plane and integration over that plane would provide them +with the same information as a three-dimensional integral. However, as pointed out in (Rabus et al 2021), such an approach +implies that it does not determine the dose enhancement, or the number of lesions produced by GNPs. Instead, such an approach + + + +3 + + +determines these quantities in the case where the GNPs are replaced by cylindrical rods of gold, that have the same circular +cross section as the GNPs but a length equal to the thickness of the nucleus. The resulting integration value greatly overestimates +the number of lethal lesions and therefore leads to an underestimation of cell survival. +Whether the results of (Lin et al 2015) suffer from this deficiency cannot be judged, as their paper does not include detailed +information on how they actually proceeded. +4. Dependence of dose per ionization on GNP size +In Section 3.2 of (Lin et al 2015), the authors comment on the dependence of the dose contribution from electrons produced +in ionizations in the GNP on the GNP size, which can be seen in their Fig. 4. Their explanation is, “For the same energy +absorbed by a single GNP, the secondary electrons generated in a large GNP are more likely to lose their energy before reaching +the surface. Such self-absorption contributes to the lower dose deposited around the GNP by one ionization event for larger +GNPs.” +The main reason for the difference in dose contribution between different GNP sizes is that the mass of a water shell of the +same thickness around GNPs of different size increases with the square of the GNP radius. Therefore, one would expect the +dose at the surface of a 2 nm GNP to be 625 times higher than at the surface of a 50 nm GNP. That the authors only find an +increase by a factor 215 suggests that contrary to the authors’ claim, the higher number of interactions in a larger GNP actually +increases the dose contribution outside. +Conclusions +Most of the results shown in (Lin et al 2015) are for gold concentrations that appear unrealistically high. The trend of +decreasing survival probability with decreasing GNP size for the same amount of gold in the cells, shown in the left panel of +Fig. 8 of (Lin et al 2015), should also apply for realistic gold concentrations. If the results shown in the right panel of Fig. 8 for +2 nm GNPs apply to a concentration of 1 µM of these GNPs, the corresponding concentration of gold atoms is 250 µM or +50 mg/L, which corresponds to a gold mass fraction of 5×10-5. Therefore, the curve for 2 nm GNPs in the right panel of Fig. 8 +presumably indicates a realistic magnitude of effects from GNPs during proton irradiation, if the authors’ calculations are not +compromised by the potential problem described in Section 3. It should be noted, however, that even if their calculations of +cell survival are correct, the 2 nm GNP data shown in the right panel of Fig. 8 only apply to the case that survival is determined +solely by physical dose enhancement and not by other factors, such as chemical and biological effects of GNPs. +References +Hubbell J H and Seltzer S M 2004 Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV +to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest (version 1.4). [Online] Available at: +https://www.nist.gov/pml/x-ray-mass-attenuation-coefficients (Gaithersburg, MD: National Institute of Standards and +Technology) +Jones B L, Krishnan S and Cho S H 2010 Estimation of microscopic dose enhancement factor around gold nanoparticles by Monte Carlo +calculations AIP Conference Proceedings 37 3809–16 +Lin Y, McMahon S J, Paganetti H and Schuemann J 2015 Biological modeling of gold nanoparticle enhanced radiotherapy for proton +therapy Physics in Medicine and Biology 60 4149–68 +Rabus H, Li W B, Villagrasa C, Schuemann J, Hepperle P A, de la Fuente Rosales L, Beuve M, Maria S D, Klapproth A P, Li C Y, +Poignant F, Rudek B and Nettelbeck H 2021 Intercomparison of Monte Carlo calculated dose enhancement ratios for gold +nanoparticles irradiated by X-rays: Assessing the uncertainty and correct methodology for extended beams Physica Medica 84 +241–53 + + diff --git a/HNA0T4oBgHgl3EQfBv_D/content/tmp_files/load_file.txt b/HNA0T4oBgHgl3EQfBv_D/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c5a8f544bb5ec37b1b539d580b02039cb895d772 --- /dev/null +++ b/HNA0T4oBgHgl3EQfBv_D/content/tmp_files/load_file.txt @@ -0,0 +1,123 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf,len=122 +page_content='1 Comment on “Biological modeling of gold nanoparticle enhanced radiotherapy for proton therapy” by Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' [Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 60 (2015) 4149–4168] Hans Rabus 1 1 Physikalisch-Technische Bundesanstalt (PTB), Berlin, Germany E mail: hans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='rabus@ptb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='de Abstract In their article published in Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Biol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 60 (2015) 4149–4168, Lin et al studied the radiosensitizing effect of gold nanoparticles (GNPs) using radiation transport simulations and a biological model for the survival of irradiated cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' This comment points out several caveats to the methodlogy used by Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' that may not be evident to readers and may contribute to confusion in the literature about the radiation effects of gold nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The two main caveats are the high mass fraction of gold considered and a potential problem with the modified local effect model used to predict cell survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Keywords: gold nanoparticle, radiotherapy, proton therapy, local effect, model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Gold concentration In the paper of Lin et al (2015), the main studied nanoparticle size and concentration of GNPs are 50 nm and 1 µM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Assuming that the mass density of gold in the GNPs is that of bulk gold, namely \uf072Au = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='32 g/cm3, a 50 nm GNP contains (50×10-7 cm)3×π/6×19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='32 g/cm3/(196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='97 g/mol)×6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='022×1023 mol-1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='81×106 (1) gold atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Thus, a concentration of 1 µM GNPs corresponds to a concentration of gold atoms of about 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='8 mol/L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' This implies a mass density of gold in solution of 750 g/L, which corresponds to a mass fraction of gold of about 43%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' When irradiated with a 50 kVp photon spectrum, most photons have energies in the range where the mass-energy absorption coefficients of gold and water differ by two orders of magnitude (Hubbell and Seltzer 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Therefore, a photon fluence that produces an absorbed dose of 1 Gy in water in the absence of the GNPs results in an average dose of about 40 Gy when the GNPs are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' So it is not a big surprise that negligibly small survival rates are predicted for the 50 kVp spectrum!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' For these low-energy photons, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' (2015) also investigated the dependence on GNP concentration in the range between 10 nM and 1 μM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' From the argument presented above, a GNP concentration of 10 nM corresponds to a mass fraction of gold of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='75%, which is still high but closer to the range of realistic values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' For the linac spectrum and protons, on the other hand, the increase in average absorbed dose is much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Here, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' (2015) studied concentrations between 100 nM and 10 μM, corresponding to mass densities of gold in solution between 75 g/L and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='5 kg/L and mass fractions between 7% and 88%!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' These are definitely unrealistically high values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Inconsistencies in the description of the simulation setup Apart from the issue of high GNP concentration, the data in the “Materials and Methods” section of the paper appear contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The paper states, “A concentration of 1 µM using 50 nm diameter GNPs results in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='4×105 GNPs for the Nucleus, CellHomo and Cytoplasm geometries (based on a cylindrical volume of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='5 µm diameter and 2 µm thickness).” The three geometries refer to the cases where the GNPs are located only in the cell nucleus, uniformly distributed throughout the cell, and only in the cytoplasm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' It is obviously impossible for the same number of GNPs to correspond to the same concentration in all three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' For a given concentration, the number of GNPs must be different for the cell and for the cell nucleus, simply because the cell has a larger volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' A cylinder with a diameter of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='5 μm and a height of 2 µm has a volume Vc of Vc = (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='5 µm)2×π/4×2 µm = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='86×102 µm3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='86×10-13 L (2) At a concentration cGNP of nanoparticles of 1 µM, the number NGNP,c of GNPs in the cell is given by NGNP,c = cGNP ×Vc×NA = 1×10-6 mol/L × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='86×10-13 L × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='02×1023 mol-1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='72×105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Conversely, if the number of GNPs in the nucleus, NGNP,n, is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='4×105 and cGNP = 1 µM, then the volume Vn of the nucleus is Vn = NGNP,n / (cGNP × NA) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='4×105 / (1×10-6 mol/L × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='022×1023 mol-1) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='33×10-13 L = 233 µm3 (3) An 8 µm diameter circle has an area of (8 µm)2\uf0b4\uf070/4 = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='3 µm2, so a cylindrical cell nucleus of volume Vn = 233 µm3 has a height of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='64 µm, which exceeds the cell’s assumed thickness of 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' If the nucleus is assumed to be spherical with a diameter of 8 µm, its volume Vn is Vn = (8 µm)3×π/6 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='68×102 µm3 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='68×10-13 L (4) and NGNP,n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='4×105 corresponds to a GNP concentration of cGNP = NGNP,n/Vn/NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='4×105 / (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='68×10-13 L × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='022×1023 mol-1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='87 µM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' (5) It should be noted that a sphere with a diameter of 8 µm will not fit into a cylinder 2 µm high, and that the volumes given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 2 and 4 are similar but not identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' It therefore remains unclear what geometry and concentration of GNPs was actually used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Local effect model Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='3 of (Lin et al 2015) describes a variant of the local effect model (LEM), called GNP-LEM, which uses a dose distribution composed of the dose contribution from interactions in water and the localized additional dose contribution around GNPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The paper states that the latter dose contribution is obtained “by multiplying the dose from a single ionizing event by the number of GNPs, the interaction probability per Gray and the prescribed dose” and that “The GNP-LEM developed in this study was implemented in 2D, where the volume integration is reduced to an area integration over the cell nucleus.” It is not clear what these two statements actually mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The first statement suggests that the spatial arrangement of the GNPs was not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The second statement suggests that GNPs are treated in analogy to ion beams in the original LEM, where the dose distribution has a cylindrical symmetry around the ion trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' If one then performs the integral over a plane perpendicular to this trajectory, one obtains the number of lesions produced per pathlength of the ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' For ions with low energy loss in the nucleus and a nucleus with cylindrical shape irradiated along the cylinder axis, the total number of lesions is obtained by multiplying the cylinder height with the number of lesions produced per pathlength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' How this can be applied to GNPs is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' In this context, it should be mentioned that the formula given in the article of Lin et al (2015) for the total number of lethal lesions (second formula on page 4149) is incorrect because the logarithm of the survival probability (appearing in the first formula on page 4149) is missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The correct formula is ������� = � � �� ������,�,��� � �� � (6) Since the procedure used calculate the integral is not described in sufficient detail, it is not possible to assess whether or not “area integration over the cell nucleus” gives a correct evaluation of the total number of induced lesions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' In conjunction with the first unclear statement, there is a possibility that Lin et al (2015) implicitly assumed (as did Jones et al (2010)) that a two- dimensional projection of the dose distributions around GNPs onto a plane and integration over that plane would provide them with the same information as a three-dimensional integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' However, as pointed out in (Rabus et al 2021), such an approach implies that it does not determine the dose enhancement, or the number of lesions produced by GNPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Instead, such an approach 3 determines these quantities in the case where the GNPs are replaced by cylindrical rods of gold, that have the same circular cross section as the GNPs but a length equal to the thickness of the nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The resulting integration value greatly overestimates the number of lethal lesions and therefore leads to an underestimation of cell survival.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Whether the results of (Lin et al 2015) suffer from this deficiency cannot be judged, as their paper does not include detailed information on how they actually proceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Dependence of dose per ionization on GNP size In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='2 of (Lin et al 2015), the authors comment on the dependence of the dose contribution from electrons produced in ionizations in the GNP on the GNP size, which can be seen in their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Their explanation is, “For the same energy absorbed by a single GNP, the secondary electrons generated in a large GNP are more likely to lose their energy before reaching the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Such self-absorption contributes to the lower dose deposited around the GNP by one ionization event for larger GNPs.” The main reason for the difference in dose contribution between different GNP sizes is that the mass of a water shell of the same thickness around GNPs of different size increases with the square of the GNP radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Therefore, one would expect the dose at the surface of a 2 nm GNP to be 625 times higher than at the surface of a 50 nm GNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' That the authors only find an increase by a factor 215 suggests that contrary to the authors’ claim, the higher number of interactions in a larger GNP actually increases the dose contribution outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Conclusions Most of the results shown in (Lin et al 2015) are for gold concentrations that appear unrealistically high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' The trend of decreasing survival probability with decreasing GNP size for the same amount of gold in the cells, shown in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 8 of (Lin et al 2015), should also apply for realistic gold concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' If the results shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 8 for 2 nm GNPs apply to a concentration of 1 µM of these GNPs, the corresponding concentration of gold atoms is 250 µM or 50 mg/L, which corresponds to a gold mass fraction of 5×10-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' Therefore, the curve for 2 nm GNPs in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 8 presumably indicates a realistic magnitude of effects from GNPs during proton irradiation, if the authors’ calculations are not compromised by the potential problem described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' It should be noted, however, that even if their calculations of cell survival are correct, the 2 nm GNP data shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' 8 only apply to the case that survival is determined solely by physical dose enhancement and not by other factors, such as chemical and biological effects of GNPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' References Hubbell J H and Seltzer S M 2004 Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z = 1 to 92 and 48 Additional Substances of Dosimetric Interest (version 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content=' [Online] Available at: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf'} +page_content='gov/pml/x-ray-mass-attenuation-coefficients (Gaithersburg,' 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a/HtAyT4oBgHgl3EQfrvl-/content/tmp_files/2301.00566v1.pdf.txt b/HtAyT4oBgHgl3EQfrvl-/content/tmp_files/2301.00566v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..74f7c413e8aaa88ce8302ff473880d23e9e2d7da --- /dev/null +++ b/HtAyT4oBgHgl3EQfrvl-/content/tmp_files/2301.00566v1.pdf.txt @@ -0,0 +1,3210 @@ +Quantum speed limit for complex dynamics +Mao Zhang, Huai-Ming Yu, and Jing Liu∗ +National Precise Gravity Measurement Facility, MOE Key Laboratory of Fundamental Physical Quantities Measurement, +School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China +Quantum speed limit focuses on the minimum time scale for a fixed mission and hence is im- +portant in quantum information where fast dynamics is usually beneficial. Recently an operational +definition of quantum speed limit (OQSL) was proposed, which reveals the intrinsic minimum time +for time-independent Hamiltonians. However, a general method to evaluate the OQSL for time- +dependent Hamiltonians, especially when noises are involved, is still in lack. Hereby we provide the +expression of OQSL for a certain type of time-dependent Hamiltonians and propose a three-step +(classification-regression-calibration) methodology based on machine learning for the evaluation of +OQSL in complex dynamics. +Quantum speed limit (QSL) is a fundamental topic in +quantum mechanics focusing on the characterization of +minimum time for quantum states to fulfill certain known +targets. In the year of 1945, Mandelstam and Tamm pro- +vided the first lower bound for this minimum time based +on the uncertainty relation [1]. +In 1996 Braunstein et +al. extended the lower bound to time-dependent Hamil- +tonians utilizing the generalized uncertainty relation [2] +where the time-average variance was applied. In 1998, +Margolus and Levitin [3] provided another bound based +on the mean energy. After these pioneer works, the topic +of QSL entered a period of rapid development in the next +20 years, especially in 2010s [4–40]. +The target in QSL could be defined via different tools, +such as the Bures metric or various types of fidelity [8– +13], relative purity [14, 15], Bloch angle [16–19], gauge +invariant distances [20, 21], and Wigner-Yanase informa- +tion [22]. Throughout this paper, the target Θ ∈ (0, π] +is defined via the Bloch angle θ(t,⃗r) = arccos +� ⃗r·⃗r(t) +|⃗r||⃗r(t)| +� +between a Bloch vector ⃗r and its evolved vector ⃗r(t). Re- +cently, an operational definition of quantum speed limit +(OQSL) was proposed [18] based on the definition of +reachable state set S := {⃗r |θ(t,⃗r) = Θ, ∃t}. With this +set, the OQSL is defined by +τ := min +⃗r∈S t +subject to θ(t,⃗r) = Θ. +(1) +Compared to lower-bound-type QSLs, the advantages of +OQSL are that it can reveal information that whether a +state can fulfill the target, and it is always attainable [18]. +However, for complex dynamics these advantages come +at a price of high computational complexity, which is not +only due to the optimization in the definition, but also +the preliminary assumption that S is known. Regarding +the fact that complex dynamics is a non-negligible sce- +nario in the study of QSL [23–26], finding methods for +the evaluation of OQSL that are friendly to the compu- +tational complexity is critical, and thus the major moti- +vation of this paper. +In many cases, the complexity of dynamics comes +from the time dependency of the Hamiltonian. +The +OQSL for a general time-dependent Hamiltonian is dif- +ficult to obtain analytically. +However, for the time- +dependent Hamiltonians with time-independent eigen- +states, the OQSL can be derived analytically. +In the +energy space, these Hamiltonians can be expressed by +H(t) = � +i Ei(t) |Ei⟩ ⟨Ei|, where the eigenstate |Ei⟩ is +time-independent for any i and the eigenvalue Ei(t) de- +pends on time. For such Hamiltonians, we present the +following theorem. +Theorem 1. +For a N-dimensional time-dependent +Hamiltonian whose eigenstates are all time-independent, +the OQSL τ satisfies the equation +ˆ τ +0 +[Emax(t) − Emin(t)] dt = Θ, +(2) +where Emax(t) and Emin(t) are the maximum and min- +imum energies of the Hamiltonian at time t. +Further +denote the p-dimensional set {|Emin⟩} and q-dimensional +set {|Emax⟩} as the sets of eigenstates with respect to +Emin(t) and Emax(t), then the optimal states to reach +the OQSL are +� +i +1 +N |Ei⟩ ⟨Ei| + +� +|Ek⟩∈{|Emin⟩}, +|El⟩∈{|Emax⟩} +ξkl |Ek⟩ ⟨El| + ξ∗ +kl |El⟩ ⟨Ek| , +where the matrix ξ (with klth entry ξkl) satisfies N 2ξ†ξ ≤ +11q with 11q the q-dimensional identity matrix. +The proof is given in Ref. [41]. As a matter of fact, this +theorem covers Theorem 1 in Ref. [18] due to the fact +that Eq. (2) reduces to τ = Θ/(Emax − Emin) when the +eigenvalues are time-independent. As a simple demon- +stration, consider the Hamiltonian H(t) = f(t)σz with +σz the Pauli Z matrix and f(t) a time-dependent func- +tion. The other two Pauli matrices are denoted by σx +and σy. It is obvious that the eigenstates of this Hamil- +tonian are independent of time. Hence the corresponding +OQSL is given in the theorem above. In the case that +| +´ t +0 f(t1)dt1| is upper bounded by cf, S is fully deter- +mined by the value of cf, which leads to the following +no-go theorem. +arXiv:2301.00566v1 [quant-ph] 2 Jan 2023 + +2 +No-go theorem. For the Hamiltonian H(t) = f(t)σz +where f(t) satisfies | +´ t +0 f(t1)dt1| ≤ cf, no state can fulfill +the target Θ if cf < Θ/2. +In the case that cf ≥ Θ/2, S is symmetric about the z +axis in the Bloch sphere, similar to the time-independent +Hamiltonians [18]. Therefore, S can be fully expressed +by the angle between the Bloch vector and z axis (de- +noted by α). +More specifically, when cf ∈ [Θ/2, π/2], +S = {⃗r |α ∈ [αf, π − αf]} with αf = arcsin +� +sin(Θ/2) +sin cf +� +, +and S = {⃗r |α ∈ [Θ/2, π − Θ/2]} when cf > π/2. Fur- +thermore, the OQSL satisfies +´ τ +0 |f(t)|dt = Θ/2. +A +physical example here is f(t) = −gµBB cos(ωt)/2 [42] +with g the Lande factor, µB the electron magnetic mo- +ment and B cos(ωt) a periodic magnetic field. +Due to +the fact | +´ t +0 f(t1)dt1| ≤ gµBB/(2ω), S is determined +by the ratio between B and ω. The OQSL reads τ = +arcsin +� +ωΘ +gµBB +� +/ω, and the optimal states are the states in +the xy plane. It is obvious that τ ≤ π/(2ω) as arcsin(·) +is always less than or equal to π/2. This upper bound +is nothing but the time when the first degenerate point +occurs, which leads to an interesting phenomenon that +all targets can be fulfilled before the first degenerate point +occurs with the states in the xy plane. In the case that a +bounded control u(t) (|u(t)| ≤ ub) is invoked, f(t) be- +comes u(t) − gµBB cos(ωt)/2 and the upper bound of +| +´ t +0 f(t1)dt1| can always overcome π/2 at a long enough +time. Hence, in this case S = {⃗r |α∈[Θ/2, π −Θ/2]} and +the OQSL satisfies +´ τ +0 |gµBB cos(ωt)/2 − u(t)|dt = Θ/2. +The minimum τ with respect to u(t) (denoted by τmin) +satisfies the equation gµBB sin(ωτmin)/(2ω) + ubτmin = +Θ/2, and τmin ≈ Θ/(gµBB + 2ub) for a small ω. The +calculation details are in Ref. [41]. +Another practical scenario to apply Theorem 1 is the +one-dimensional Ising model with a longitudinal field, +where two boundary conditions (periodic and open) ex- +ist. +Let us first consider the case of periodic bound- +ary condition, in which the Hamiltonian reads H/J = +− �n +j=1 σz +j σz +j+1 − �n +j=1 g(t)σz +j with σz +n+1 = σz +1. +Here +J > 0 is the interaction strength of the nearest-neighbor +coupling, and g(t) is a global time-dependent longitudi- +nal field. +σz +j is the Pauli Z matrix for jth spin. +The +spin number n ≥ 3. In this case, the minimum energy is +−n[1+|g(t)|], and the maximum energy is n − η[2−|g(t)|] +when |g(t)| < 2 and n[|g(t)|−1] when |g(t)| ≥ 2. Here +η:=[1+(−1)n+1]/2. If |g(t)|≥2 for all time t, the OQSL +satisfies the equation +´ τ +0 |g(t)|dt = Θ/(2n). Due to the +fact that +´ τ +0 |g(t)|dt ≥ +´ τ +0 2dt = 2τ, one can immedi- +ately finds that τ ≤ Θ/(4n). If |g(t)| < 2 all the time, +Eq. (2) reduces to 2 (n − η) τ + (n + η) +´ τ +0 |g(t)|dt = Θ. +In this case τ ∈ +� Θ +4n, +Θ +2n−2η +� +since +´ τ +0 |g(t)|dt ∈ [0, 2τ]. +For a g(t) that is not always bounded by 2, the inte- +gration in Eq. (2) needs to be calculated part by part +and the rigorous solution may not easy to be acquired +in general. However, in some cases a good approxima- +⋯ +⋯ +⋯ +⋯ +fail +success +⋯ +⋯ +⋯ +⋯ +training set +classification +⋯ +⋯ +trained +training set +trained +regression +OQSL: +and +AC2nicjVHLSsNAFD2Nr1pfUXHlJlgEVyURUZdFNy4r2 +FZoS5nE0QbzYjIRSujGnbj1B9zqB4l/oH/hnTEFtYhOSHLm3HvOzL3XTQI/lb9WjKmpmdm58rzlYXFpeUVc3WtlcaZ8HjTi4NYnLs5YEf8ab0ZcDPE8FZ6Aa87V4fq3j7hovUj6MzOUx4L2RXkX/pe0wS1Tc3upJl/bwbMjkQYR5wJqLRqG9W7ZqtlzUJn +AJUaxGbL6giwvE8JAhBEcESTgAQ0pPBw5sJMT1kBMnCPk6zjFChbQZXHKYMRe0/eKdp2CjWivPFOt9uiUgF5BSgvbpIkpTxBWp1k6nmlnxf7mnWtPdbch/d3CKyRWYkDsX7px5n91qhaJSxzqGnyqKdGMqs4rXDLdFXVz60tVkhwS4hS+oLg7GnluM+W1q +S6dtVbpuNvOlOxau8VuRne1S1pwM7PcU6C1m7N2a85p3vV+lEx6jI2sYUdmucB6jhBA03yzvGIJzwbXePWuDPuP1ONUqFZx7dlPHwAYLeY8A=⌧learn +calibration +rigorous dynamics +ACyHicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdVl0I64qmLbQFkm0zqYJmEy +Urpxh9wq18m/oH+hXfGFNQiOiHJmXPvOTP3iAJRaoc57Vgzc0vLC4Vl0srq2vrG+XNrUYaZ5Jxj8VhLFuBn/JQRNxTQoW8lUjuD4OQN4PbMx1v3nGZiji6UqOEd4f+IBJ9wXxFlNcZq87kulxqo5Z9ixwc1Bvupx+QUd9BCDIcMQHBEU4RA+UnracOEgIa6LMXGSkDBxjglKpM0oi1OGT+wtfQe +0a+dsRHvtmRo1o1NCeiUpbeyRJqY8SVifZpt4Zpw1+5v32Hjqu43oH+ReQ2IVboj9SzfN/K9O16LQx4mpQVBNiWF0dSx3yUxX9M3tL1UpckiI07hHcUmYGeW0z7bRpKZ23VvfxN9Mpmb1nuW5Gd71LWnA7s9xzoLGQdU9qrqXh5XaT7qInawi32a5zFqOEcdHnkLPOIJz9aFlVj31ugz1Srkm18W9 +bDBwEQkVc={t} +rigorous dynamics +optimal state: +AC2nicjVHLSsNAFD2Nr1pfVXHlJlgEVyURUZeiG5cV7AP +aUibptA3mxWQiSOjGnbj1B9zqB4l/oH/hnTEFtYhOSHLm3HvOzL3XiX0vkZb1WjBmZufmF4qLpaXldW18vpGI4lS4fK6G/mRaDks4b4X8r0pM9bseAscHzedK7OVLx5zUXiReGlvIl5N2D0Bt4LpNE9cpbHTGKelknYHIkgsznTITjca9csaqWXuY0sHNQb5q +UfkFHfQRwUWKABwhJGEfDAk9bdiwEBPXRUacIOTpOMcYJdKmlMUpgxF7Rd8h7do5G9JeSZa7dIpPr2ClCZ2SRNRniCsTjN1PNXOiv3NO9Oe6m439Hdyr4BYiRGxf+kmf/VqVokBjWNXhU6wZVZ2bu6S6K+rm5peqJDnExCncp7g7GrlpM+m1iS6dtVbpuNv +OlOxau/muSne1S1pwPbPcU6Dxn7VPqzaFweVk9N81EVsYwd7NM8jnOAcNdTJO8MjnvBsdIxb4864/0w1CrlmE9+W8fABXkmY7w=⇢learn +AC2HicjVHLSsNAFD2Nr1pf0S7dBIvgqiQi6rLoxmUFW0UrJUmnbWiSCZOJUErBnbj1B9zqF4l/oH/hnTEFtYhOSHLm3HvOzL3XS8Iglb9WjBmZufmF4qLpaXldU1c32jmfJM+Kzh85CLC89NWRjErCEDGbKL +RDA38kJ27g2OVfz8hok04PGZHCbsOnJ7cdANfFcS1TbLdHn7VErcmVfRCOeyPG4bVbsq2XNQ2cHFSQrzo3X9BCBxw+MkRgiCEJh3CR0nMFBzYS4q4xIk4QCnScYwSaTPKYpThEjugb492Vzkb015plrt0ykhvYKUFrZJwylPEFanWTqeaWfF/uY90p7qbkP6e7lXRKxEn9i/dJPM/+pULRJdHOoaAqop0Yyqzs9dMt0VdXPrS1WSHBLiFO5QXBD2tXLSZ0trUl276q2r4286U7Fq7+e5Gd7VLWnAzs9xToPmbtXZrzqne5XaUT7qIjaxhR2a5wFqOEdDfIe4hFPeDYujVvjzrj/TDUKuaMb8t4+A8gpgc⇢opt +ACznicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdVl047KCfUBbJEmn7dBpEiaTQinFrT/gVj9L/AP9C+M +KahFdEKSM+ec2fuvX4seKIc5zVnLS2vrK7l1wsbm1vbO8XdvXoSpTJgtSASkWz6XsIED1lNcSVYM5bMG/mCNfzhlY43xkwmPApv1SRmnZHXD3mPB54iqtXuMqE8uy0H0V2x5JQds+xF4GaghGxVo+IL2ugiQoAUIzCEUIQFPCT0tODCQUxcB1PiJCFu4gwzFMibkoqRwiN2SN8+7VoZG9Je50yMO6BTBL2SnDaOyBORThLWp9kmnprMmv0t9Tk1Heb0N/P +co2IVRgQ+5dvrvyvT9ei0MOFqYFTbFhdHVBliU1XdE3t79UpShDTJzGXYpLwoFxzvtsG09iate9Uz8zSg1q/dBpk3xrm9JA3Z/jnMR1E/K7lnZvTktVS6zUedxgEMc0zPUcE1qiZj/iCc9W1RpbM+v+U2rlMs8+vi3r4QPFWZOa�⇢ +⌧opt +AC2HicjVHLSsNAFD3GV62vapdugkVwVdIq6LoxmUF+8C2lEk6bU +PzYjIRSim4E7f+gFv9IvEP9C+8M6agFtEJSc6ce8+ZufakefG0rJeF4zFpeWV1cxadn1jc2s7t7Nbj8NEOLzmhF4omjaLuecGvCZd6fFmJDjzbY837NG5ijduIjdMLiS4h3fDYI3L7rMElUN5dvS5Z0J2fyaHwJ2Ekp9NurmAVLb3MeVBKQHpqoa5F7TRQwgHCXxwBJCEPTDE9L +RQgoWIuA4mxAlCro5zTJElbUJZnDIYsSP6DmjXStmA9soz1mqHTvHoFaQ0cUCakPIEYXWaqeOJdlbsb94T7anuNqa/nXr5xEoMif1LN8v8r07VItHqa7BpZoizajqnNQl0V1RNze/VCXJISJO4R7FBWFHK2d9NrUm1rWr3jIdf9OZilV7J81N8K5uSQMu/RznPKiXi6WjYvnyuFA5S +0edwR72cUjzPEF6iRt5jPOIJz8a1cWvcGfefqcZCqsnj2zIePgA+RJgb +Figure 1. CRC methodology to learn the OQSL for complex +dynamics. The three steps are classification (gray box), re- +gression (orange box), and calibration (blue box). +tion can still be obtained since τ is usually small. Take +g(t) = B cos(ωt) as an example, where B and ω are the +amplitude and frequency. In this case, if ω is not very +large, then τ ≈ Θ/[2(n − η) + B(n + η)] when B <2 and +τ ≈ Θ/(2Bn) when B ≥ 2, which are nothing but the +OQSLs with respect to the constant field g(t) = B. +In the case of open boundary condition, the Hamilto- +nian reads − �n−1 +j=1 σz +j σz +j+1−�n +j=1 g(t)σz +j . The minimum +energy is −n[1+|g(t)|]+1, and the maximum energy is +n+η|g(t)|−1 when |g(t)| ≤ 1, n−(2−η)(2−|g(t)|)+1 +when |g(t)| ∈ (1, 2), and n[|g(t)| − 1] + 1 when |g(t)| ≥ 2. +For g(t)=B cos(ωt) with a not very large ω, an interest- +ing phenomenon occurs when B < 2 and n is even. The +OQSL in this case approximates to Θ/[n(B+2)−2] when +B ≤ 1, and Θ/[n(B + 2) + 2(B − 2)] when B ∈ (1, 2), +which are different from the OQSL under the periodic +boundary condition. These two OQSLs, as well as their +difference, are quite robust to global and local dephasing. +Therefore, the OQSL may be used to detect whether an +even-numbered spin ring is ruptured, especially when the +number is not very large. More details are in Ref. [41]. +CRC methodology. The brute-force search is the most +common method for the numerical evaluation of OQSL +and is easy to execute for simple dynamics. +However, +when the evaluation of dynamics for one state is too +time-consuming, the entire brute-force search would be +impossible to finish as it usually requires executing thou- +sand and even million rounds of dynamics. +In recent +years, machine learning has been successfully applied +to quantum physics for the simulation of complex dy- +namics, such as the theoretical dynamics of many-body +systems [43–45] and realistic dynamics of experimental + +t2 +t3miniti,t2,·.·jmin(t}3 +systems [46, 47]. With the help of trained neural net- +works, the computing time to evaluate the dynamics sig- +nificantly reduces compared to the rigorous calculation. +Therefore, such learning techniques could be powerful +tools to evaluate the OQSL. Hereby we provide a three- +step methodology (CRC methodology) based on learning +to evaluate the OQSL for complex dynamics. The three +steps are (1) classification; (2) regression; and (3) cal- +ibration, as illustrated in Fig. 1. +As a matter of fact, +classification and regression are two terminologies in su- +pervised learning. Classification is a problem to identify +the categories of objects and regression is to predict some +values related to the objects. +The reachable state set S is crucial in the evaluation of +OQSL. It is not only essential for the further calculation +of OQSL, but also reveals information that whether a +state is capable to fulfill the target. Hence, the first step +(classification) in CRC methodology is to find S. In this +step, a reasonable number of quantum states and cor- +responding binary labels (0 or 1) consist of the training +set. Quantum states and binary labels are the input and +output of the neural network. In our calculation, label +1 (0) represents the state is in (not in) S. The perfor- +mance of the trained network can be tested via a test set. +After the training and performance verification, a large +number of random states are input into the network to +construct S according to the outputs. In the following +the learned reachable state set in this step is denoted by +Slearn. +The second step is regression. In this step, a subset +of Slearn and the corresponding time to reach the target +consist of the training set. The time to reach the target +is extracted from the rigorous dynamics. Notice that it is +possible some states in this subset cannot fulfill the target +and need to be removed from the training set since Slearn +could be slightly different from S in practice. After the +training and performance verification, all states in Slearn +will be input into the trained network, and the minimum +output (τlearn) and corresponding states (ρlearn) are ex- +tracted. +In principle τlearn could be treated as an approximation +of OQSL. However, if the methodology stops here then +the accuracy of learned OQSL would be strongly affected +by the residuals, namely, the differences between the true +and predicted values. In the meantime, ρlearn may not be +the actual optimal state in the neighborhood due to the +existence of residuals. To further improve the methodol- +ogy’s performance, we introduce the third step: calibra- +tion. In this step, a reasonable region around ρlearn in the +state space is picked, and the dynamics of enough ran- +dom states in this region are calculated rigorously. Then +the minimum time to reach the target in this region (τopt) +and corresponding state (ρopt) are picked out. τopt is the +final evaluated value of OQSL in the methodology. +To verify the validity of CRC methodology, we ap- +ply it in the Landau-Zener model where the reachable +0 +2 +4 +6 + [units of v] +0.0 +0.4 +0.8 +1.2 +v +opt +noiseless dynamics +noisy dynamics +controlled noiseless dynamics +controlled noisy dynamics +/(2 ) +Figure 2. The OQSL as a function of ∆ in the cases of noise- +less dynamics (solid black line), noisy dynamics (red circles), +controlled noiseless dynamics (blue squares), and controlled +noisy dynamics (yellow triangles). The cyan dotted line rep- +resents Θ/(2∆). The target Θ = π/2. +state set and OQSL have been thoroughly discussed via +brute-force search among about one million states [18], +and thus the methodology’s performance is easy to be +tested. The Hamiltonian for the Landau-Zener model is +H = ∆σx + vtσz with ∆ and v two time-independent +parameters. In the step of classification, three training +sets with different numbers of data are used to train the +network and about one million states are used as the test +set. +The scores (correctness of prediction) are no less +than 99.59%, 97.83%, and 98.00% for all training sets in +the cases of ∆ = 0, 1, and 2. In the step of regression, +the mean square errors of learning are on the scale of +10−5 for ∆ = 0, 2, and no larger than 1.22 × 10−4 for +∆ = 1. In the last step, the region for calibration is cho- +sen as [αlearn−0.1, αlearn+0.1] and [φlearn−0.1, φlearn+0.1] +where αlearn and φlearn are the spherical coordinates of +ρlearn, i.e., cos(αlearn) = Tr(ρlearnσz) and cos(φlearn) = +Tr(ρlearnσx)/ sin(αlearn). The results of calibration show +that in this case ρlearn is just ρopt for all values of ∆, and +the corresponding τopt coincides with the exact OQSL ob- +tained from the brute-force search. The validity of CRC +methodology is then verified [41]. +One advantage of CRC methodology is that it can deal +with controlled dynamics, where the brute-force-search +evaluation is usually difficult to realize due to the com- +plexity of twofold optimizations. In the meantime, CRC +methodology can also deal with noisy scenarios where +the rigorous dynamics is usually more time-consuming +than the unitary counterpart. Let us still consider the +Landau-Zener model with the control Hamiltonian ⃗u · ⃗σ. +Here ⃗u = (ux, uy, uz) is the vector of control amplitudes +and ⃗σ = (σx, σy, σz) is the vector of Pauli matrices. +All control amplitudes are assumed to be in the regime +[−√v, √v]. +Both the noiseless and noisy scenarios are +studied. In the noisy scenario, the dynamics is governed +by the master equation ∂tρ = −i[H, ρ]+γ(σzρσz−ρ) with + +4 +101 +102 +n +0.0 +0.5 +1.0 +ratio +nonzero entry number: 2 +nonzero entry number: 3 +nonzero entry number: 10 +Figure 3. Ratio of states that can fulfill the target Θ = π/2 +in different categories. +The red pentagrams, green crosses, +and blue triangles represent the ratios for the states with 2, +3, and 10 nonzero entries. The dash-dotted red, dotted green, +and dashed blue lines represent the corresponding fitting func- +tions. +γ the decay rate, which is taken as 0.5√v as a demon- +stration. In this example, the evaluation of OQSL for +∆ = 0 via brute-force search among one million states +on a daily-use computer costs more than 830 days, which +reduces to 30 days when the CRC methodology is ap- +plied [? ]. The result of CRC methodology shows that +all states in the state space can fulfill the target Θ = π/2 +under control in both noisy and noiseless cases. Further- +more, the OQSL is very robust to the dephasing in both +noncontrolled and controlled cases, as shown in Fig. 2. In +the meantime, the controls can significantly reduce the +OQSL when ∆ is not very large. However, this improve- +ment becomes limited with the increase of ∆. An inter- +esting phenomenon is that regardless of the existence of +both noise and controls, the OQSL always converges to +Θ/(2∆), which is nothing but the OQSL for the Hamilto- +nian ∆σx in the absence of noise [18]. This phenomenon +on speed limit is difficult to be revealed by lower-bound- +type QSLs not only due to their dependence on both +initial states and time, but also the lousy attainability +when controls are involved. +Another example we studied is the transverse Ising +model with a periodic external field. +The Hamilto- +nian is H/J = −�n +j=1 σz +j σz +j+1−�n +j=1 g(t)σx +j with g(t) = +B cos(ωt). +In the demonstration, the amplitude B is +taken as 0.5 and the frequency ω/J = 1. Because of the +enormous state space (2n), it is difficult to construct a +training set that is general enough for the CRC method- +ology, especially when n is large. To feasibly apply the +CRC methodology, we need to analyze the state struc- +ture first and reduce the state space for the study. A +simple way to categorize the states is based on the num- +ber of nonzero entries in a certain basis, such as the basis +{|↑⟩ , |↓⟩}⊗n considered as follows. |↑⟩ (|↓⟩) is the eigen- +state of σz with respect to the eigenvalue 1 (−1). More- +over, here we consider the noiseless dynamics and hence +only pure states need to be studied. The ratios of reach- +able states for the target Θ = π/2 in the categories of +2 (red pentagrams), 3 (green crosses), and 10 nonzero +entries (blue triangles) are given in Fig. 3. The ratio in +each category is obtained from 2000 random states. It +can be seen that basically all states in each category can +fulfill the target when n is large, which is reasonable as +more target directions exist when the dimension is high. +Moreover, the ratio increases with the rise of the nonzero +entry number. More interestingly, the ratio in each cate- +gory basically fits the function 1/(1+anbe−cnd), and the +parameters a, b, c, d can be found in Ref. [41]. The gen- +eral behaviors of the ratio and the physical mechanism +behind it are still open questions that require further in- +vestigation. The minimum time to reach the target for all +states in each category is also investigated and the spe- +cific results are given in Ref. [41], which indicates that +in this example we only need to focus on the states with +few nonzero entries for the study of OQSL. +Next we perform the CRC methodology in the case of +n = 10. The methodology is applied to the categories +of states with 2 to 5 nonzero entries. Here we present +the result in the category of 2 nonzero entries, and oth- +ers are given in Ref. [41]. +22500 and 7500 states and +corresponding labels are used as the training and test +sets for the classification. The best score of the trained +network we obtained is 94.55%. Then about one million +states are input into this network, and the result shows +that 7.71% states can fulfill the target, close to the re- +sult (5.15%) obtained from 2000 random states. In the +regression process, 22500 and 7500 states consist of the +training and test sets. +The best mean square error is +8.95×10−4 and the corresponding τlearn is 0.24, close to +the true evolution time (0.19) of ρlearn. +About 10000 +states in the neighborhood of ρlearn are used in the cali- +bration and the final result is 0.18. Combing the results +of the other three categories, the final value of OQSL ob- +tained from the CRC methodology is 0.18, which can be +realized by some states with 2 nonzero entries. +Conclusion. 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L. Lehmann and G. Casella, Theory of Point Estima- +tion (Springer, New York, 1998). +[56] M. Zhang, H.-M. Yu, H. Yuan, X. Wang, R. Demkowicz- +Dobrzański, and J. Liu, QuanEstimation: +An open- +source toolkit for quantum parameter estimation, Phys. +Rev. Research 4, 043057 (2022). +[57] M. Innes, Don’t Unroll Adjoint: +Differentiating SSA- +form programs, arXiv:1810.07951. +Appendix A: The OQSL for time-dependent +Hamiltonian with time-independent eigenstates +1. +Proof of Theorem 1 +Consider the time-dependent Hamiltonian of the form +H(t) = +� +i +Ei(t) |Ei⟩ ⟨Ei| , +(A1) +where the energies are assumed to be ordered ascend- +ingly, i.e., E0(t) ≤ E1(t) ≤ · · · ≤ EN−1(t) (not all the +equalities are saturated simultaneously) and |Ei⟩ is in- +dependent of the time for any subscript i. +With this +Hamiltonian, the OQSL τ satisfies +ˆ τ +0 +EN−1(t) − E0(t)dt = Θ, +(A2) +where EN−1(t) and E0(t) are the highest and lowest ener- +gies of the Hamiltonian at time t. The proof is as follows. +In the Bloch representation, any N-dimensional den- +sity matrix ρ can be expressed by +ρ = 1 +N +� +11 + +� +N(N − 1) +2 +⃗r · ⃗λ +� +, +(A3) +where ⃗λ is the vector of SU(N) generators, ⃗r is the Bloch +vector satisfying |⃗r| ≤ 1 and 11 is the identity matrix. +In the case of unitary dynamics, any SU(N) generator +satisfies U(t)λiU †(t) = � +j Cij(t)λj with U(t) a unitary +operator, then the dynamics of ⃗r can be written as ⃗r(t) = +CT(t)⃗r. Utilizing the relation Tr(λiλj) = 2δij with δij +the Kronecker delta function, Cij(t) can be further solved +as Cij(t) = Tr(U(t)λiU †(t)λj)/2. +With the Hamiltonian (A1), the unitary operator can +be expressed by +U(t) = e−i � +m +´ t +0 Em(t1)dt1|Em⟩⟨Em| += +� +m +e−i +´ t +0 Em(t1)dt1 |Em⟩ ⟨Em| , +(A4) +which indicates +Cij(t) = 1 +2 +� +mn +ei +´ t +0 Em(t1)−En(t1)dt1[λi]∗ +mn[λj]mn +(A5) +with [λj]mn the mnth entry of λj. In the energy basis +{|E0⟩ , |E1⟩ , . . . , |EN−1⟩}, C(t) has the same structure +with the time-dependent Hamiltonian [18], i.e., +C(t) = +N−1 +� +n=1 +V (n, t), +(A6) +where V (n, t) = +�n−1 +� +i=0 +M(∆ni) +� � 1 with +M(x) = +� cos x − sin x +sin x +cos x +� +(A7) +and ∆ni = +´ t +0 En(t1)−Ei(t1)dt1. Then the angle between +the initial and evolved Bloch vectors is +cos θ = ⃗r(t) · ⃗r +|⃗r|2 += ⃗rTC(t)⃗r +|⃗r|2 +. +(A8) +Utilizing Eq. (A6), it can be further calculated as +cos θ=1 − +1 +|⃗r|2 +N−1 +� +n=1 +n−1 +� +i=0 +[1−cos(∆ni)](r2 +n2+2i−1+r2 +n2+2i) + +7 +with ri the ith element of ⃗r. Hence, the set S can be +directly expressed by +S = +� +⃗r +�� 1 − cos Θ = +1 +|⃗r|2 +N−1 +� +n=1 +n−1 +� +i=0 +[1 − cos(∆ni)] +× +� +r2 +n2+2i−1 + r2 +n2+2i +� +, ∃t +� +. +(A9) +To further obtain the OQSL, the two-step proof strat- +egy used in Appendix B in Ref. [18] needs to be applied. +Define +f(t) := +1 +|⃗r|2 +N−1 +� +n=1 +n−1 +� +i=0 +[1 − cos(∆ni)](r2 +n2+2i−1 + r2 +n2+2i). +Substituting the equation +ˆ τ +0 +EN−1(t) − E0(t)dt = Θ +(A10) +into the expression of f(t), it can be seen that +∂f(t) +∂t +��� +t=τ ≥ 0, +(A11) +which indicates τ is in the first monotonic increasing +regime of f(t). In the meantime, it can also be found +that f(τ) ≤ 1 − cos Θ. If the solution of f(t) = 1 − cos Θ +is not in the first increasing regime of f(t), then t is +obviously larger than τ; if this solution is in the first in- +creasing regime, then due to f(τ) ≤ 1 − cos Θ = f(t) one +can also see that t ≥ τ. Hence, τ is a lower bound of the +time to reach the target angle. +Now we discuss the optimal probe states to reach the +OQSL. To let the equation 1 − cos Θ = f(τ) holds, the +term r2 +n2+2i−1 + r2 +n2+2i for the subscripts n, i satisfying +∆ni ̸= +´ τ +0 EN−1(t) − E0(t)dt has to vanish. Further as- +sume the degeneracy of the ground states and highest ex- +cited states are p and q, namely, E0(t) = E1(t) = · · · = +Ep−1(t) and EN−q(t) = EN−q+1(t) = · · · = EN−1(t), +then it is easy to see that r2 +n2+2i−1 + r2 +n2+2i can only be +nonzero when n ∈ [N − q, N − 1] and i ∈ [0, p − 1], which +indicates that the optimal state is of the form +N−1 +� +i=0 +1 +N |Ei⟩ ⟨Ei| + +� +k∈[0,p−1], +l∈[N−q,N−1] +ξkl |Ek⟩ ⟨El| + ξ∗ +kl |El⟩ ⟨Ek| , +(A12) +where ξkl = +� +N−1 +2N (rl2+2k−1 − irl2+2k). In the energy +basis {|E0⟩ , |E1⟩ , . . . , |EN−1⟩}, the state above can be +written as +1 +N 11 + +� +� +0 +0 +ξ +0 +· · · 0 +ξ† +0 +0 +� +� , +(A13) +where 11 is a N-dimensional identity matrix, and ξ is a +p by q matrix with klth entry ξkl. +To make sure the +density matrix is positive-semidefinite, according to the +Schur complement theorem ξ needs to satisfy +ξ†ξ ≤ +1 +N 2 11q, +(A14) +where 11q is a q-dimensional identity matrix. The theorem +is then proved. +■ +2. +Example: two-level systems +Here we take a two-level system as a demonstration of +Theorem 1. Consider the Hamiltonian +H(t) = f(t)σz, +(A15) +where f(t) is a function of time t, and σz is a Pauli Z +matrix. In the Bloch representation, the evolved Bloch +vector can be solved as +rx(t) = rx cos +� +2 +ˆ t +0 +f(t1)dt1 +� +− ry sin +� +2 +ˆ t +0 +f(t1)dt1 +� +, +ry(t) = rx sin +� +2 +ˆ t +0 +f(t1)dt1 +� ++ ry cos +� +2 +ˆ t +0 +f(t1)dt1 +� +, +rz(t) = rz, +where (rx, ry, rz)T = ⃗r is the Bloch vector of the initial +state. Based on this dynamics, the angle between the +initial and evolved states is +cos θ = +cos +� +2 +´ t +0 f(t1)dt1 +� +(r2 +x + r2 +y) + r2 +z +|⃗r|2 +, +(A16) +which indicates that the time to reach the target angle +Θ satisfies the following equation +sin2 +�ˆ t +0 +f(t1)dt1 +� += +|⃗r|2 +|⃗r|2 − r2z +sin2 +�Θ +2 +� +. +(A17) +Rewrite ⃗r into +⃗r = η(sin α cos ϕ, sin α sin ϕ, cos α)T +(A18) +with η ∈ [0, 1], α ∈ [0, π] and ϕ ∈ [0, 2π], and Eq. (A17) +reduces to +sin2 α = +sin2 � Θ +2 +� +sin2 �´ t +0 f(t1)dt1 +�. +(A19) +Now consider that | +´ t +0 f(t1)dt1| is upper bounded by cf, +then in the case that cf < Θ/2, sin2 �´ t +0 f(t1)dt1 +� +is al- +ways less than sin2(Θ/2), which gives sin2 α > 1. This +means no state can fulfill the target as sin2 α is always +equal or less than 1. In the case that cf ∈ [Θ/2, π/2], +sin2 +�ˆ t +0 +f(t1)dt1 +� +≤ sin2 cf ≤ 1. +(A20) + +8 +Hence, +sin2 α ≥ sin2 � Θ +2 +� +sin2 cf +, +(A21) +indicating that the states that can fulfill the target sat- +isfies α ∈ [αf, π − αf] with +αf = arcsin +� +sin +� Θ +2 +� +sin cf +� +. +(A22) +In the case that cf > π/2, sin2[ +´ t +0 f(t1)dt1] can reach +all the values between 0 and 1, and sin2 α ≥ sin2(Θ/2), +therefore, the states satisfies α ∈ [Θ/2, π − Θ/2]. In a +word, the set S can be expressed by +S = +� +� +� +� +� +∅, +cf < Θ +2 , +{⃗r | α ∈ [αf, π − αf]}, +cf ∈ [ Θ +2 , π +2 ], +{⃗r | α ∈ [ Θ +2 , π − Θ +2 ]}, +cf > π +2 . +(A23) +Here ∅ is the empty set, and in the second and third +circumstances η ∈ (0, 1] and ϕ ∈ [0, 2π]. +With respect to the OQSL, due to the fact that the +eigenvalues are always f(t) and −f(t), the maximum and +minimum ones are always |f(t)| and −|f(t)|, respectively. +Based on Theorem 1, the OQSL τ then satisfies +ˆ τ +0 +|f(t)|dt = Θ +2 . +(A24) +A physical example for the Hamiltonian (A15) is the +energy splitting coming from Zeeman effect, i.e., +f(t) = −gµB +2 B(t) +(A25) +with g the Lande factor and µB the electron magnetic +moment. B(t) is the time-dependent magnetic field. For +a periodic magnetic field B(t) = B cos(ωt) with B, ω > 0, +it is easy to see +���� +ˆ t +0 +f(t1)dt1 +���� = +���� +gµBB +2ω +sin(ωt) +���� ≤ gµBB +2ω +. +(A26) +According to Eq. (A23), the set S reads +S = +� +� +� +� +� +∅, +gµBB +2ω +< Θ +2 , +{⃗r | α ∈ [αf, π − αf]}, +gµBB +2ω +∈ [ Θ +2 , π +2 ], +{⃗r | α ∈ [ Θ +2 , π − Θ +2 ]}, +gµBB +2ω +> π +2 . +(A27) +Here η ∈ (0, 1], ϕ ∈ [0, 2π] and +αf = arcsin +� +� +sin +� Θ +2 +� +sin +� +gµBB +2ω +� +� +� . +(A28) +Utilizing Theorem 1, Eq. (A24) can be written as +ˆ τ +0 +| cos(ωt)|dt = +Θ +gµBB . +(A29) +In the case that gµBB +2ω +< Θ +2 , τ = ∞ as no states can reach +the target. Hence we only consider the non-trivial case +that gµBB +2ω +≥ Θ +2 , which means +Θ +gµBB ≤ 1 +ω, and therefore +´ τ +0 | cos(ωt)|dt ≤ 1/ω, namely, +´ τ +0 | cos(ωt)|d(ωt) ≤ 1. +The integration of | cos(ωt)| is only less or equal to 1 +when ωt ≤ π/2, in which regime cos(ωt) is always non- +negative, hence, the integration is equivalent to be per- +formed on cos(ωt). Finally, the equation above can be +rewritten into +ˆ τ +0 +cos(ωt)dt = +Θ +gµBB , +(A30) +which immediately gives the analytical expression of τ as +below +τ = 1 +ω arcsin +� ωΘ +gµBB +� +. +(A31) +An interesting fact in this case is that the first degener- +ate point shows at t = π/(2ω), and the OQSL is always +less or equal to this time, indicating that the target Θ, +regardless of its value, can always be reached before this +first degeneracy point. +Next we consider a controlled case that +f(t) = −gµB +2 B cos(ωt) + u(t), +(A32) +where |u(t)| ≤ ub is a bounded control. Since +���� +ˆ t +0 +−gµB +2 B cos(ωt1) + u(t1)dt1 +���� += +���� +gµBB +2ω +sin(ωt) − +ˆ t +0 +u(t1)dt1 +���� +≤gµBB +2ω ++ +���� +ˆ t +0 +u(t1)dt1 +���� +≤gµBB +2ω ++ ubt, +(A33) +which can be larger than π/2 for a long enough time, in +this case +S = +� +⃗r +�� α ∈ [Θ +2 , π − Θ +2 ] +� +. +(A34) +The OQSL here satisfies +ˆ τ +0 +���� +gµBB +2 +cos(ωt) − u(t) +���� dt = Θ +2 . +(A35) +Then the minimum value of τ (denoted by τmin) can be +solved via the problem +τmin = min +u(t) τ, +subject to +�´ τ +0 | gµBB +2 +cos(ωt)−u(t)|dt = Θ +2 , +|u(t)| ≤ ub. + +9 +This problem can be solved by maximizing the function +| 1 +2gµBB cos(ωt) − u(t)| under the constraint that its in- +tegration is fixed. Since cos(ωt) is a monotonic function +within the regime [0, π/(2ω)], one could have +ˆ +π +2ω +0 +���� +gµBB +2 +cos(ωt) − u(t) +���� dt +≤ +ˆ +π +2ω +0 +�gµBB +2 +cos(ωt) + ub +� +dt +=gµBB +2ω ++ π +2ω ub. +(A36) +Notice the condition to make sure S ̸= ∅ is gµBB +2ω +≥ Θ +2 . In +this case, the upper bound of +´ +π +2ω +0 +| gµBB +2 +cos(ωt)−u(t)|dt +is larger than Θ/2, indicating that the integration will +reach Θ/2 before the time π/(2ω) with proper controls. +Hence, τmin must be less than π/(2ω). Under this con- +dition, the maximum value of | 1 +2gµBB cos(ωt) − u(t)| is +attained when u(t) ≡ −ub due to the fact that cos(ωt) is +a monotonic function here. Therefore, τmin satisfies the +equation +gµBB +2ω +sin(ωτmin) + ubτmin = Θ +2 . +(A37) +If ω is small, τmin approximates to +τmin ≈ +Θ +gµBB + 2ub +. +(A38) +3. +Example: one-dimensional Ising model with a +longitudinal field +a. +Periodic boundary condition +In the following we consider the one-dimensional Ising +model with a longitudinal field. The Hamiltonian of this +system reads +H/J = − +n +� +j=1 +σz +j σz +j+1 − +n +� +j=1 +g(t)σz +j , +(A39) +where J > 0 is the interaction strength of the nearest- +neighbor coupling, and g(t) is a global time-dependent +longitudinal field. σz +j is the Pauli Z matrix for jth spin. +The Hamiltonian satisfies the periodic boundary condi- +tion σz +n+1 = σz +1. +Here we only consider the case that +n ≥ 3. +Now we calculate the maximum and minimum eigen- +values of H/J. +Since the Hamiltonian only contains +the Pauli Z matrix, it is naturally a diagonal matrix in +the space consisting of the eigenspaces of σz +j for all j. +Denote |↑j⟩ and |↓j⟩ as the eigenstates of σz +j with re- +spect to the eigenvalues 1 and −1, then the eigenvalues +of −σz +j σz +j+1 − g(t)σz +j are 1 + g(t), 1 − g(t), −1 − g(t), +and −1 + g(t), and the corresponding eigenstates are +… +… +… +… +… +… +… +… +flip +…… +AC2XicjVHLTsMwEJyGVymv8rhxCVRInKoEIeBYwYVjkehDalHlpG6 +JmiaR4BK4cANceUHuMIPIf4A/oK1SWgQmArznh2Z+z1OpHvxdKyXjPGxOTU9Ex2Njc3v7C4lF9eqcZhIlxecUM/FHWHxdz3Al6RnvR5PRKc9R2f15zekYrXLriIvTA4lYOIn/VZN/A6nskUa382nVzg2YSMSHCy6ZgQdfnrXzBKlp6mOPATkEB6SiH+Rc0UYIFwn64AgCftgiGk2 +YMNCRNwZhsQJQp6Oc9wgR9qEsjhlMGJ7tHZp10jZgPbKM9Zql07x6ROkNLFmpDyBGF1mqnjiXZW7G/eQ+2p7jagv5N69YmVOCf2L90o8786VYtEBwe6Bo9qijSjqnNTl0S/irq5+aUqSQ4RcQq3KS4Iu1o5emdTa2Jdu3pbpuNvOlOxau+muQne1S2pwfbPdo6D6k7R3ivaJ7uF0mHa6i +zWsYlt6uc+SjhGRXyvsIjnvBsNIxb4864/0w1MqlmFd+G8fABSwKXsQ=|"i +AC23icjVHLSsNAFD2Nr/qOCm7cRIvgqiQi6lJ047KCbYW2y +CQdazDNhMnEUqord+LWH3Cr/yP+gf6Fd8YIahGdIZMz595zZu5cP4nCVLnuS8EaGR0bnyhOTk3PzM7N2wuLtVRkMuDVQERCnvgs5VEY86oKVcRPEslZ14943b840PH6JZdpKOJj1U94q8s6cXgWBkwRdWovXzVXabZFL2ZSil5TsrgT8VO75JZdM5xh4OWghHxUhP2MJt +oQCJChC4YinAEhpRmAx5cJMS1MCBOEgpNnOMaU6TNKItTBiP2gtYO7Ro5G9Ne6ZGHdApEX2SlA7WSMoTxLWpzkmnhlnzf7mPTCe+m59+vu5V5dYhXNi/9J9Zv5Xp2tROMOuqSGkmhLD6OqC3CUzr6Jv7nypSpFDQpzGbYpLwoFRfr6zYzSpqV2/LTPxV5OpWb0P8tw +Mb/qW1GDvZzuHQW2z7G2XvaOt0t5+3uoiVrCGDernDvZwiAq5H2FBziyWpZN9atdfeRahVyzRK+Dev+HZvgmJg=|#i +Figure 4. Schematic of obtaining any eigenstate of H/J by +flipping any number of |↑⟩ (black up arrow) into |↓⟩ (red down +arrow) in the state |↑⟩⊗n. +|↓j↑j+1⟩, |↑j↓j+1⟩, |↑j↑j+1⟩, and |↓j↓j+1⟩. +The eigen- +values of H/J can be obtained by the summation of +a certain number of these four terms. +For instance, +the eigenvalue with respect to the eigenstate |↑⟩⊗n is +�n +j=1[−1 − g(t)] = n[−1 − g(t)]. +Regarding the minimum eigenvalue of H/J, it is easy +to see that the minimum eigenvalue of −σz +j σz +j+1 − g(t)σz +j +is −1 − g(t) when g(t) ≥ 0 and −1 + g(t) when g(t) ≤ 0, +namely, −1 − |g(t)|. Therefore, the minimum eigenvalue +of H/J is +Emin,p = −n [1 + |g(t)|] , +(A40) +which can be attained by the eigenstate |↑⟩⊗n when +g(t) ≥ 0 and |↓⟩⊗n when g(t) ≤ 0. +Next, we calculate the maximum eigenvalue. For an +eigenstate ⊗n +j=1 |aj⟩ (aj = ↑, ↓), denote the number of +|↑j↑j+1⟩, |↓j↓j+1⟩, |↓j↑j+1⟩, and |↑j↓j+1⟩ (j ∈ [1, N]) are +x1, x2, x3, and x4, respectively. +For example, for the +state |↑↓↑⟩, x1 = 1, x2 = 0, and x3 = x4 = 1. Notice +that any eigenstate of H/J can be obtained by flipping +any number of |↑⟩ in the state |↑⟩⊗n into |↓⟩. As long +as the number of flipped spins is less than n, no matter +how many spins are flipped, there always exists a pair +of |↑↓⟩ and |↓↑⟩ at the boundary of the flipped spins, as +shown in Fig. 4. For example, assume a flip occurs at the +jth spin and k spins are flipped. Then the state of the +(j − 1)th and jth spins must be |↑j−1↓j⟩, and that of the +(j +k−1)th and (j +k)th spins must be |↓j+k−1↑j+k⟩. If +all the spins are flipped, no |↑↓⟩ and |↓↑⟩ exist in the state. +The simultaneous existence of |↑↓⟩ and |↓↑⟩ in the flip in- +dicates that x3 always equals to x4. Utilizing x1, x2, x3, +and the condition x1 + x2 + 2x3 = n, the eigenvalue of +H/J can be expressed by −[2+g(t)]x1+[−2+g(t)]x2+n. +Hence, the calculation of the maximum eigenvalue is + +10 +equivalent to a linear optimization problem: the maxi- +mization of −[2 + g(t)]x1 + [−2 + g(t)]x2 + n under some +constraints on x1, x2, and x3. It is easy to see that the +natural constraints on x1, x2, and x3 are 0 ≤ x1, x2 ≤ n +and 0 ≤ x3 ≤ ⌊n/2⌋. +Here ⌊·⌋ is the floor function. +Combing the equation x1 + x2 + 2x3 = n, the condition +0 ≤ x3 ≤ ⌊n/2⌋ is equivalent to 0 ≤ x1 + x2 ≤ n when n +is even and 1 ≤ x1 +x2 ≤ n when n is odd, which can be +unified as 1 +2[1+(−1)n+1] ≤ x1+x2 ≤ n. This condition is +fully contained by the constraint 0 ≤ x1, x2 ≤ n. Hence, +the full linear optimization problem can be expressed by +max +x1,x2 − [2 + g(t)] x1 + [−2 + g(t)] x2 + n, +subject to +� +η ≤ x1 + x2 ≤ n, +x1, x2 ∈ N. +(A41) +Here η := +1 +2[1 + (−1)n+1] and N is the set of natural +numbers. +To solve this problem, four cases have to be discussed: +(1) g(t) ≤ −2, (2) −2 < g(t) ≤ 0, (3) 0 < g(t) < 2, and +(4) g(t) ≥ 2. In the case that g(t) ≤ −2, the coefficients +−[2 + g(t)] ≥ 0 and −2 + g(t) ≤ 0, indicating that the +maximum eigenvalue is obtained when x1 is largest and +x2 vanishes, i.e., x1 = n, x2 = 0. +The corresponding +maximum eigenvalue is n[−g(t) − 1]. In the case that +g(t) ∈ (−2, 0], both coefficients −[2 + g(t)] and −2 + g(t) +are negative, and the maximum eigenvalue is attained +by the lower bounds of x1 and x2. +If n is even, the +minimum value of x1 and x2 are both zero, which leads to +the maximum value n. If n is odd, the maximum value is +attained by x1 = 1, x2 = 0 due to the fact that −[2+g(t)] +is larger than −2 + g(t). The corresponding maximum +value is n − 2 − g(t). In the case that g(t) ∈ (0, 2), the +situation is similar to the second one. +The maximum +value is n and attained by x1 = x2 = 0 when n is even. +For an odd n, the maximum value is n − 2 + g(t), which +can be attained by x1 = 0, x2 = 1. +In the last case +that g(t) ≥ 2, −[2 + g(t)] ≤ 0 and −2 + g(t) ≥ 0. The +maximum value is n[g(t)−1], which is attained by x1 = 0, +x2 = n. In summary, the maximum eigenvalue of H/J is +of the form +Emax,p = +� +n − η [2 − |g(t)|] , +|g(t)| < 2, +n [|g(t)| − 1] , +|g(t)| ≥ 2. +(A42) +Now we consider a specific case that g(t) = B cos(ωt), +where B is a positive amplitude and ω is the frequency. +The OQSL τ is solved via the equation +ˆ τ +0 +Emax,p(t) − Emin,p(t)dt = Θ. +(A43) +When B < 2, |g(t)| is always less than 2, which means +Emax,p always takes the form n − η [2 − |g(t)|], and +Eq. (A43) reduces to +2 (n − η) τ + (n + η) +ˆ τ +0 +|g(t)|dt = Θ. +(A44) +(a) +ACyXicjVHLSsNAFD2Nr1pf +VZdugkVwVRIRdVl0I7ipYB/QljJpzU2L5OJWIsrf8Ct/pj4B/oX3hmnoBbRCUnOnHvPmbn3OrHvpcKyXnPGzOzc/EJ+sbC0vLK6VlzfqKdRlri85kZ+lDQdlnLfC3lNeMLnzTjhLHB83nCGJzLeuO +FJ6kXhRjFvBOwQej1PZcJouptwbKu3S2WrLKljkNbA1K0KsaFV/QRg8RXGQIwBFCEPbBkNLTg0LMXEdjIlLCHkqznGPAmkzyuKUwYgd0ndAu5ZmQ9pLz1SpXTrFpzchpYkd0kSUlxCWp5kqnilny +f7mPVae8m4j+jvaKyBW4JLYv3STzP/qZC0CfRypGjyqKVaMrM7VLpnqiry5+aUqQ4xcRL3KJ4QdpVy0mdTaVJVu+wtU/E3lSlZuXd1boZ3eUsasP1znNOgvle2D8r2+X6pcqxHncWtrFL8zxEBae +okbeV3jE56NM+PauDXuPlONnNZs4tsyHj4AU3uRcw=⌧1 +ACyXicjVHLTsJAFD3UF+IL +demkZi4Ii0x6pLoxsQNJgImQMi0DjSl+3UiMSVP+BWf8z4B/oX3hlLohKj07Q9c+49Z+be60SeSKRlveaMmdm5+YX8YmFpeWV1rbi+0UjCNHZ53Q29ML5wWMI9EfC6FNLjF1HMme94vOkMj1W8ec +PjRITBuRxFvOzQSD6wmWSqEZbsrRb6RZLVtnSy5wGdgZKyFYtL6gjR5CuEjhgyOAJOyBIaGnBRsWIuI6GBMXExI6znGPAmlTyuKUwYgd0ndAu1bGBrRXnolWu3SKR29MShM7pAkpLyasTjN1PNXOi +v3Ne6w91d1G9HcyL59YiUti/9JNMv+rU7VI9HGoaxBU6QZVZ2buaS6K+rm5peqJDlExCnco3hM2NXKSZ9NrUl07aq3TMfdKZi1d7NclO8q1vSgO2f45wGjUrZ3i/bZ3ul6lE26jy2sI1dmucBqjh +BDXyvsIjnvBsnBrXxq1x95lq5DLNJr4t4+EDVduRdA=⌧2 +ACyHicjVHLSsNAFD2Nr1pfVZdugkVwFRIRdSOUuhFXFUwr1CJ +Oq1D82IyUpx4w+41S8T/0D/wjtjCmoRnZDkzLn3nJl7r5+GPJO2/VoyZmbn5hfKi5Wl5ZXVter6RitLchEwN0jCRFz6XsZCHjNXchmy1QwL/JD1vaHJyrevmUi40l8IUcp60beIOZ9HniSKLdx7Fj2dbVmW7Ze5jRwClBDsZpJ9QVX6CFBgBwRGJIwiE8ZPR04MBGSlwXY+I +EIa7jDPeokDanLEYZHrFD+g5o1ynYmPbKM9PqgE4J6RWkNLFDmoTyBGF1mqnjuXZW7G/eY+2p7jaiv194RcRK3BD7l26S+V+dqkWijyNdA6eaUs2o6oLCJdUTc3v1QlySElTuEexQXhQCsnfTa1JtO1q956Ov6mMxWr9kGRm+Nd3ZIG7Pwc5zRo7VnOgeWc79fqjWLUZWxhG7 +s0z0PUcYomXPLmeMQTno0zIzXujNFnqlEqNJv4toyHD2rNkE=B = 1.0 +ACyHicjVHLSsNAFD2Nr1pfVZdugkVwFRIVdSOUuhFXFewDap +EkndaheTGZKW48Qfc6peJf6B/4Z0xBbWITkhy5tx7zsy910sCnkrbfi0YM7Nz8wvFxdLS8srqWnl9o5nGmfBZw4+DWLQ9N2UBj1hDchmwdiKYG3oBa3nDUxVv3TKR8ji6lKOEdUN3EPE+91JVKN2sm/Z1+WKbdl6mdPAyUEF+arH5RdcoYcYPjKEYIgCQdwkdLTgQMbC +XFdjIkThLiOM9yjRNqMshluMQO6TugXSdnI9orz1SrfToloFeQ0sQOaWLKE4TVaOZ9pZsb95j7WnutuI/l7uFRIrcUPsX7pJ5n91qhaJPo51DZxqSjSjqvNzl0x3Rd3c/FKVJIeEOIV7FBeEfa2c9NnUmlTXrnr6vibzlSs2vt5boZ3dUsasPNznNOguWc5h5ZzcVCp +1vJRF7GFbezSPI9QxRnqaJA3xyOe8GycG4lxZ4w+U41CrtnEt2U8fABvkZBDB = 3.0 +ACyXicjVHLSsNAFD2Nr1p +fVZdugkVwVRIRdVl0I7ipYB/QljJpzU2L5OJWIsrf8Ct/pj4B/oX3hmnoBbRCUnOnHvPmbn3OrHvpcKyXnPGzOzc/EJ+sbC0vLK6VlzfqKdRlri85kZ+lDQdlnLfC3lNeMLnzTjhLHB83nCGJzLe +uOFJ6kXhRjFvBOwQej1PZcJouptwbKu3S2WrLKljkNbA1K0KsaFV/QRg8RXGQIwBFCEPbBkNLTg0LMXEdjIlLCHkqznGPAmkzyuKUwYgd0ndAu5ZmQ9pLz1SpXTrFpzchpYkd0kSUlxCWp5kqn +ilnyf7mPVae8m4j+jvaKyBW4JLYv3STzP/qZC0CfRypGjyqKVaMrM7VLpnqiry5+aUqQ4xcRL3KJ4QdpVy0mdTaVJVu+wtU/E3lSlZuXd1boZ3eUsasP1znNOgvle2D8r2+X6pcqxHncWtrFL8zx +EBaeokbeV3jE56NM+PauDXuPlONnNZs4tsyHj4AU3uRcw=⌧1 +ACyXicjVHLTsJAFD3UF+I +LdemkZi4Ii0x6pLoxsQNJgImQMi0DjSl+3UiMSVP+BWf8z4B/oX3hlLohKj07Q9c+49Z+be60SeSKRlveaMmdm5+YX8YmFpeWV1rbi+0UjCNHZ53Q29ML5wWMI9EfC6FNLjF1HMme94vOkMj1W8 +ecPjRITBuRxFvOzQSD6wmWSqEZbsrRb6RZLVtnSy5wGdgZKyFYtL6gjR5CuEjhgyOAJOyBIaGnBRsWIuI6GBMXExI6znGPAmlTyuKUwYgd0ndAu1bGBrRXnolWu3SKR29MShM7pAkpLyasTjN1P +NXOiv3Ne6w91d1G9HcyL59YiUti/9JNMv+rU7VI9HGoaxBU6QZVZ2buaS6K+rm5peqJDlExCnco3hM2NXKSZ9NrUl07aq3TMfdKZi1d7NclO8q1vSgO2f45wGjUrZ3i/bZ3ul6lE26jy2sI1dmuc +BqjhBDXyvsIjnvBsnBrXxq1x95lq5DLNJr4t4+EDVduRdA=⌧2 +(b) +Figure 5. Validity of the approximation with the changes of +(a) the amplitude B and (b) the frequency ω for n = 10 (solid +red lines), n = 15 (solid circle cyan lines), n = 20 (dashed blue +lines), and n = 55 (dash-dotted green lines). ω/J = 1 in (a), +and in (b) B = 1.0 and B = 3.0 for the upper and lower +panels, respectively. +For a not very large ω, +´ τ +0 |g(t)|dt = +B +ω sin(ωτ) ≈ Bτ. +Hence, +τ ≈ +Θ +2(n − η) + B(n + η) =: τ1. +(A45) +When B > 2, the relation between |g(t)| and 2 is not +fixed at different time. However, for a not very large ω, +τ is still very small in this case, which means Emax,p takes +the form n[g(t)−1] before the time τ, and +´ τ +0 |g(t)|dt still +approximates to Bτ. Therefore, according to Eq. (A43), +τ approximates to +τ ≈ +Θ +2Bn =: τ2. +(A46) +The validity of approximation is numerically tested +with the changes of amplitude B and frequency ω for +different spin number n. As shown in Fig. 5(a), the per- +formance of approximation is very well for different values +of B when ω is not extremely large [ω/J = 1 in the plot]. + +11 +As to the frequency ω, the approximation is valid when +ω is no larger than around 10 for both B = 1.0 [upper +panel in Fig. 5(b)] and B = 3.0 (lower panel). As a mat- +ter of fact, τ1 and τ2 are nothing but the OQSLs for the +constant external field g(t) = B. Hence, the validity of +approximation for a large regime of ω indicates that the +OQSL is way more sensitive to the amplitude than the +frequency as long as the frequency is not extremely large. +b. +Open boundary condition +Next we consider the case of the open boundary con- +dition. The corresponding Hamiltonian reads +H/J = − +n−1 +� +j=1 +σz +j σz +j+1 − +n +� +j=1 +g(t)σz +j . +(A47) +In this case, the minimum eigenvalue of −σz +j σz +j+1−g(t)σz +j +is −1−g(t) [−1+g(t)] for g(t) ≥ 0 [g(t) ≤ 0], which leads +to the minimum eigenvalue of H/J +Emin,o = −n [1 + |g(t)|] + 1. +(A48) +The minimum eigenvalue can be attained by the eigen- +state |↑⟩⊗n [|↓⟩⊗n] for g(t) ≥ 0 [g(t) < 0]. +To calculate the maximum eigenvalue, we rewrite the +Hamiltonian into the form +H/J = Hp + σz +nσz +1, +(A49) +where Hp is the Hamiltonian under the periodic bound- +ary condition. Now let us denote Emax,p and |Emax,p⟩ as +the maximum eigenvalue and corresponding eigenstate +of Hp, which is actually already obtained in the previous +discussion. +Notice that the eigenstates of Hp are also +eigenstates of σz +nσz +1, and the corresponding eigenvalues +can only be 1 and −1. Hence, if |Emax,p⟩ also corresponds +to the eigenvalue 1, i.e., σz +nσz +1 |Emax,p⟩ = |Emax,p⟩, then +the maximum energy for the entire Hamiltonian is just +Emax,p + 1. +As a matter of fact, this is just the case +for any n in the regime |g(t)| ≥ 2, and for odd n in the +regime |g(t)| < 2. Hence, the maximum eigenvalue Emax +for these cases reads +Emax,o = +� +n [|g(t)| − 1] + 1, +|g(t)| ≥ 2, +n + |g(t)| − 1, +|g(t)| < 2 and n is odd. +For an even n in the regime |g(t)| < 2, Emax,p − 1 = +n−1 may not be the maximum eigenvalue anymore. An- +other possible candidate must be among the eigenval- +ues of which the corresponding eigenstate |Ec⟩ satisfies +σz +nσz +1 |Ec⟩ = |Ec⟩. +It is obvious that we only need to +find the maximum eigenvalues in this case and compare +it with Emax,p − 1. This maximization problem can still +be formulated as a linear optimization problem as follows +max +x1,x2 − [2 + g(t)] x1 + [−2 + g(t)] x2 + n + 1, +subject to +� +� +� +� +� +2 ≤ x1 + x2 ≤ n, +x1, x2 ∈ N, +|g(t)| ≤ 2. +(A50) +The constraint x1 + x2 ≥ 2 comes from the fact that +σz +nσz +1 |Ec⟩ = |Ec⟩ is equivalent to require x1 ≥ 1 or x2 ≥ +1, and x1 + x2 + 2x3 = n requires x1 + x2 has to be an +even number when n is even. Hence, x1 + x2 has to be +no smaller than 2. Since both the coefficients −[2 + g(t)] +and −2 + g(t) are nonpositive in this case, the maximum +value must be attained by x1 = 2, x2 = 0 or x1 = 0, x2 = +2. +Therefore, in this case the maximum eigenvalue is +n+2|g(t)|−3. Next we need to compare the value between +n − 1 and n + 2|g(t)| − 3. +As a matter of fact, it is +easy to see when n − 1 is larger when |g(t)| < 1 and +n + 2|g(t)| − 3 is larger when |g(t)| > 1. In summary, the +maximum eigenvalue Emax,o under the open boundary +condition reads +� +� +� +� +� +� +� +� +� +n − 1, +|g(t)| ≤ 1 and n is even, +n + 2|g(t)| − 3, +1 < |g(t)| < 2 and n is even, +n + |g(t)| − 1, +|g(t)| < 2 and n is odd, +n [|g(t)| − 1] + 1, +|g(t)| ≥ 2. +(A51) +Utilizing the symbol η = [1 + (−1)n+1]/2, the equation +above can be rewritten into +Emax,o = +� +� +� +� +� +n + η|g(t)| − 1, +|g(t)| ≤ 1, +n − (2 − η)[2 − |g(t)|] + 1, +1 < |g(t)| < 2, +n [|g(t)| − 1] + 1, +|g(t)| ≥ 2. +(A52) +Next we calculate the OQSL. In the case that |g(t)| ≤ +1, τ satisfies the equation +(n + η) +ˆ τ +0 +|g(t)|dt + (2n − 2)τ = Θ. +(A53) +It is easy to see that here +´ τ +0 |g(t)|dt is less than τ, indi- +cating that +τ ≥ +Θ +3n − 2 + η . +(A54) +When 1 < |g(t)| < 2, τ satisfies +(n + 2 − η) +ˆ τ +0 +|g(t)|dt + (2n − 4 + 2η)τ = Θ, +(A55) +which gives +Θ +4n < τ < +Θ +3n − 2 + η +(A56) + +12 +due to the fact that τ < +´ τ +0 |g(t)|dt < 2τ. When |g(t)| ≥ +2, the OQSL satisfies +2n +ˆ τ +0 +|g(t)|dt = Θ, +(A57) +which means τ ≤ Θ/(4n). +Let us still consider a specific form of g(t) that g(t) = +B cos(ωt). Similar to the case with the periodic boundary +condition, the approximated expressions of OQSL can +also be analytically obtained utilizing the approximation +´ τ +0 |g(t)|dt ≈ Bτ for a not very large ω. In the regime +B ≥ 2, the OQSL is the same with τ2 [Eq. (A46)]. A +more interesting phenomenon occurs in the regime B < +2, where the OQSL is different from τ1 [Eq. (A45)] for an +even n. Specifically, the OQSL is +τ ≈ +Θ +nB + 2n − 2 =: τ3 +(A58) +when B ≤ 1, and it is +τ ≈ +Θ +nB + 2n + 2B − 4 +(A59) +when 1 < B < 2. The maximum gap between the OQSLs +for periodic and open boundary conditions happens at +the point B = 0, i.e., when no external field exists. In +this case, the OQSL can be rigorously solved and the +difference is +τ3 − τ1 = +Θ +2n(n − 1) =: ∆τ. +(A60) +The optimal states to realize τ1 and τ3 are in the form of +Eq. (A13). One thing that should be noticed is that the +dimension of ξ in the case of periodic boundary condi- +tion could be different from that in the case of the open +boundary condition due to the different degeneracy of +minimum and maximum energies in these two cases. +c. +Robustness analysis +The dependence on the boundary condition indicates +that the OQSL may be used to detect whether an even- +numbered spin ring is ruptured. To do that, one needs +to prepare the optimal states in Eq. (A12) and then +measure Tr(ρ0ρt) and Tr(ρ2 +t) at time τ3 and τ1, which +can be realized via techniques like randomized measure- +ments [48, 49]. Here ρ0 and ρt are the initial state and +evolved state at time t. After the measurement, the Bloch +angle can be calculated via the equation +cos(θ(t)) = +Tr(ρ0ρt) − 2−n +� +[Tr(ρ2 +0) − 2−n] [Tr(ρ2 +t) − 2−n] +. +(A61) +If the target is fulfilled at time τ3, then the ring is rup- +tured, and it is complete if the target is fulfilled at the +time τ1. +A more interesting fact is that the evolution time for +the states in Eq. (A12) is robust to the global and lo- +cal dephasing. The global dephasing is described by the +master equation +∂tρt = −i[H, ρt] + γg +� +JzρtJz − 1 +2 +� +ρt, J2 +z +�� +(A62) +with γg the decay rate and Jz = 1 +2 +�n +j=1 σz +j , and the local +dephasing is described by +∂tρt = −i[H, ρt] + +n +� +j=1 +γl,j +� +σz +j ρtσz +j − ρt +� +, +(A63) +where γl,j is the decay rate for jth spin. +Now we analytically discuss this robustness under +global and local dephasing. We need to emphasize that +the optimal states [Eq. (A12)] in the noiseless case may +not keep optimal when global and local dephasing are in- +volved, and the corresponding evolution time to reach the +target may also not be the OQSL anymore. The analysis +of OQSL under the noise requires the CRC methodology. +Here we only discuss the robustness of the evolution time +for the states in Eq. (A12). +Recall that the states in Eq. (A12) can be written into +Eq. (A13) in the basis {|E0⟩ , |E1⟩ , · · · , |E2n−1⟩}. With- +out the external field, the degeneracy of ground states +and the highest energy levels are both two. In the mean- +time, due to the fact that σz +j (for any j) and Jz are both +diagonal in this basis, we are allowed to denote Jz = +diag(A, . . . , G) with A and G 2-dimensional diagonal ma- +trices, and σz +j = diag(Cj, . . . , Dj) with Cj and Dj 2- +dimensional diagonal matrices. Utilizing these notations, +the master equation for global dephasing [Eq. (A62)] re- +duces to the evolution of the block ξ as follows +∂tξt = i(Emax − Emin)ξt + γgAξtG − γg +2 +� +ξtG2 + A2ξt +� +, +(A64) +where ξt is the evolved block at time t, and the one for +local dephasing [Eq. (A63)] reduces to +∂tξt = i(Emax − Emin)ξt + +� +j +γl,j (CjξtDj − ξt) . (A65) +As long as the specific forms of A, G, Cj, and Dj are +known, the dynamics can be easily solved. Next, we show +the calculations of these blocks. +It is not difficult to see that σz +j is easy to be expressed in +the basis {|↑⟩ , |↓⟩}⊗n, and the specific forms of σz +j (diago- +nal values) for different values of j are shown in Fig. 6(a), +where ⃗1k (−⃗1k) represents a k-dimensional vector with +all entries 1 (−1). To find the expressions of Cj and Dj, +we need to know the entry positions of minimum and +maximum energies for the Hamiltonian − � +j σz +j σz +j+1 and +extract the values of σz +j in the same positions to recon- +struct Cj and Dj. The expression of −σz +j σz +j+1 in the basis +{|↑⟩ , |↓⟩}⊗n for different values of j are given in Fig. 6(b). + +13 +−1!!"# +1!!"# +1!!"# +−1!!"# +−1!!"# +1!!"# +1!!"# +−1!!"# +−1!!"$ +1!!"$ +1!!"$ +−1!!"$ +1 +2 +𝑘 +… … +… … +−1 1 +1 −1 +−1 1 +1 −1 +−1 1 −1 1 +1 −1 1 −1 +… … +… … +… … +… … +1!!"%"& +1!!"%"& +1!!"%"& +… … +… … +… … +… … +… … +… … +1!!"$ +1!!"# +−1!!"# +−1!!"' +1!!"' +1!!"' +−1!!"' +−1!!"( +1!!"( +−1!!"( +−1!!"( +1!!"( +1!!"( +−1!!"( +(d) +2 +3 +4 +−1! +… … +… … +… … +… … +… … +… … +… … +1! +1!!"( +… … +1 +(b) +1!!"$ +−1!!"$ +1!!"$ +−1!!"$ +−1!!"& +1 +2 +𝑘 +… … +… … +1 +1 −1−1 +1 +1 −1−1 +1 −1 1 −1 +1 −1 1 −1 +… … +… … +… … +… … +1!!"% +−1!!"% +−1!!"% +1!!"% +−1!!"% +1!!"% +−1!!"% +… … +… … +… … +… … +… … +… … +(a) +1!!"$ +−1!!"# +1!!"' +1!!"' +−1!!"' +(c) +2 +3 +4 +… … +… … +… … +… … +… … +… … +… … +−1! +… … +1 +−1!!"# +1!!"' +−1!!"( +1 +−1 +1 +1 +−1!!"%"& +−1!!"%"& +−1!!"%"& +−1!!"%"& +1!!"%"& +1!!"& +… … +… … +… … +… … +−1!!"# +1!!"' +−1!!"( +−1! +1!!"% +1st block +2nd block +3rd block +4th block +AC6HicjVHLSsQwFD3W1/gedemOAiKMLSjqEvRjcs +RHBUclTRmnGpfpKkwlHvzp249Qfc6peIf6B/4U2s4APRlLYn5zkpt4SeCnyn +Ge6zev6BwdLQ8Mjo2PhEeXJqN40zyUWDx0Es9z2WisCPREP5KhD7iRQs9AKx5 +51v6vrehZCpH0c7qpOIw5CdRn7L50wRdVyuNFuS8dzt5kvd+dpRtFhbaIZMtWY +q/aliJTsdEnlVB0z7J/ALUAFxajH5Sc0cYIYHBlCERQhAMwpPQcwIWDhLhD5MR +JQr6pC3QxTN6MVIUjNhz+p7S7KBgI5rzNS4Oa0S0CvJaWOPDHpJG9m3qmU +nW7G/ZucnUe+vQ3yuyQmIV2sT+5ftQ/tene1FoYc304FNPiWF0d7xIycyp6J3b +n7pSlJAQp/EJ1SVhbpwf52wbT2p612fLTP3FKDWr57zQZnjVu6QLdr9f50+wW6u +6K1V3e7myvlFcdQkzmMU83ecq1rGFOhqUfYV7PODROrOurRvr9l1q9RSeaXwZ1t +0bjQ6d8Q=1 +3(2n + 2)th entry +AC6HicjVHLSsQwFD3W1/gedemOAiKMLSjqEvRjcsRHBUclTRmnGpfpKkwlHvzp249Qfc6p +eIf6B/4U2s4APRlLYn5zkpt4SeCnynGe6zev6BwdLQ8Mjo2PhEeXJqN40zyUWDx0Es9z2WisCPREP5KhD7iRQs9AKx51v6vrehZCpH0c7qpOIw5CdRn7L50wRdVyuNFuS8dzt5kvd+dpRtFhbaIZMtWYq/aliJTsdEnlVB0z7J/ALUAFxajH5Sc0cYIYHBlCERQhAMwpPQcwIWDhLhD5MRJQr6pC3QxTN6MVIUjNhz+p7S7KBgI5 +rzNS4Oa0S0CvJaWOPDHpJG9m3qmUnW7G/ZucnUe+vQ3yuyQmIV2sT+5ftQ/tene1FoYc304FNPiWF0d7xIycyp6J3bn7pSlJAQp/EJ1SVhbpwf52wbT2p612fLTP3FKDWr57zQZnjVu6QLdr9f50+wW6u6K1V3e7myvlFcdQkzmMU83ecq1rGFOhqUfYV7PODROrOurRvr9l1q9RSeaXwZ1t0bjQ6d8Q=1 +3(2n + 2)th entry +Figure 6. Schematic for the search of entry positions of the minimum and maximum energies. (a) The diagonal entry distribution +for σz +j ; (b) The diagonal entry distribution for −σz +j σz +j+1. [(c),(d)] The second blocks for σz +j and −σz +j σz +j+1 for the search of the +maximum energy. +In this diagram, searching the entry positions of the min- +imum and maximum energies is equivalent to searching +a column with the most number of −1 and 1. It can be +seen that the entries of −σz +j σz +j+1 for all values of j are +symmetric, indicating that the entire diagram can be di- +vided into four blocks, where the first and fourth (second +and third) blocks are mirror symmetric. The positions +with respect to the minimum energy are easy to locate +since only the first and last entries of −σz +j σz +j+1 are always +−1 for all values of j. Hence, their summation (summa- +tion of the column in dashed-red boxes) would also be +the minimum. In the meantime, the first and last entries +of σz +j are always 1 and −1 for all values of j, indicating +that Cj = σz. Moreover, due to the fact that Jz is half +of the summation of all σz +j , the entry positions in Jz that +correspond to the minimum energy are also the first and +last entries, which means A = diag(n/2, −n/2) = nσz/2. +For the sake of finding the entry positions of the max- +imum energy, we need to locate the position where the +entry is always 1 for any value of j, namely, a column in +the diagram where all entries are 1. It is obvious that +it can only exist in the second and third blocks. +Due +to the symmetry, we only need to consider the second +block. As shown in Fig. 6(d), a significant feature in this +block is that the overlap between the positions of ⃗1 in +the jth and (j + 1)th lines halves. More specifically to +say, compared to the position of ⃗1 in the jth line, only +the left (right) half in the same position keeps being 1 in +the (j + 1)th line if j is odd (even). For example, in the +first line (j = 1) all entries are 1, and hence the length +of ⃗1 is 2n−2. In the second line (j = 2), only the left +half keeps being one, and the length of ⃗1 becomes 2n−3. +Similarly, in the third line (j = 3) only the right half +keeps being 1 compared to the position of ⃗1 in the sec- +ond line. Utilizing this feature, one can find that when +n is even, the 1 +3(2n − 1)th and 1 +3(2n + 2)th entries keep +being 1 in the (n − 2)th line. Notice that the entry num- +ber here starts from the beginning of all diagonal entries +of −σz +j σz +j+1, not the beginning of the second block. And +in the (n − 1)th line, the +1 +3(2n + 2)th entry is 1. +In +the case of open boundary condition, this is the last line +and the position is located. In the case of the periodic +boundary condition, one more line of −σz +nσz +1 needs to +be considered. Luckily, this position of −σz +nσz +1 is also 1 +when n is even. Therefore, the maximum energy is at +the 1 +3(2n + 2)th entry under both boundary conditions. +Due to the symmetry, the 1 +3(2n+1 + 1)th entry, which is +in the third block, is also maximum. +Now we locate the values of 1 +3(2n + 2)th and 1 +3(2n+1 + + +n-2n-176 +2+1114 +1)th entries in σz +j , which is irrelevant to the boundary +condition. The block of entries in σz +j with respect to the +second block in Fig. 6(b) is given in Fig. 6(c). As shown +in this diagram, the 1 +3(2n + 2)th entry is 1 for an odd j +and −1 for an even j, namely, it is (−1)j+1. Similarly, +one can find that the 1 +3(2n+1+1)th entry is (−1)j. Hence, +Dj = (−1)j+1σz. In the meantime, both 1 +3(2n +2)th and +1 +3(2n+1+1)th entries are zero in Jz when n is even, which +means G = 0. +In summary, we have found that A = nσz/2, G = 0, +Cj = σz, and Dj = (−1)j+1σz. Utilizing these expres- +sions, Eqs. (A64) and (A65) can be further written into +∂tξt = +� +i(Emax − Emin) − n2γg +8 +� +ξt, +(A66) +and +∂tξt = i(Emax − Emin)ξt + +� +j +γl,j +� +(−1)j+1σzξtσz − ξt +� +. +(A67) +Equation (A66) can be easily solved as +ξt = e +� +i(Emax−Emin)− +n2γg +8 +� +ξ, +(A68) +and Eq. (A67) can be solved as +[ξt]00(11) =ei(Emax−Emin)−�n +j=1 γl,j[1+(−1)j][ξ]00(11), +[ξt]01(10) =ei(Emax−Emin)−�n +j=1 γl,j[1−(−1)j][ξ]01(10). +Here [·]ab represents the abth entry (a, b = 0, 1). +Next we calculate cos(θ(t)). Notice that Eq. (A61) can +be expressed by +cos(θ(t)) = +Re +� +Tr(ξξ† +t ) +� +� +Re (Tr(ξξ†)) Re +� +Tr(ξtξ† +t ) +�, +(A69) +where Re(·) represents the real part. In the case of global +dephasing [Eq. (A66)], the expression above reduces to +cos(θ(t)) = cos ((Emax − Emin)t) , +(A70) +which is irrelevant to the decay rate γ. Hence, the evo- +lution time to reach the target for the optimal states in +Eq. (A13) is indeed robust to the global dephasing in +both periodic and open boundary conditions, indicating +that their difference is also robust. +In the case of local dephasing, Eq. (A69) can be ex- +pressed by +cos(θ(t)) += cos ((Emax − Emin)t) +ς1 + ς2e−2tγall +√ς1 + ς2 +√ς1 + ς2e−4tγall , +where ς1 = |[ξt]00|2 + |[ξt]11|2, ς2 = |[ξt]01|2 + |[ξt]10|2, +and γall = �n +j=1 γl,j(−1)j. +If the values of all decay +AC2nicjVHLSsQwFD1T3+NrVFy5KQ6Cq6HVQd0IohuXCs4YEXSmNFgpy1pKkiZjTtx6w+41Q8S/0D/wptYwQeiKW1Pzr3nJPfeMI1kpj3vueIMDA4 +Nj4yOVcnJqemazOz7SzJFRctnkSJ6oQsE5GMRUtLHYlOqgTrhZE4DC92TPzwUqhMJvGBvkrFcY+dxbIrOdNEndTmg4Nzodlm0FWMF6v9otkPUnlSq3sNzy73J/BLUEe59pLaEwKcIgFHjh4EYmjCERgyeo7gw0NK3DEK4hQhaeMCfVRJm1OWoAxG7AV9z2h3VLIx7Y1nZtWcTonoVaR0sUSahPIUYXOa+O5dTbsb96F9TR3u6J/WHr1iNU4J/Yv3Ufmf3WmFo0uNmwNkmpKLWOq46VLbrtibu5+qkqTQ0qcwacUV4S5VX702bWazNZuests/MVmGtb +seZmb49Xckgbsfx/nT9BeafhrDX+/Wd/aLkc9igUsYpnmuY4t7GIPLfIucI8HPDqBc+3cOLfvqU6l1Mzhy3Lu3gBVY5gX +⇥ = 3 +4⇡ +Figure 7. The variety of the gap between the maximum and +minimum values of the evolution time to reach the target +Θ among 100 random states with random values of {γl,j} ∈ +(0, 1). The insets present the ratios of 10000 states at different +evolution time to reach the target Θ = 3π/4 for periodic +(red dots) and open (green pentagrams) boundary conditions. +n = 10 in all plots. +rates {γl,j} are very close, for example γl,j ≈ γ for +any j, then γall ≈ 0 and cos(θ(t)) still approximates to +cos ((Emax − Emin)t), which is also irrelevant to the de- +cay rates, and thus in this case the evolution time, as +well as the time difference, are also robust to the local +dephasing. In the case that the values of {γl,j} are not +close, Eq. (A69) is indeed dependent on the decay rates. +However, since ς1 + ς2e−2tγall is always positive at finite +time, the evolution time is still irrelevant to γall for the +target Θ = π/2 and hence robust to the local dephasing. +For a general target, we have tested 100 random states +in Eq. (A13) with random values of {γl,j} ∈ (0, 1) for +each target in the case of n = 10, and the gap between +the maximum and minimum values of the evolution time +for these 100 states are given in Fig. 7. It can be seen +that the robustness is quite good when the target is no +larger than π/2, and it is indeed compromised when Θ is +larger than π/2. Even for those targets with large gaps, +the evolution time for different states could concentrate +on some specific values, namely, the distribution of states +in the gap has a sharp peak. For example, the insets of +Fig. 7 show the distributions of 10000 states for periodic +(red dots) and open (green pentagrams) boundary con- +ditions in the case of Θ = 3π/4. +It can be seen that +the distributions for both periodic and open boundary +conditions have a sharp peak at the minimum values, in- +dicating that the evolution time is still relatively robust +for most states. + +15 +Appendix B: Learning the OQSL in Landau-Zener +model +1. +Verification of the validity of CRC methodology +Here we present the process of learning the OQSL in +the Landau-Zener model and show the validity of the +CRC methodology. The Hamiltonian of this model is +H = ∆σx + vtσz, +(B1) +where ∆ and v are two time-independent parameters. σx +and σz are the Pauli matrices. The OQSL in this model +has been thoroughly discussed in Ref. [18], in which the +set S is obtained via the brute-force search among around +one million pure states. The reason why only pure states +are considered here is due to the fact that unitary evo- +lution does not affect the purity and in the Bloch rep- +resentation all states in the same direction can/cannot +reach the target simultaneously. The dynamics is solved +via QuTiP [50, 51]. The full evolution is truncated at +vt = 10, namely, the state is treated to not be in S +if it cannot reach the target within the truncated time. +The Bloch vector of the initial state is parameterized by +⃗r = (sin α cos φ, sin α sin φ, cos α)T with α ∈ [0, π] and +φ ∈ [0, 2π). +In the step of classification, a multilayer neural net- +work with two inputs (α and φ) and one output (1 or 0) +is created with a hyperbolic tangent function as the ac- +tivation function. The output result 1/0 represents that +the input initial state can/cannot realize the given tar- +get, respectively. Supervised learning is performed via +Scikit-learn [52]. The network contains five to six hidden +layers each with about 200 to 250 neurons. The Cross- +Entropy loss function [53] is used as the loss function, +and Adam [54] is applied in the updates of the network. +The test set contains all the initial states (around one +million states) used in the brute-force search. The per- +formance of training for different values of ∆ are given +in Fig. 8(a). The first, second, and third rows represent +the learned S for ∆ = 0, ∆ = 1, and ∆ = 2 (in the +units of √v). The solid blue and dashed red lines repre- +sent the boundaries between S and its complementary set +obtained via supervised learning and brute-force search. +Different numbers of the training set, including 15000, +22500, and 30000, have also been tested and compared, +as shown in the first (15000), second (22500), and third +(30000) columns in Fig. 8(a). The percentage numbers +in the plots are the scores of learning, i.e., the correctness +of the network’s output. It can be seen that the perfor- +mance of 15000 training data is better than the others in +the case of ∆ = 0, and 22500 training data present the +best performance in the cases of ∆ = 1 and ∆ = 2. One +should notice that all the parameters of the network are +manually tuned case by case, and the slight difference in +the performance may not be fully due to the difference +in the training data number. In the case of 22500 train- +ing data, the correctness is around 99%, indicating that +about 0.99 million states are correctly classified into S +and its complementary set. Therefore, the neural net- +work indeed works for the classification in this example. +The second step is the regression process, in which +basically the same neural network is created but with +rectified linear unit function as the activation function. +The loss function is taken as the square error loss func- +tion [55]. The training data are sorted by the evolution +time to reach the target from smallest to largest. Sim- +ilar to the classification process, all the states in S are +used to test the performance of the network. Notice that +S here is the exact reachable state set obtained via the +brute-force search since we need to check the validity of +the network. The performance of regression is presented +for different values of ∆ and training data number in +Fig. 8(b). All the plots in this figure are semi-logarithmic +(x axis). The first, second, and third rows represent the +results for ∆ = 0, ∆ = 1, and ∆ = 2. The first, second, +and third columns represent the results for 15000, 22500, +and 30000 training data. The number in the plots are the +mean square errors of learning, i.e., 1 +m +�m +i=1 +� +t(i) +pre−t(i) +ext +�2. +Here t(i) +pre and t(i) +ext are the predicted time obtained via +learning and exact time obtained via brute-force search +for the ith state. +The order of states in the figure is +sorted by the evolution time obtained in the brute-force +search from smallest to largest, and the learned time is +plotted using the same order of states. Notice that these +states are not exactly the same for different values of ∆ +due to the dependence of S on ∆. It can be seen that +the performance of learning (solid blue lines) is good for +all values of ∆, especially when the training data num- +ber is 22500 and 30000. Basically the mean square errors +of learning in these two cases for all values of ∆ are in +the scale of 10−5. Hence, the network also works for the +regression in this example. +As a matter of fact, in practice the reachable state +set used in the regression process is the one obtained in +the classification process (denoted by Slearn). Hence, al- +though it is reasonable to use the true S to check the +validity of the regression, Slearn has to be applied to test +if the OQSL obtained from CRC methodology is rea- +sonable. The performance of regression with respect to +Slearn for 22500 training data is given in Fig. 8(c) for dif- +ferent values of ∆. The results for the other two training +data numbers are not shown here due to their similar- +ity. Since the training set chosen in Fig. 8(b) is also a +subset of Slearn, we can directly use it as the training set +in this case and the trained network is then the same. +The states in the plots are sorted by the evolution time +to reach the target from smallest to largest. As shown in +this figure, the trend of learned time basically coincides +with the exact time in Fig. 8(b). One should notice that +in fact these two lines cannot be compared directly as the + +16 +(a) +(b) +training data +number: 15000 +training data +number: 22500 +training data +number: 30000 +4.75×10-5 +4.04×10-5 +1.44×10-5 +1.22×10-4 +6.66×10-5 +5.52×10-5 +5.77×10-5 +3.96×10-5 +3.13×10-5 +99.63% +99.60% +99.59% +97.83% +98.98% +98.96% +98.00% +99.29% +99.02% +Predicted time +1.2552 +0.7310 +0.3859 +True time +1.2534 +0.7374 +0.3907 +Learned OQSL +1.2534 +0.7374 +0.3907 +Exact OQSL +1.2534 +0.7374 +0.3907 +(e) +(c) +(d) +AC2nicjVHLSsNAFD2Nr1pfUXHlJlgEVyURUZdFNy4r2FZ +oS5nE0QbzYjIRSujGnbj1B9zqB4l/oH/hnTEFtYhOSHLm3HvOzL3XTQI/lb9WjKmpmdm58rzlYXFpeUVc3WtlcaZ8HjTi4NYnLs5YEf8ab0ZcDPE8FZ6Aa87V4fq3j7hovUj6MzOUx4L2RXkX/pe0wS1Tc3upJl/bwbMjkQYR5wJqLRqG9W7ZqtlzUJnAJUaxG +bL6giwvE8JAhBEcESTgAQ0pPBw5sJMT1kBMnCPk6zjFChbQZXHKYMRe0/eKdp2CjWivPFOt9uiUgF5BSgvbpIkpTxBWp1k6nmlnxf7mnWtPdbch/d3CKyRWYkDsX7px5n91qhaJSxzqGnyqKdGMqs4rXDLdFXVz60tVkhwS4hS+oLg7GnluM+W1qS6dtVbpuNv 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(a) Comparison between the set S (brute-force search) and Slearn (learning) with different values of ∆ and different +training data number. The first, second, and third rows represent the results for ∆ = 0, ∆ = 1, and ∆ = 2, respectively. The +first, second, and third columns represent the results for 15000, 22500, and 30000 training data, respectively. The solid blue +(dashed red) lines represent the boundaries between S (Slearn) and its complementary set. The percentage numbers in the plots +are the scores of the learning. (b) Comparison of the evolution time to reach the target obtained from the regression process +(solid blue lines) and the exact time obtained from the brute-force search (dashed red lines). Here the input states in the +regression are the ones in S. The numbers in the plots are the mean square errors of learning. (c) The practical performance +of the regression process where the input states are those in Slearn. (d) Results of the calibration process. The region for +calibration is taken as [αlearn − 0.1, αlearn + 0.1] and [φlearn − 0.1, φlearn + 0.1]. The black dots represent (αlearn, φlearn), the +"optimal" points obtained in the regression. (e) Table of the predicted time (τlearn) obtained in the practical regression process, +the corresponding true time, and finally learned OQSL after the calibration process for different values of ∆. The exact OQSL +is obtained via brute-force search. In all plots v is set to be 1 and the target angle Θ = π/2. +states are not exactly the same. Utilizing the result of the +regression, the "optimal" state ρlearn and corresponding +predicted time τlearn can be located. The rigorous evo- +lution time of ρlearn to reach the target (true time) is +given in the table in Fig. 8(e). It can be seen that the +predicted time τlearn is very close to the true time for all +values of ∆. The errors in all cases are on the scale of +10−3, indicating that the regression process works well +in this example. +Furthermore, the true time of ρlearn +coincides with the exact OQSL obtained via brute-force +search, which means ρlearn is indeed an optimal state in +this case. +The last process is calibration. The core of this pro- +cess is to calculate the rigorous dynamics of the states +around ρlearn and find the exact minimum time in this +region. This process guarantees the finally obtained time +is the rigorous minimum time in this region. In this ex- +ample, the values of (α, φ) for the "optimal" states [de- +noted by (αlearn, φlearn)] in the cases of ∆ = 0, ∆ = 1, +and ∆ = 2 are (1.57, 1.69), (2, 78, 0.20), and (1.92, 4.78), +respectively [black dots in Fig. 8(d)]. +The region to +perform the calibration is [αlearn − 0.1, αlearn + 0.1] and +[φlearn−0.1, φlearn+0.1]. The rigorous dynamics of about +10000 states in this region are calculated. The results +are given in Fig. 8(d) and the corresponding minimum +time (learned OQSL) is given in the table in Fig. 8(e). + +suoervised +learning +orute-force +search17 +Algorithm 1: auto-GRAPE +Initialize the control amplitude uk(t) for all t and k; +for episode=1, M do +Receive initial state ρ0; +for t = 1, T do +Evolve with the control ρt = e∆tLtρt−1; +Calculate ft = +NTr(ρ0ρt)−1 +� +[NTr(ρ2 +0)−1][NTr(ρ2 +t )−1] and +save it; +end for +Calculate the objective function f = � +t ft. +Calculate the gradient +δf +δuk(t) with the automatic +differentiation method for all t and k. +for t = 1, T do +for k = 1, K do +Update control uk(t)←uk(t)+ϵ +δf +δuk(t). +end for +end for +end for +Save the controls {uk}. +The consistency between the learned OQSL and the ex- +act OQSL proves that the final result is indeed the exact +OQSL in this example. The validity of the CRC method- +ology is then confirmed. +2. +Learning the OQSL in the controlled system +Next, we apply the CRC methodology to search the +OQSL in the controlled Landau-Zener model. The full +Hamiltonian of this model reads +H = ∆σx + vtσz + ⃗u · ⃗σ, +(B2) +where ⃗σ = (σx, σy, σz) is the vector of Pauli matrices, +and ⃗u = (ux, uy, uz) is the vector of control amplitudes. +We first discuss the generation of controls for a specific +initial state to reach the target at the minimum time. +The controls are generated via the auto-GRAPE [56] with +the objective function +f = +ˆ T +0 +cos (θ(t)) dt, +(B3) +where T is a reasonably long time (truncated time in our +calculation), and θ(t) is the angle between the Bloch vec- +tors of initial state ρ0 and its evolved state ρt, which sat- +isfies the equation ∂tρt = Ltρt with Lt a time-dependent +superoperator. Notice that in the Bloch representation +the density matrix can be expressed by Eq. (A3). Then +cos(θ(t)) can be calculated by +cos(θ(t)) = +NTr(ρ0ρt) − 1 +� +[NTr(ρ2 +0) − 1] [NTr(ρ2 +t) − 1] +. +(B4) +In this case, the dynamics is unitary and only pure +states need to be calculated, then cos(θ(t)) reduces to +ACzHicjVHLSsNAFD2Nr1pfVZdugkV +wVRIRdVl040oq2FpS0m0xqaF5NJoYRu/QG3+l3iH+hfeGdMQS2iE5KcOfeM3PvdWPfS6RlvRaMhcWl5ZXiamltfWNzq7y90yiVDeYJEfiZbrJNz3Qt6QnvR5KxbcCVyf37qjCxW/HXOReF4Iycx7wbOMPQGHnMkU +XedMWeZmPbsXrliVS29zHlg56CfNWj8gs6CMCQ4oAHCEkYR8OEnrasGEhJq6LjDhByNxjilKpE0pi1OGQ+yIvkPatXM2pL3yTLSa0Sk+vYKUJg5IE1GeIKxOM3U81c6K/c07057qbhP6u7lXQKzEPbF/6WaZ/9WpWiQ +GONM1eFRTrBlVHctdUt0VdXPzS1WSHGLiFO5TXBmWjnrs6k1ia5d9dbR8TedqVi1Z3luind1Sxqw/XOc86B5VLVPqvb1caV2no+6iD3s45DmeYoaLlFHg7wDPOIJz8aVIY3MmH6mGoVcs4tvy3j4A1Vku8=~r1 +ACzHicjVHLSsNAFD2Nr1pfVZdugkV +wVZIi6rLoxpVUsA9pS0m0xqaF5NJoZRu/QG3+l3iH+hfeGdMQS2iE5KcOfeM3PvdWPfS6RlveaMpeWV1bX8emFjc2t7p7i710iVDBeZ5EfiZbrJNz3Ql6XnvR5KxbcCVyfN93RpYo3x1wkXhTeyknMu4EzDL2BxJ1 +F1nzNlUzHqVXrFklS29zEVgZ6CEbNWi4gs6CMCQ4oAHCEkYR8OEnrasGEhJq6LKXGCkKfjHDMUSJtSFqcMh9gRfYe0a2dsSHvlmWg1o1N8egUpTRyRJqI8QVidZup4qp0V+5v3VHuqu03o72ZeAbES98T+pZtn/lenapE +Y4FzX4FNsWZUdSxzSXVX1M3NL1VJcoiJU7hPcUGYaeW8z6bWJLp21VtHx90pmLVnmW5Kd7VLWnA9s9xLoJGpWyflu2bk1L1Iht1Hgc4xDHN8wxVXKGOnkHeMQTno1rQxpTY/aZauQyzT6+LePhAw+1kvA=~r2 +minimum +time +Figure 9. Performance of controls for two randomly generated +initial states ⃗r1 and ⃗r2 with different values of T (in the unit +of v), including T = 0.3 (green dots), T = 0.4 (dashed blue +lines), and T = 0.5 (solid red lines). +2Tr(ρ0ρt) − 1. In the numerical calculation, the evolu- +tion time is usually discretized into many equally spaced +time points ({ti}), and thus we can use the discrete form +f = +� +i +cos(θ(ti)) +(B5) +as the objective function instead. The time interval here +is neglected since it does not affect the final performance. +Auto-GRAPE is a gradient-based algorithm where the +gradient is evaluated via automatic differentiation [56]. +In Ref. [56] the quantum metrological quantities like +quantum Fisher information are taken as the objective +function, here in this paper we take Eq. (B5) as the ob- +jective function. The corresponding pseudocode is given +in Algorithm 1. In one episode, the initial state is evolved +to time T and the objective function is calculated. Then +the gradients δf/δuk(t) for all t and k are evaluated via +automatic differentiation, which is realized with the Julia +package Zygote [57]. At last, all the control amplitudes +are updated simultaneously according to the evaluation +of gradients. In practice, Adam [54] could be applied to +further improve efficiency. +To test the validity of the objective function [Eq. (B5)], +the performance of corresponding controls are demon- +strated in Fig. 9 for two randomly generated initial states, +of which the Bloch vectors are (0.22, 0.20, −0.96)T := ⃗r1 +and (0.95, −0.15, −0.29)T := ⃗r2. Three different values +of T (in the unit of v), including T = 0.3 (green dots), +T = 0.4 (dashed blue lines), and T = 0.5 (solid red lines) +are tested. As shown in the figure, the optimal controls +for T = 0.4 and 0.5 can let the states reach the target an- +gle (dotted black line) at the same time, confirming that +this found time (black dots) is indeed minimum. In the +meantime, if T is smaller than the minimum time, for ex- +ample T = 0.3, the states cannot reach the target angle +during the entire evolution, which also corroborates that + +18 +ACy3icjVHLSsNAFD2Nr1pfVZdugkVwVRIRdSMUdeFGqGAf0BaZTKc1mCYhmQi1uvQH3Op +/iX+gf+GdcQpqEZ2Q5My59yZe68XB34qHec1Z01Nz8zO5ecLC4tLyvF1bV6GmUJFzUeBVHS9FgqAj8UNenLQDTjRLCBF4iGd32s4o0bkaR+F7IYSw6A9YP/Z7PmSq2T4RgWSHzmWx5JQdvexJ4BpQglnVqPiCNrqIwJFhAIEQknAhpSeFlw4iInrYERcQsjXcYF7FMibkUqQghF7Td8+7VqGDWmvcqbazemUgN6EnDa2yBO +RLiGsTrN1PNOZFftb7pHOqe42pL9ncg2Ilbgi9i/fWPlfn6pFocDXYNPNcWaUdVxkyXTXVE3t79UJSlDTJzCXYonhLl2jvtsa0+qa1e9ZTr+pWKVXtutBne1S1pwO7PcU6C+k7Z3Su757ulypEZdR4b2MQ2zXMfFZyipqe4yOe8GydWal1a919Sq2c8azj27IePgDMGJIE� = 0 +ACy3icjVHLSsNAFD2Nr1pfVZdugkVwVRIRdSMUdeFGqGAf0BaZpNMaTJMwMxFqdekP 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CRC methodology for controlled noiseless dynamics. (a) The left column: The test set performance of the regression +process for ∆ = 0, 1, 2, 3, 4, 5, 6 (top to bottom). +The right column: The result of regression for ∆ = 0, 1, 2, 3, 4, 5, 6 (top +to bottom). +The solid blue and dashed red lines represent the learned time and exact time obtained from regression and +rigorous dynamics, respectively. The numbers in the left column are the mean square errors of learning. The x axes in both +columns are in the logarithmic scales. [(b1)-(b7)] Results of calibration for ∆ = 0, 1, 2, 3, 4, 5, 6. The regime for calibration is +[αlearn − 0.1, αlearn + 0.1] and [φlearn − 0.1, φlearn + 0.1]. The target Θ is taken as π/2 in all plots. +the found time is minimum as the states cannot reach the +target before this time. Hence, the validity of the objec- +tive function and corresponding controls are confirmed. +Moreover, the consistency of performance for T = 0.4 +and T = 0.5 shows that the choice of T does not affect +the result of minimum time as long as it is larger than +the minimum time. +Next, we perform the CRC methodology for ∆ = +0, 1, 2, 3, 4, 5, 6 (in the units of √v) in both noiseless and +noisy cases. In the noiseless case, the result of classifica- +tion shows that all states in the state space can fulfill the +target. This phenomenon is reasonable in physics due to +the full controllability of ⃗u · ⃗σ, which means the controls +can realize the rotation of a state from any angle. Thus, +any state can fulfill the target in finite time under this +control Hamiltonian even without the free Hamiltonian +∆σx + vtσz. +In the step of regression, the data number of train- +ing and test sets are 22500 and 7500. +Similar to the +noncontrolled case, the mean square errors between the +learned time (solid blue lines) and exact time (dashed +red lines) in the test set are still on the scales of 10−5 +and 10−6 for all values of ∆, as shown in the left col- +umn in Fig. 10(a). Utilizing this learned regression net- +work, about one million states are input and correspond- +ing learned time (solid blue lines) is given in the right +column in Fig. 10(a). +The minimum time τlearn for +∆ = 0, 1, 2, 3, 4, 5, 6 are 0.4154, 0.3080, 0.2315, 0.1830, +0.1486, 0.1261, and 0.1113, respectively. +In the last step, the region for calibration is still taken +as [αlearn− 0.1, αlearn + 0.1] and [φlearn − 0.1, φlearn + 0.1]. +The results of calibration are given in Figs. 10(b1)- +10(b7). +As shown in the plots, the optimal state ρopt +coincides with ρlearn in the cases of ∆ = 0, 1, 4. How- +ever, the position of ρopt slightly moves away from ρlearn +in other cases, which proves the necessity of the step of +calibration. +In the noisy case, the dephasing is invoked and the +dynamics of the density matrix is governed by the master +equation +∂tρt = −i[H, ρt] + γ(σzρtσz − ρt), +(B6) +where the Hamiltonian is in Eq. (B2) and γ is the decay +rate, which is taken as 0.5√v in the following calculation. +The CRC methodology has been applied in this case for +∆ = 0, 1, 2, 3, 4, 5, 6 (in the units of √v). The result of the +classification here is the same as that in the noiseless case, +i.e., all states can fulfill the target under control. The +results of regression and calibration are given in Fig. 11. +Similar to the noiseless case, the mean square errors of +regression are still on the scales of 10−5 and 10−6, as +shown in Fig. 11(a). The minimum time τlearn for ∆ = +0, 1, 2, 3, 4, 5, 6 are 0.3961, 0.2968, 0.2242, 0.1783, 0.1440, +0.1196, and 0.1063, respectively. 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CRC methodology for controlled noisy dynamics. (a) The left column: The test set performance of the regression +process for ∆ = 0, 1, 2, 3, 4, 5, 6 (top to bottom). +The right column: The result of regression for ∆ = 0, 1, 2, 3, 4, 5, 6 (top +to bottom). +The solid blue and dashed red lines represent the learned time and exact time obtained from regression and +rigorous dynamics, respectively. The numbers in the left column are the mean square errors of learning. The x axes in both +columns are in the logarithmic scales. [(b1)-(b7)] Results of calibration for ∆ = 0, 1, 2, 3, 4, 5, 6. The regime for calibration is +[αlearn − 0.1, αlearn + 0.1] and [φlearn − 0.1, φlearn + 0.1]. The target Θ is taken as π/2 in all plots. +region for calibration is still taken as [αlearn−0.1, αlearn+ +0.1] and [φlearn−0.1, φlearn+0.1], as shown in Figs. 11(b1)- +11(b7) for different values of ∆. The results show that +ρopt are either the same with ρlearn, or very close to it, +indicating that both regression and calibration processes +are effective in this case. +Appendix C: The OQSL for transverse Ising model +In this section we show the OQSL in the case of the +one-dimensional transverse Ising model, of which the +Hamiltonian is +H/J = − +n +� +j=1 +σz +j σz +j+1 − g(t) +n +� +j=1 +σx +j , +(C1) +where J is the interaction strength between the qubits, +and g(t) is the time-dependent strength of the external +field. Here we consider that g(t) = B cos(ωt). +Because of the enormous state space for this Hamil- +tonian, it is not easy to set up good training sets that +are general enough for the CRC methodology. To feasi- +bly apply the CRC methodology, we need to analyze the +state structure first and reduce the state space for the +study. A simple way to categorize the states is based on +the number of nonzero entries in a certain basis. There- +fore, we analyze the evolution time to reach the target +0 +25 +50 +75 +100 +n +0.1 +0.2 +0.3 +0.4 +0.5 +minimum time +nonzero entry number: 2 +nonzero entry number: 3 +nonzero entry number: 10 +Figure 12. +The minimum time to reach the target as a +function of spin number n for 2000 random states with 2 +nonzero entries (dotted-red-pentagram line), 3 nonzero entries +(dashed-green-cross line), and 10 nonzero entries (solid-blue- +triangle line). The parameters are set as Θ = π/2, B = 0.5, +and ω/J = 1. +for the states with different numbers of nonzero entries +in the basis {|↑⟩ , |↓⟩}⊗n. Here |↑⟩ (|↓⟩) is the eigenvalue +of σz with respect to the eigenvalue 1 (−1). The evolu- +tion time to reach the target has been calculated for 2000 + +20 +nonzero entry number +classification +regression +calibration +score +ratio +mean square error +τlearn +true value +optimal time +2 +94.55% +7.71% +8.95×10−4 +0.24 +0.19 +0.18 +3 +89.67% +5.85% +2.06×10−2 +0.18 +0.25 +0.24 +4 +85.44% +6.73% +1.69×10−2 +0.52 +0.37 +0.36 +5 +81.52% +9.48% +1.54×10−2 +0.54 +0.24 +0.24 +Table I. Results of CRC methodology in the categories of 2, 3, 4, and 5 nonzero entries. +AC2nicjVHLSsNAFD2Nr1pfVXHlJlgEVyURUZeiG5cV7APaUibptA3mxWQiSOjGnbj1B9zqB4l/oH/hnTEFtYhOSH +Lm3HvOzL3XiX0vkZb1WjBmZufmF4qLpaXldW18vpGI4lS4fK6G/mRaDks4b4X8r0pM9bseAscHzedK7OVLx5zUXiReGl +vIl5N2D0Bt4LpNE9cpbHTGKelknYHIkgsznTITjca9csaqWXuY0sHNQb5qUfkFHfQRwUWKABwhJGEfDAk9bdiwEBPXRU +acIOTpOMcYJdKmlMUpgxF7Rd8h7do5G9JeSZa7dIpPr2ClCZ2SRNRniCsTjN1PNXOiv3NO9Oe6m439Hdyr4BYiRGxf+km +mf/VqVokBjWNXhU6wZVZ2bu6S6K+rm5peqJDnExCncp7g7GrlpM+m1iS6dtVbpuNvOlOxau/muSne1S1pwPbPcU6Dxn +7VPqzaFweVk9N81EVsYwd7NM8jnOAcNdTJO8MjnvBsdIxb4864/0w1CrlmE9+W8fABXkmY7w=⇢learn +Figure 13. Calibration result in the category of states with 2 +nonzero entries. The green dot is the position of ρlearn. +random states in each category, and the minimum time +is given in Fig. 12. It can be seen that the minimum time +for the states with 2 (dotted-red-pentagram line) and 3 +(dashed-green-cross line) nonzero entries is always lower +than that for the states with 10 (solid-blue-triangle line) +nonzero entries when n is no larger than 100. This phe- +nomenon indicates that in this example we only need to +focus on the states with few nonzero entries for the study +of OQSL. +In the meantime, we found an interesting phenomenon. +The ratio of reachable states in the 2000 random states +basically fits the function +1 +1 + anbe−cnd . +(C2) +The parameters a, b, c, d are 1.132, 1.309, 0.005, 1.826 for +the category of 2 nonzero entries, and 0.450, 1.654, 0.034, +1.355 for the category of 3 nonzero entries, and 0.613, +0.842, 0.007, 1.754 for the category of 10 nonzero entries. +The true ratio in this case and the physical mechanism +behind it are still open questions and need to be further +investigated in the future. +Next, we perform the CRC methodology to evaluate +the OQSL. Since we only need to focus on the states with +few nonzero entries, the CRC methodology is applied in +the categories of states with 2, 3, 4, and 5 nonzero en- +tries. The results are given in Table I. In all cases, 22500 +and 7500 states and corresponding results (0 or 1) con- +sist of the training and test sets in the classification pro- +cess. The best score of the trained network we obtain +is 94.55%, 89.67%, 85.44%, and 81.52% in the categories +of 2, 3, 4, and 5 nonzero entries. The results show that +7.71%, 5.85%, 6.73%, and 9.48% states can fulfill the tar- +get in these categories. In the regression process, we also +use 22500 and 7500 states and the corresponding evo- +lution time to reach the target as the training and test +sets. The best mean square errors of the trained network +we obtain are 8.95 × 10−4, 0.0206, 0.0169, and 0.0154 +in the categories of 2, 3, 4, and 5 nonzero entries, and +corresponding values of τlearn are 0.24, 0.18, 0.52, and +0.54. The true values of the evolution time of ρlearn are +0.19, 0.25, 0.37, and 0.24. The gap between τlearn and +the true values are majorly affected by the mean square +errors, and it becomes difficult to obtain a good mean +square error with the increase of the nonzero entry num- +ber. +About 10000 random states in the neighborhood +of ρlearn are used in the process of calibration. +These +states share the same positions of nonzero entries with +ρlearn and the differences of the norms and phases be- +tween them and ρlearn are less than 0.1. The calibration +in the category of 2 nonzero entries is shown in Fig. 13. +The x and y axes are the norms of the nonzero entries +and the z axis is the phase difference between these two +entries. The green dot is the position of ρlearn. The other +three categories are not shown since the parameters are +larger than 3. After the calibration, the optimal evolu- +tion time in these categories is 0.18, 0.24, 0.36, and 0.24. +Hence, the final evaluation of OQSL in this example is +0.18, which can be realized by some states with 2 nonzero +entries. + diff --git a/HtAyT4oBgHgl3EQfrvl-/content/tmp_files/load_file.txt b/HtAyT4oBgHgl3EQfrvl-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..04c6a49cad8eb449eef2edf905ea7f240fe13f4a --- /dev/null +++ b/HtAyT4oBgHgl3EQfrvl-/content/tmp_files/load_file.txt @@ -0,0 +1,1939 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf,len=1938 +page_content='Quantum speed limit for complex dynamics Mao Zhang, Huai-Ming Yu, and Jing Liu∗ National Precise Gravity Measurement Facility, MOE Key Laboratory of Fundamental Physical Quantities Measurement, School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Quantum speed limit focuses on the minimum time scale for a fixed mission and hence is im- portant in quantum information where fast dynamics is usually beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Recently an operational definition of quantum speed limit (OQSL) was proposed, which reveals the intrinsic minimum time for time-independent Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, a general method to evaluate the OQSL for time- dependent Hamiltonians, especially when noises are involved, is still in lack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hereby we provide the expression of OQSL for a certain type of time-dependent Hamiltonians and propose a three-step (classification-regression-calibration) methodology based on machine learning for the evaluation of OQSL in complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Quantum speed limit (QSL) is a fundamental topic in quantum mechanics focusing on the characterization of minimum time for quantum states to fulfill certain known targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the year of 1945, Mandelstam and Tamm pro- vided the first lower bound for this minimum time based on the uncertainty relation [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In 1996 Braunstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' extended the lower bound to time-dependent Hamil- tonians utilizing the generalized uncertainty relation [2] where the time-average variance was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In 1998, Margolus and Levitin [3] provided another bound based on the mean energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' After these pioneer works, the topic of QSL entered a period of rapid development in the next 20 years, especially in 2010s [4–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The target in QSL could be defined via different tools, such as the Bures metric or various types of fidelity [8– 13], relative purity [14, 15], Bloch angle [16–19], gauge invariant distances [20, 21], and Wigner-Yanase informa- tion [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Throughout this paper, the target Θ ∈ (0, π] is defined via the Bloch angle θ(t,⃗r) = arccos � ⃗r·⃗r(t) |⃗r||⃗r(t)| � between a Bloch vector ⃗r and its evolved vector ⃗r(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Re- cently, an operational definition of quantum speed limit (OQSL) was proposed [18] based on the definition of reachable state set S := {⃗r |θ(t,⃗r) = Θ, ∃t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' With this set, the OQSL is defined by τ := min ⃗r∈S t subject to θ(t,⃗r) = Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (1) Compared to lower-bound-type QSLs, the advantages of OQSL are that it can reveal information that whether a state can fulfill the target, and it is always attainable [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, for complex dynamics these advantages come at a price of high computational complexity, which is not only due to the optimization in the definition, but also the preliminary assumption that S is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Regarding the fact that complex dynamics is a non-negligible sce- nario in the study of QSL [23–26], finding methods for the evaluation of OQSL that are friendly to the compu- tational complexity is critical, and thus the major moti- vation of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In many cases, the complexity of dynamics comes from the time dependency of the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The OQSL for a general time-dependent Hamiltonian is dif- ficult to obtain analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, for the time- dependent Hamiltonians with time-independent eigen- states, the OQSL can be derived analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the energy space, these Hamiltonians can be expressed by H(t) = � i Ei(t) |Ei⟩ ⟨Ei|, where the eigenstate |Ei⟩ is time-independent for any i and the eigenvalue Ei(t) de- pends on time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For such Hamiltonians, we present the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For a N-dimensional time-dependent Hamiltonian whose eigenstates are all time-independent, the OQSL τ satisfies the equation ˆ τ 0 [Emax(t) − Emin(t)] dt = Θ, (2) where Emax(t) and Emin(t) are the maximum and min- imum energies of the Hamiltonian at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Further denote the p-dimensional set {|Emin⟩} and q-dimensional set {|Emax⟩} as the sets of eigenstates with respect to Emin(t) and Emax(t), then the optimal states to reach the OQSL are � i 1 N |Ei⟩ ⟨Ei| + � |Ek⟩∈{|Emin⟩}, |El⟩∈{|Emax⟩} ξkl |Ek⟩ ⟨El| + ξ∗ kl |El⟩ ⟨Ek| , where the matrix ξ (with klth entry ξkl) satisfies N 2ξ†ξ ≤ 11q with 11q the q-dimensional identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The proof is given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As a matter of fact, this theorem covers Theorem 1 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [18] due to the fact that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (2) reduces to τ = Θ/(Emax − Emin) when the eigenvalues are time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As a simple demon- stration, consider the Hamiltonian H(t) = f(t)σz with σz the Pauli Z matrix and f(t) a time-dependent func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The other two Pauli matrices are denoted by σx and σy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It is obvious that the eigenstates of this Hamil- tonian are independent of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence the corresponding OQSL is given in the theorem above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that | ´ t 0 f(t1)dt1| is upper bounded by cf, S is fully deter- mined by the value of cf, which leads to the following no-go theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='00566v1 [quant-ph] 2 Jan 2023 2 No-go theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For the Hamiltonian H(t) = f(t)σz where f(t) satisfies | ´ t 0 f(t1)dt1| ≤ cf, no state can fulfill the target Θ if cf < Θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that cf ≥ Θ/2, S is symmetric about the z axis in the Bloch sphere, similar to the time-independent Hamiltonians [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, S can be fully expressed by the angle between the Bloch vector and z axis (de- noted by α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' More specifically, when cf ∈ [Θ/2, π/2], S = {⃗r |α ∈ [αf, π − αf]} with αf = arcsin � sin(Θ/2) sin cf � , and S = {⃗r |α ∈ [Θ/2, π − Θ/2]} when cf > π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Fur- thermore, the OQSL satisfies ´ τ 0 |f(t)|dt = Θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' A physical example here is f(t) = −gµBB cos(ωt)/2 [42] with g the Lande factor, µB the electron magnetic mo- ment and B cos(ωt) a periodic magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Due to the fact | ´ t 0 f(t1)dt1| ≤ gµBB/(2ω), S is determined by the ratio between B and ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The OQSL reads τ = arcsin � ωΘ gµBB � /ω, and the optimal states are the states in the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It is obvious that τ ≤ π/(2ω) as arcsin(·) is always less than or equal to π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' This upper bound is nothing but the time when the first degenerate point occurs, which leads to an interesting phenomenon that all targets can be fulfilled before the first degenerate point occurs with the states in the xy plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that a bounded control u(t) (|u(t)| ≤ ub) is invoked, f(t) be- comes u(t) − gµBB cos(ωt)/2 and the upper bound of | ´ t 0 f(t1)dt1| can always overcome π/2 at a long enough time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, in this case S = {⃗r |α∈[Θ/2, π −Θ/2]} and the OQSL satisfies ´ τ 0 |gµBB cos(ωt)/2 − u(t)|dt = Θ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The minimum τ with respect to u(t) (denoted by τmin) satisfies the equation gµBB sin(ωτmin)/(2ω) + ubτmin = Θ/2, and τmin ≈ Θ/(gµBB + 2ub) for a small ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The calculation details are in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Another practical scenario to apply Theorem 1 is the one-dimensional Ising model with a longitudinal field, where two boundary conditions (periodic and open) ex- ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Let us first consider the case of periodic bound- ary condition, in which the Hamiltonian reads H/J = − �n j=1 σz j σz j+1 − �n j=1 g(t)σz j with σz n+1 = σz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here J > 0 is the interaction strength of the nearest-neighbor coupling, and g(t) is a global time-dependent longitudi- nal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' σz j is the Pauli Z matrix for jth spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The spin number n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this case, the minimum energy is −n[1+|g(t)|], and the maximum energy is n − η[2−|g(t)|] when |g(t)| < 2 and n[|g(t)|−1] when |g(t)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here η:=[1+(−1)n+1]/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' If |g(t)|≥2 for all time t, the OQSL satisfies the equation ´ τ 0 |g(t)|dt = Θ/(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Due to the fact that ´ τ 0 |g(t)|dt ≥ ´ τ 0 2dt = 2τ, one can immedi- ately finds that τ ≤ Θ/(4n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' If |g(t)| < 2 all the time, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (2) reduces to 2 (n − η) τ + (n + η) ´ τ 0 |g(t)|dt = Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this case τ ∈ � Θ 4n, Θ 2n−2η � since ´ τ 0 |g(t)|dt ∈ [0, 2τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For a g(t) that is not always bounded by 2, the inte- gration in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (2) needs to be calculated part by part and the rigorous solution may not easy to be acquired in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' in some cases a good approxima- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='fail ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='success ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='training set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⋯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='trained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='training set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='trained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='regression ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='OQSL: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='AC2nicjVHLSsNAFD2Nr1pfUXHlJlgEVyURUZdFNy4r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' CRC methodology to learn the OQSL for complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The three steps are classification (gray box), re- gression (orange box), and calibration (blue box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' tion can still be obtained since τ is usually small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Take g(t) = B cos(ωt) as an example, where B and ω are the amplitude and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this case, if ω is not very large, then τ ≈ Θ/[2(n − η) + B(n + η)] when B <2 and τ ≈ Θ/(2Bn) when B ≥ 2, which are nothing but the OQSLs with respect to the constant field g(t) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case of open boundary condition, the Hamilto- nian reads − �n−1 j=1 σz j σz j+1−�n j=1 g(t)σz j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The minimum energy is −n[1+|g(t)|]+1, and the maximum energy is n+η|g(t)|−1 when |g(t)| ≤ 1, n−(2−η)(2−|g(t)|)+1 when |g(t)| ∈ (1, 2), and n[|g(t)| − 1] + 1 when |g(t)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For g(t)=B cos(ωt) with a not very large ω, an interest- ing phenomenon occurs when B < 2 and n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The OQSL in this case approximates to Θ/[n(B+2)−2] when B ≤ 1, and Θ/[n(B + 2) + 2(B − 2)] when B ∈ (1, 2), which are different from the OQSL under the periodic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' These two OQSLs, as well as their difference, are quite robust to global and local dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, the OQSL may be used to detect whether an even-numbered spin ring is ruptured, especially when the number is not very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' More details are in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' CRC methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The brute-force search is the most common method for the numerical evaluation of OQSL and is easy to execute for simple dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, when the evaluation of dynamics for one state is too time-consuming, the entire brute-force search would be impossible to finish as it usually requires executing thou- sand and even million rounds of dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In recent years, machine learning has been successfully applied to quantum physics for the simulation of complex dy- namics, such as the theoretical dynamics of many-body systems [43–45] and realistic dynamics of experimental t2 t3miniti,t2,·.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='·jmin(t}3 systems [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' With the help of trained neural net- works, the computing time to evaluate the dynamics sig- nificantly reduces compared to the rigorous calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, such learning techniques could be powerful tools to evaluate the OQSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hereby we provide a three- step methodology (CRC methodology) based on learning to evaluate the OQSL for complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The three steps are (1) classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (2) regression;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' and (3) cal- ibration, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As a matter of fact, classification and regression are two terminologies in su- pervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Classification is a problem to identify the categories of objects and regression is to predict some values related to the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The reachable state set S is crucial in the evaluation of OQSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It is not only essential for the further calculation of OQSL, but also reveals information that whether a state is capable to fulfill the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the first step (classification) in CRC methodology is to find S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this step, a reasonable number of quantum states and cor- responding binary labels (0 or 1) consist of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Quantum states and binary labels are the input and output of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In our calculation, label 1 (0) represents the state is in (not in) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The perfor- mance of the trained network can be tested via a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' After the training and performance verification, a large number of random states are input into the network to construct S according to the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the following the learned reachable state set in this step is denoted by Slearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The second step is regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this step, a subset of Slearn and the corresponding time to reach the target consist of the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The time to reach the target is extracted from the rigorous dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Notice that it is possible some states in this subset cannot fulfill the target and need to be removed from the training set since Slearn could be slightly different from S in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' After the training and performance verification, all states in Slearn will be input into the trained network, and the minimum output (τlearn) and corresponding states (ρlearn) are ex- tracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In principle τlearn could be treated as an approximation of OQSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, if the methodology stops here then the accuracy of learned OQSL would be strongly affected by the residuals, namely, the differences between the true and predicted values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the meantime, ρlearn may not be the actual optimal state in the neighborhood due to the existence of residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To further improve the methodol- ogy’s performance, we introduce the third step: calibra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this step, a reasonable region around ρlearn in the state space is picked, and the dynamics of enough ran- dom states in this region are calculated rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Then the minimum time to reach the target in this region (τopt) and corresponding state (ρopt) are picked out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' τopt is the final evaluated value of OQSL in the methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To verify the validity of CRC methodology, we ap- ply it in the Landau-Zener model where the reachable 0 2 4 6 [units of v] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='2 v opt noiseless dynamics noisy dynamics controlled noiseless dynamics controlled noisy dynamics /(2 ) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The OQSL as a function of ∆ in the cases of noise- less dynamics (solid black line), noisy dynamics (red circles), controlled noiseless dynamics (blue squares), and controlled noisy dynamics (yellow triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The cyan dotted line rep- resents Θ/(2∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The target Θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' state set and OQSL have been thoroughly discussed via brute-force search among about one million states [18], and thus the methodology’s performance is easy to be tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The Hamiltonian for the Landau-Zener model is H = ∆σx + vtσz with ∆ and v two time-independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the step of classification, three training sets with different numbers of data are used to train the network and about one million states are used as the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The scores (correctness of prediction) are no less than 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='59%, 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='83%, and 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='00% for all training sets in the cases of ∆ = 0, 1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the step of regression, the mean square errors of learning are on the scale of 10−5 for ∆ = 0, 2, and no larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='22 × 10−4 for ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the last step, the region for calibration is cho- sen as [αlearn−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='1, αlearn+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='1] and [φlearn−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='1, φlearn+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='1] where αlearn and φlearn are the spherical coordinates of ρlearn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', cos(αlearn) = Tr(ρlearnσz) and cos(φlearn) = Tr(ρlearnσx)/ sin(αlearn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The results of calibration show that in this case ρlearn is just ρopt for all values of ∆, and the corresponding τopt coincides with the exact OQSL ob- tained from the brute-force search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The validity of CRC methodology is then verified [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' One advantage of CRC methodology is that it can deal with controlled dynamics, where the brute-force-search evaluation is usually difficult to realize due to the com- plexity of twofold optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the meantime, CRC methodology can also deal with noisy scenarios where the rigorous dynamics is usually more time-consuming than the unitary counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Let us still consider the Landau-Zener model with the control Hamiltonian ⃗u · ⃗σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here ⃗u = (ux, uy, uz) is the vector of control amplitudes and ⃗σ = (σx, σy, σz) is the vector of Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' All control amplitudes are assumed to be in the regime [−√v, √v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Both the noiseless and noisy scenarios are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the noisy scenario, the dynamics is governed by the master equation ∂tρ = −i[H, ρ]+γ(σzρσz−ρ) with 4 101 102 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='0 ratio nonzero entry number: 2 nonzero entry number: 3 nonzero entry number: 10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Ratio of states that can fulfill the target Θ = π/2 in different categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The red pentagrams, green crosses, and blue triangles represent the ratios for the states with 2, 3, and 10 nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The dash-dotted red, dotted green, and dashed blue lines represent the corresponding fitting func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' γ the decay rate, which is taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='5√v as a demon- stration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this example, the evaluation of OQSL for ∆ = 0 via brute-force search among one million states on a daily-use computer costs more than 830 days, which reduces to 30 days when the CRC methodology is ap- plied [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The result of CRC methodology shows that all states in the state space can fulfill the target Θ = π/2 under control in both noisy and noiseless cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Further- more, the OQSL is very robust to the dephasing in both noncontrolled and controlled cases, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the meantime, the controls can significantly reduce the OQSL when ∆ is not very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, this improve- ment becomes limited with the increase of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' An inter- esting phenomenon is that regardless of the existence of both noise and controls, the OQSL always converges to Θ/(2∆), which is nothing but the OQSL for the Hamilto- nian ∆σx in the absence of noise [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' This phenomenon on speed limit is difficult to be revealed by lower-bound- type QSLs not only due to their dependence on both initial states and time, but also the lousy attainability when controls are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Another example we studied is the transverse Ising model with a periodic external field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The Hamilto- nian is H/J = −�n j=1 σz j σz j+1−�n j=1 g(t)σx j with g(t) = B cos(ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the demonstration, the amplitude B is taken as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='5 and the frequency ω/J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Because of the enormous state space (2n), it is difficult to construct a training set that is general enough for the CRC method- ology, especially when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To feasibly apply the CRC methodology, we need to analyze the state struc- ture first and reduce the state space for the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' A simple way to categorize the states is based on the num- ber of nonzero entries in a certain basis, such as the basis {|↑⟩ , |↓⟩}⊗n considered as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' |↑⟩ (|↓⟩) is the eigen- state of σz with respect to the eigenvalue 1 (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' More- over, here we consider the noiseless dynamics and hence only pure states need to be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The ratios of reach- able states for the target Θ = π/2 in the categories of 2 (red pentagrams), 3 (green crosses), and 10 nonzero entries (blue triangles) are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The ratio in each category is obtained from 2000 random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It can be seen that basically all states in each category can fulfill the target when n is large, which is reasonable as more target directions exist when the dimension is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Moreover, the ratio increases with the rise of the nonzero entry number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' More interestingly, the ratio in each cate- gory basically fits the function 1/(1+anbe−cnd), and the parameters a, b, c, d can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The gen- eral behaviors of the ratio and the physical mechanism behind it are still open questions that require further in- vestigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The minimum time to reach the target for all states in each category is also investigated and the spe- cific results are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [41], which indicates that in this example we only need to focus on the states with few nonzero entries for the study of OQSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Next we perform the CRC methodology in the case of n = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The methodology is applied to the categories of states with 2 to 5 nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here we present the result in the category of 2 nonzero entries, and oth- ers are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 22500 and 7500 states and corresponding labels are used as the training and test sets for the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The best score of the trained network we obtained is 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='55%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Then about one million states are input into this network, and the result shows that 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='71% states can fulfill the target, close to the re- sult (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='15%) obtained from 2000 random states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the regression process, 22500 and 7500 states consist of the training and test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The best mean square error is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='95×10−4 and the corresponding τlearn is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='24, close to the true evolution time (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='19) of ρlearn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' About 10000 states in the neighborhood of ρlearn are used in the cali- bration and the final result is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Combing the results of the other three categories, the final value of OQSL ob- tained from the CRC methodology is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='18, which can be realized by some states with 2 nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this paper we focus on the evaluation of OQSL for time-dependent Hamiltonians and complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' We first provide the expression of OQSL for a type of time-dependent Hamiltonians whose eigenstates are time-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For complex dynamics, such as the controlled dissipation and dynamics of many-body systems, the CRC methodology is introduced and demon- strated for the evaluation of the OQSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The authors thank Yuqian Xu for helpful discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 12175075).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 5 ∗ liujingphys@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='cn [1] L.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 12, 2825 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [53] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Baum and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Wilczek, Supervised learning of prob- ability distributions by neural networks (American Insti- tute of Physics, New York, 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [54] D.' metadata={'source': 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+page_content=' Lehmann and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Casella, Theory of Point Estima- tion (Springer, New York, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [56] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Yuan, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Wang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Demkowicz- Dobrzański, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Liu, QuanEstimation: An open- source toolkit for quantum parameter estimation, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Research 4, 043057 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [57] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Innes, Don’t Unroll Adjoint: Differentiating SSA- form programs, arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='07951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Appendix A: The OQSL for time-dependent Hamiltonian with time-independent eigenstates 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Proof of Theorem 1 Consider the time-dependent Hamiltonian of the form H(t) = � i Ei(t) |Ei⟩ ⟨Ei| , (A1) where the energies are assumed to be ordered ascend- ingly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', E0(t) ≤ E1(t) ≤ · · · ≤ EN−1(t) (not all the equalities are saturated simultaneously) and |Ei⟩ is in- dependent of the time for any subscript i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' With this Hamiltonian, the OQSL τ satisfies ˆ τ 0 EN−1(t) − E0(t)dt = Θ, (A2) where EN−1(t) and E0(t) are the highest and lowest ener- gies of the Hamiltonian at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The proof is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the Bloch representation, any N-dimensional den- sity matrix ρ can be expressed by ρ = 1 N � 11 + � N(N − 1) 2 ⃗r · ⃗λ � , (A3) where ⃗λ is the vector of SU(N) generators, ⃗r is the Bloch vector satisfying |⃗r| ≤ 1 and 11 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case of unitary dynamics, any SU(N) generator satisfies U(t)λiU †(t) = � j Cij(t)λj with U(t) a unitary operator, then the dynamics of ⃗r can be written as ⃗r(t) = CT(t)⃗r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Utilizing the relation Tr(λiλj) = 2δij with δij the Kronecker delta function, Cij(t) can be further solved as Cij(t) = Tr(U(t)λiU †(t)λj)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' With the Hamiltonian (A1), the unitary operator can be expressed by U(t) = e−i � m ´ t 0 Em(t1)dt1|Em⟩⟨Em| = � m e−i ´ t 0 Em(t1)dt1 |Em⟩ ⟨Em| , (A4) which indicates Cij(t) = 1 2 � mn ei ´ t 0 Em(t1)−En(t1)dt1[λi]∗ mn[λj]mn (A5) with [λj]mn the mnth entry of λj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the energy basis {|E0⟩ , |E1⟩ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' , |EN−1⟩}, C(t) has the same structure with the time-dependent Hamiltonian [18], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', C(t) = N−1 � n=1 V (n, t), (A6) where V (n, t) = �n−1 � i=0 M(∆ni) � � 1 with M(x) = � cos x − sin x sin x cos x � (A7) and ∆ni = ´ t 0 En(t1)−Ei(t1)dt1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Then the angle between the initial and evolved Bloch vectors is cos θ = ⃗r(t) · ⃗r |⃗r|2 = ⃗rTC(t)⃗r |⃗r|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A8) Utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A6), it can be further calculated as cos θ=1 − 1 |⃗r|2 N−1 � n=1 n−1 � i=0 [1−cos(∆ni)](r2 n2+2i−1+r2 n2+2i) 7 with ri the ith element of ⃗r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the set S can be directly expressed by S = � ⃗r �� 1 − cos Θ = 1 |⃗r|2 N−1 � n=1 n−1 � i=0 [1 − cos(∆ni)] × � r2 n2+2i−1 + r2 n2+2i � , ∃t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A9) To further obtain the OQSL, the two-step proof strat- egy used in Appendix B in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [18] needs to be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Define f(t) := 1 |⃗r|2 N−1 � n=1 n−1 � i=0 [1 − cos(∆ni)](r2 n2+2i−1 + r2 n2+2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Substituting the equation ˆ τ 0 EN−1(t) − E0(t)dt = Θ (A10) into the expression of f(t), it can be seen that ∂f(t) ∂t ��� t=τ ≥ 0, (A11) which indicates τ is in the first monotonic increasing regime of f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the meantime, it can also be found that f(τ) ≤ 1 − cos Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' If the solution of f(t) = 1 − cos Θ is not in the first increasing regime of f(t), then t is obviously larger than τ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' if this solution is in the first in- creasing regime, then due to f(τ) ≤ 1 − cos Θ = f(t) one can also see that t ≥ τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, τ is a lower bound of the time to reach the target angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Now we discuss the optimal probe states to reach the OQSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To let the equation 1 − cos Θ = f(τ) holds, the term r2 n2+2i−1 + r2 n2+2i for the subscripts n, i satisfying ∆ni ̸= ´ τ 0 EN−1(t) − E0(t)dt has to vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Further as- sume the degeneracy of the ground states and highest ex- cited states are p and q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' E0(t) = E1(t) = · · · = Ep−1(t) and EN−q(t) = EN−q+1(t) = · · · = EN−1(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' then it is easy to see that r2 n2+2i−1 + r2 n2+2i can only be nonzero when n ∈ [N − q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' N − 1] and i ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' p − 1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' which indicates that the optimal state is of the form N−1 � i=0 1 N |Ei⟩ ⟨Ei| + � k∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='p−1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' l∈[N−q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='N−1] ξkl |Ek⟩ ⟨El| + ξ∗ kl |El⟩ ⟨Ek| ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A12) where ξkl = � N−1 2N (rl2+2k−1 − irl2+2k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the energy basis {|E0⟩ , |E1⟩ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' , |EN−1⟩}, the state above can be written as 1 N 11 + � � 0 0 ξ 0 · · 0 ξ† 0 0 � � , (A13) where 11 is a N-dimensional identity matrix, and ξ is a p by q matrix with klth entry ξkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To make sure the density matrix is positive-semidefinite, according to the Schur complement theorem ξ needs to satisfy ξ†ξ ≤ 1 N 2 11q, (A14) where 11q is a q-dimensional identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The theorem is then proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' ■ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Example: two-level systems Here we take a two-level system as a demonstration of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Consider the Hamiltonian H(t) = f(t)σz, (A15) where f(t) is a function of time t, and σz is a Pauli Z matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the Bloch representation, the evolved Bloch vector can be solved as rx(t) = rx cos � 2 ˆ t 0 f(t1)dt1 � − ry sin � 2 ˆ t 0 f(t1)dt1 � , ry(t) = rx sin � 2 ˆ t 0 f(t1)dt1 � + ry cos � 2 ˆ t 0 f(t1)dt1 � , rz(t) = rz, where (rx, ry, rz)T = ⃗r is the Bloch vector of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Based on this dynamics, the angle between the initial and evolved states is cos θ = cos � 2 ´ t 0 f(t1)dt1 � (r2 x + r2 y) + r2 z |⃗r|2 , (A16) which indicates that the time to reach the target angle Θ satisfies the following equation sin2 �ˆ t 0 f(t1)dt1 � = |⃗r|2 |⃗r|2 − r2z sin2 �Θ 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A17) Rewrite ⃗r into ⃗r = η(sin α cos ϕ, sin α sin ϕ, cos α)T (A18) with η ∈ [0, 1], α ∈ [0, π] and ϕ ∈ [0, 2π], and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A17) reduces to sin2 α = sin2 � Θ 2 � sin2 �´ t 0 f(t1)dt1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A19) Now consider that | ´ t 0 f(t1)dt1| is upper bounded by cf, then in the case that cf < Θ/2, sin2 �´ t 0 f(t1)dt1 � is al- ways less than sin2(Θ/2), which gives sin2 α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' This means no state can fulfill the target as sin2 α is always equal or less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that cf ∈ [Θ/2, π/2], sin2 �ˆ t 0 f(t1)dt1 � ≤ sin2 cf ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A20) 8 Hence, sin2 α ≥ sin2 � Θ 2 � sin2 cf , (A21) indicating that the states that can fulfill the target sat- isfies α ∈ [αf, π − αf] with αf = arcsin � sin � Θ 2 � sin cf � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A22) In the case that cf > π/2, sin2[ ´ t 0 f(t1)dt1] can reach all the values between 0 and 1, and sin2 α ≥ sin2(Θ/2), therefore, the states satisfies α ∈ [Θ/2, π − Θ/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In a word, the set S can be expressed by S = � � � � � ∅, cf < Θ 2 , {⃗r | α ∈ [αf, π − αf]}, cf ∈ [ Θ 2 , π 2 ], {⃗r | α ∈ [ Θ 2 , π − Θ 2 ]}, cf > π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A23) Here ∅ is the empty set, and in the second and third circumstances η ∈ (0, 1] and ϕ ∈ [0, 2π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' With respect to the OQSL, due to the fact that the eigenvalues are always f(t) and −f(t), the maximum and minimum ones are always |f(t)| and −|f(t)|, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Based on Theorem 1, the OQSL τ then satisfies ˆ τ 0 |f(t)|dt = Θ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A24) A physical example for the Hamiltonian (A15) is the energy splitting coming from Zeeman effect, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', f(t) = −gµB 2 B(t) (A25) with g the Lande factor and µB the electron magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' B(t) is the time-dependent magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For a periodic magnetic field B(t) = B cos(ωt) with B, ω > 0, it is easy to see ���� ˆ t 0 f(t1)dt1 ���� = ���� gµBB 2ω sin(ωt) ���� ≤ gµBB 2ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A26) According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A23), the set S reads S = � � � � � ∅, gµBB 2ω < Θ 2 , {⃗r | α ∈ [αf, π − αf]}, gµBB 2ω ∈ [ Θ 2 , π 2 ], {⃗r | α ∈ [ Θ 2 , π − Θ 2 ]}, gµBB 2ω > π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A27) Here η ∈ (0, 1], ϕ ∈ [0, 2π] and αf = arcsin � � sin � Θ 2 � sin � gµBB 2ω � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A28) Utilizing Theorem 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A24) can be written as ˆ τ 0 | cos(ωt)|dt = Θ gµBB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A29) In the case that gµBB 2ω < Θ 2 , τ = ∞ as no states can reach the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence we only consider the non-trivial case that gµBB 2ω ≥ Θ 2 , which means Θ gµBB ≤ 1 ω, and therefore ´ τ 0 | cos(ωt)|dt ≤ 1/ω, namely, ´ τ 0 | cos(ωt)|d(ωt) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The integration of | cos(ωt)| is only less or equal to 1 when ωt ≤ π/2, in which regime cos(ωt) is always non- negative, hence, the integration is equivalent to be per- formed on cos(ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Finally, the equation above can be rewritten into ˆ τ 0 cos(ωt)dt = Θ gµBB , (A30) which immediately gives the analytical expression of τ as below τ = 1 ω arcsin � ωΘ gµBB � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A31) An interesting fact in this case is that the first degener- ate point shows at t = π/(2ω), and the OQSL is always less or equal to this time, indicating that the target Θ, regardless of its value, can always be reached before this first degeneracy point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Next we consider a controlled case that f(t) = −gµB 2 B cos(ωt) + u(t), (A32) where |u(t)| ≤ ub is a bounded control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Since ���� ˆ t 0 −gµB 2 B cos(ωt1) + u(t1)dt1 ���� = ���� gµBB 2ω sin(ωt) − ˆ t 0 u(t1)dt1 ���� ≤gµBB 2ω + ���� ˆ t 0 u(t1)dt1 ���� ≤gµBB 2ω + ubt, (A33) which can be larger than π/2 for a long enough time, in this case S = � ⃗r �� α ∈ [Θ 2 , π − Θ 2 ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A34) The OQSL here satisfies ˆ τ 0 ���� gµBB 2 cos(ωt) − u(t) ���� dt = Θ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A35) Then the minimum value of τ (denoted by τmin) can be solved via the problem τmin = min u(t) τ, subject to �´ τ 0 | gµBB 2 cos(ωt)−u(t)|dt = Θ 2 , |u(t)| ≤ ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 9 This problem can be solved by maximizing the function | 1 2gµBB cos(ωt) − u(t)| under the constraint that its in- tegration is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Since cos(ωt) is a monotonic function within the regime [0, π/(2ω)], one could have ˆ π 2ω 0 ���� gµBB 2 cos(ωt) − u(t) ���� dt ≤ ˆ π 2ω 0 �gµBB 2 cos(ωt) + ub � dt =gµBB 2ω + π 2ω ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A36) Notice the condition to make sure S ̸= ∅ is gµBB 2ω ≥ Θ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this case, the upper bound of ´ π 2ω 0 | gµBB 2 cos(ωt)−u(t)|dt is larger than Θ/2, indicating that the integration will reach Θ/2 before the time π/(2ω) with proper controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, τmin must be less than π/(2ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Under this con- dition, the maximum value of | 1 2gµBB cos(ωt) − u(t)| is attained when u(t) ≡ −ub due to the fact that cos(ωt) is a monotonic function here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, τmin satisfies the equation gµBB 2ω sin(ωτmin) + ubτmin = Θ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A37) If ω is small, τmin approximates to τmin ≈ Θ gµBB + 2ub .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A38) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Example: one-dimensional Ising model with a longitudinal field a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Periodic boundary condition In the following we consider the one-dimensional Ising model with a longitudinal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The Hamiltonian of this system reads H/J = − n � j=1 σz j σz j+1 − n � j=1 g(t)σz j , (A39) where J > 0 is the interaction strength of the nearest- neighbor coupling, and g(t) is a global time-dependent longitudinal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' σz j is the Pauli Z matrix for jth spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The Hamiltonian satisfies the periodic boundary condi- tion σz n+1 = σz 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here we only consider the case that n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Now we calculate the maximum and minimum eigen- values of H/J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Since the Hamiltonian only contains the Pauli Z matrix, it is naturally a diagonal matrix in the space consisting of the eigenspaces of σz j for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Denote |↑j⟩ and |↓j⟩ as the eigenstates of σz j with re- spect to the eigenvalues 1 and −1, then the eigenvalues of −σz j σz j+1 − g(t)σz j are 1 + g(t), 1 − g(t), −1 − g(t), and −1 + g(t), and the corresponding eigenstates are … … … … … … … … flip ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='AC2XicjVHLTsMwEJyGVymv8rhxCVRInKoEIeBYwYVjkehDalHlpG6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='JmiaR4BK4cANceUHuMIPIf4A/oK1SWgQmArznh2Z+z1OpHvxdKyXjPGxOTU9Ex2Njc3v7C4lF9eqcZhIlxecUM/FHWHxdz3Al6RnvR5PRKc9R2f15zekYrXLriIvTA4lYOIn/VZN/A6nskUa382nVzg2YSMSHCy6ZgQdfnrXzBKlp6mOPATkEB6SiH+Rc0UYIFwn64AgCftgiGk2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='YMNCRNwZhsQJQp6Oc9wgR9qEsjhlMGJ7tHZp10jZgPbKM9Zql07x6ROkNLFmpDyBGF1mqnjiXZW7G/eQ+2p7jagv5N69YmVOCf2L90o8786VYtEBwe6Bo9qijSjqnNTl0S/irq5+aUqSQ4RcQq3KS4Iu1o5emdTa2Jdu3pbpuNvOlOxau+muQne1S2pwfbPdo6D6k7R3ivaJ7uF0mHa6i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='zWsYlt6uc+SjhGRXyvsIjnvBsNIxb4864/0w1MqlmFd+G8fABSwKXsQ=|"i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='AC23icjVHLSsNAFD2Nr/qOCm7cRIvgqiQi6lJ047KCbYW2y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='CQdazDNhMnEUqord+LWH3Cr/yP+gf6Fd8YIahGdIZMz595zZu5cP4nCVLnuS8EaGR0bnyhOTk3PzM7N2wuLtVRkMuDVQERCnvgs5VEY86oKVcRPEslZ14943b840PH6JZdpKOJj1U94q8s6cXgWBkwRdWovXzVXabZFL2ZSil5TsrgT8VO75JZdM5xh4OWghHxUhP2MJt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='oQCJChC4YinAEhpRmAx5cJMS1MCBOEgpNnOMaU6TNKItTBiP2gtYO7Ro5G9Ne6ZGHdApEX2SlA7WSMoTxLWpzkmnhlnzf7mPTCe+m59+vu5V5dYhXNi/9J9Zv5Xp2tROMOuqSGkmhLD6OqC3CUzr6Jv7nypSpFDQpzGbYpLwoFRfr6zYzSpqV2/LTPxV5OpWb0P8tw ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='Mb/qW1GDvZzuHQW2z7G2XvaOt0t5+3uoiVrCGDernDvZwiAq5H2FBziyWpZN9atdfeRahVyzRK+Dev+HZvgmJg=|#i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Schematic of obtaining any eigenstate of H/J by flipping any number of |↑⟩ (black up arrow) into |↓⟩ (red down arrow) in the state |↑⟩⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' |↓j↑j+1⟩, |↑j↓j+1⟩, |↑j↑j+1⟩, and |↓j↓j+1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The eigen- values of H/J can be obtained by the summation of a certain number of these four terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For instance, the eigenvalue with respect to the eigenstate |↑⟩⊗n is �n j=1[−1 − g(t)] = n[−1 − g(t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Regarding the minimum eigenvalue of H/J, it is easy to see that the minimum eigenvalue of −σz j σz j+1 − g(t)σz j is −1 − g(t) when g(t) ≥ 0 and −1 + g(t) when g(t) ≤ 0, namely, −1 − |g(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, the minimum eigenvalue of H/J is Emin,p = −n [1 + |g(t)|] , (A40) which can be attained by the eigenstate |↑⟩⊗n when g(t) ≥ 0 and |↓⟩⊗n when g(t) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Next, we calculate the maximum eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For an eigenstate ⊗n j=1 |aj⟩ (aj = ↑, ↓), denote the number of |↑j↑j+1⟩, |↓j↓j+1⟩, |↓j↑j+1⟩, and |↑j↓j+1⟩ (j ∈ [1, N]) are x1, x2, x3, and x4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For example, for the state |↑↓↑⟩, x1 = 1, x2 = 0, and x3 = x4 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Notice that any eigenstate of H/J can be obtained by flipping any number of |↑⟩ in the state |↑⟩⊗n into |↓⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As long as the number of flipped spins is less than n, no matter how many spins are flipped, there always exists a pair of |↑↓⟩ and |↓↑⟩ at the boundary of the flipped spins, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For example, assume a flip occurs at the jth spin and k spins are flipped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Then the state of the (j − 1)th and jth spins must be |↑j−1↓j⟩, and that of the (j +k−1)th and (j +k)th spins must be |↓j+k−1↑j+k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' If all the spins are flipped, no |↑↓⟩ and |↓↑⟩ exist in the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The simultaneous existence of |↑↓⟩ and |↓↑⟩ in the flip in- dicates that x3 always equals to x4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Utilizing x1, x2, x3, and the condition x1 + x2 + 2x3 = n, the eigenvalue of H/J can be expressed by −[2+g(t)]x1+[−2+g(t)]x2+n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the calculation of the maximum eigenvalue is 10 equivalent to a linear optimization problem: the maxi- mization of −[2 + g(t)]x1 + [−2 + g(t)]x2 + n under some constraints on x1, x2, and x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It is easy to see that the natural constraints on x1, x2, and x3 are 0 ≤ x1, x2 ≤ n and 0 ≤ x3 ≤ ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here ⌊·⌋ is the floor function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Combing the equation x1 + x2 + 2x3 = n, the condition 0 ≤ x3 ≤ ⌊n/2⌋ is equivalent to 0 ≤ x1 + x2 ≤ n when n is even and 1 ≤ x1 +x2 ≤ n when n is odd, which can be unified as 1 2[1+(−1)n+1] ≤ x1+x2 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' This condition is fully contained by the constraint 0 ≤ x1, x2 ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the full linear optimization problem can be expressed by max x1,x2 − [2 + g(t)] x1 + [−2 + g(t)] x2 + n, subject to � η ≤ x1 + x2 ≤ n, x1, x2 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A41) Here η := 1 2[1 + (−1)n+1] and N is the set of natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To solve this problem, four cases have to be discussed: (1) g(t) ≤ −2, (2) −2 < g(t) ≤ 0, (3) 0 < g(t) < 2, and (4) g(t) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that g(t) ≤ −2, the coefficients −[2 + g(t)] ≥ 0 and −2 + g(t) ≤ 0, indicating that the maximum eigenvalue is obtained when x1 is largest and x2 vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', x1 = n, x2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The corresponding maximum eigenvalue is n[−g(t) − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that g(t) ∈ (−2, 0], both coefficients −[2 + g(t)] and −2 + g(t) are negative, and the maximum eigenvalue is attained by the lower bounds of x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' If n is even, the minimum value of x1 and x2 are both zero, which leads to the maximum value n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' If n is odd, the maximum value is attained by x1 = 1, x2 = 0 due to the fact that −[2+g(t)] is larger than −2 + g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The corresponding maximum value is n − 2 − g(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that g(t) ∈ (0, 2), the situation is similar to the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The maximum value is n and attained by x1 = x2 = 0 when n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For an odd n, the maximum value is n − 2 + g(t), which can be attained by x1 = 0, x2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the last case that g(t) ≥ 2, −[2 + g(t)] ≤ 0 and −2 + g(t) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The maximum value is n[g(t)−1], which is attained by x1 = 0, x2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In summary, the maximum eigenvalue of H/J is of the form Emax,p = � n − η [2 − |g(t)|] , |g(t)| < 2, n [|g(t)| − 1] , |g(t)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A42) Now we consider a specific case that g(t) = B cos(ωt), where B is a positive amplitude and ω is the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The OQSL τ is solved via the equation ˆ τ 0 Emax,p(t) − Emin,p(t)dt = Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A43) When B < 2, |g(t)| is always less than 2, which means Emax,p always takes the form n − η [2 − |g(t)|], and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A43) reduces to 2 (n − η) τ + (n + η) ˆ τ 0 |g(t)|dt = Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='(A44) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='(a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='ACyXicjVHLSsNAFD2Nr1pf ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='NXOiv3Ne6w91d1G9HcyL59YiUti/9JNMv+rU7VI9HGoaxBU6QZVZ2buaS6K+rm5peqJDlExCnco3hM2NXKSZ9NrUl07aq3TMfdKZi1d7NclO8q1vSgO2f45wGjUrZ3i/bZ3ul6lE26jy2sI1dmuc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='BqjhBDXyvsIjnvBsnBrXxq1x95lq5DLNJr4t4+EDVduRdA=⌧2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='(b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Validity of the approximation with the changes of (a) the amplitude B and (b) the frequency ω for n = 10 (solid red lines), n = 15 (solid circle cyan lines), n = 20 (dashed blue lines), and n = 55 (dash-dotted green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' ω/J = 1 in (a), and in (b) B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='0 and B = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='0 for the upper and lower panels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For a not very large ω, ´ τ 0 |g(t)|dt = B ω sin(ωτ) ≈ Bτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, τ ≈ Θ 2(n − η) + B(n + η) =: τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A45) When B > 2, the relation between |g(t)| and 2 is not fixed at different time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, for a not very large ω, τ is still very small in this case, which means Emax,p takes the form n[g(t)−1] before the time τ, and ´ τ 0 |g(t)|dt still approximates to Bτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A43), τ approximates to τ ≈ Θ 2Bn =: τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A46) The validity of approximation is numerically tested with the changes of amplitude B and frequency ω for different spin number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 5(a), the per- formance of approximation is very well for different values of B when ω is not extremely large [ω/J = 1 in the plot].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 11 As to the frequency ω, the approximation is valid when ω is no larger than around 10 for both B = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='0 [upper panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 5(b)] and B = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='0 (lower panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As a mat- ter of fact, τ1 and τ2 are nothing but the OQSLs for the constant external field g(t) = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the validity of approximation for a large regime of ω indicates that the OQSL is way more sensitive to the amplitude than the frequency as long as the frequency is not extremely large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Open boundary condition Next we consider the case of the open boundary con- dition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The corresponding Hamiltonian reads H/J = − n−1 � j=1 σz j σz j+1 − n � j=1 g(t)σz j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A47) In this case, the minimum eigenvalue of −σz j σz j+1−g(t)σz j is −1−g(t) [−1+g(t)] for g(t) ≥ 0 [g(t) ≤ 0], which leads to the minimum eigenvalue of H/J Emin,o = −n [1 + |g(t)|] + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A48) The minimum eigenvalue can be attained by the eigen- state |↑⟩⊗n [|↓⟩⊗n] for g(t) ≥ 0 [g(t) < 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To calculate the maximum eigenvalue, we rewrite the Hamiltonian into the form H/J = Hp + σz nσz 1, (A49) where Hp is the Hamiltonian under the periodic bound- ary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Now let us denote Emax,p and |Emax,p⟩ as the maximum eigenvalue and corresponding eigenstate of Hp, which is actually already obtained in the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Notice that the eigenstates of Hp are also eigenstates of σz nσz 1, and the corresponding eigenvalues can only be 1 and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, if |Emax,p⟩ also corresponds to the eigenvalue 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', σz nσz 1 |Emax,p⟩ = |Emax,p⟩, then the maximum energy for the entire Hamiltonian is just Emax,p + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As a matter of fact, this is just the case for any n in the regime |g(t)| ≥ 2, and for odd n in the regime |g(t)| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the maximum eigenvalue Emax for these cases reads Emax,o = � n [|g(t)| − 1] + 1, |g(t)| ≥ 2, n + |g(t)| − 1, |g(t)| < 2 and n is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For an even n in the regime |g(t)| < 2, Emax,p − 1 = n−1 may not be the maximum eigenvalue anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' An- other possible candidate must be among the eigenval- ues of which the corresponding eigenstate |Ec⟩ satisfies σz nσz 1 |Ec⟩ = |Ec⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It is obvious that we only need to find the maximum eigenvalues in this case and compare it with Emax,p − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' This maximization problem can still be formulated as a linear optimization problem as follows max x1,x2 − [2 + g(t)] x1 + [−2 + g(t)] x2 + n + 1, subject to � � � � � 2 ≤ x1 + x2 ≤ n, x1, x2 ∈ N, |g(t)| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A50) The constraint x1 + x2 ≥ 2 comes from the fact that σz nσz 1 |Ec⟩ = |Ec⟩ is equivalent to require x1 ≥ 1 or x2 ≥ 1, and x1 + x2 + 2x3 = n requires x1 + x2 has to be an even number when n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, x1 + x2 has to be no smaller than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Since both the coefficients −[2 + g(t)] and −2 + g(t) are nonpositive in this case, the maximum value must be attained by x1 = 2, x2 = 0 or x1 = 0, x2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, in this case the maximum eigenvalue is n+2|g(t)|−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Next we need to compare the value between n − 1 and n + 2|g(t)| − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As a matter of fact, it is easy to see when n − 1 is larger when |g(t)| < 1 and n + 2|g(t)| − 3 is larger when |g(t)| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In summary, the maximum eigenvalue Emax,o under the open boundary condition reads � � � � � � � � � n − 1, |g(t)| ≤ 1 and n is even, n + 2|g(t)| − 3, 1 < |g(t)| < 2 and n is even, n + |g(t)| − 1, |g(t)| < 2 and n is odd, n [|g(t)| − 1] + 1, |g(t)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A51) Utilizing the symbol η = [1 + (−1)n+1]/2, the equation above can be rewritten into Emax,o = � � � � � n + η|g(t)| − 1, |g(t)| ≤ 1, n − (2 − η)[2 − |g(t)|] + 1, 1 < |g(t)| < 2, n [|g(t)| − 1] + 1, |g(t)| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A52) Next we calculate the OQSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that |g(t)| ≤ 1, τ satisfies the equation (n + η) ˆ τ 0 |g(t)|dt + (2n − 2)τ = Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A53) It is easy to see that here ´ τ 0 |g(t)|dt is less than τ, indi- cating that τ ≥ Θ 3n − 2 + η .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A54) When 1 < |g(t)| < 2, τ satisfies (n + 2 − η) ˆ τ 0 |g(t)|dt + (2n − 4 + 2η)τ = Θ, (A55) which gives Θ 4n < τ < Θ 3n − 2 + η (A56) 12 due to the fact that τ < ´ τ 0 |g(t)|dt < 2τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' When |g(t)| ≥ 2, the OQSL satisfies 2n ˆ τ 0 |g(t)|dt = Θ, (A57) which means τ ≤ Θ/(4n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Let us still consider a specific form of g(t) that g(t) = B cos(ωt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Similar to the case with the periodic boundary condition, the approximated expressions of OQSL can also be analytically obtained utilizing the approximation ´ τ 0 |g(t)|dt ≈ Bτ for a not very large ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the regime B ≥ 2, the OQSL is the same with τ2 [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A46)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' A more interesting phenomenon occurs in the regime B < 2, where the OQSL is different from τ1 [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A45)] for an even n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Specifically, the OQSL is τ ≈ Θ nB + 2n − 2 =: τ3 (A58) when B ≤ 1, and it is τ ≈ Θ nB + 2n + 2B − 4 (A59) when 1 < B < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The maximum gap between the OQSLs for periodic and open boundary conditions happens at the point B = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', when no external field exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this case, the OQSL can be rigorously solved and the difference is τ3 − τ1 = Θ 2n(n − 1) =: ∆τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A60) The optimal states to realize τ1 and τ3 are in the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' One thing that should be noticed is that the dimension of ξ in the case of periodic boundary condi- tion could be different from that in the case of the open boundary condition due to the different degeneracy of minimum and maximum energies in these two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Robustness analysis The dependence on the boundary condition indicates that the OQSL may be used to detect whether an even- numbered spin ring is ruptured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To do that, one needs to prepare the optimal states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A12) and then measure Tr(ρ0ρt) and Tr(ρ2 t) at time τ3 and τ1, which can be realized via techniques like randomized measure- ments [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here ρ0 and ρt are the initial state and evolved state at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' After the measurement, the Bloch angle can be calculated via the equation cos(θ(t)) = Tr(ρ0ρt) − 2−n � [Tr(ρ2 0) − 2−n] [Tr(ρ2 t) − 2−n] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A61) If the target is fulfilled at time τ3, then the ring is rup- tured, and it is complete if the target is fulfilled at the time τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' A more interesting fact is that the evolution time for the states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A12) is robust to the global and lo- cal dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The global dephasing is described by the master equation ∂tρt = −i[H, ρt] + γg � JzρtJz − 1 2 � ρt, J2 z �� (A62) with γg the decay rate and Jz = 1 2 �n j=1 σz j , and the local dephasing is described by ∂tρt = −i[H, ρt] + n � j=1 γl,j � σz j ρtσz j − ρt � , (A63) where γl,j is the decay rate for jth spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Now we analytically discuss this robustness under global and local dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' We need to emphasize that the optimal states [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A12)] in the noiseless case may not keep optimal when global and local dephasing are in- volved, and the corresponding evolution time to reach the target may also not be the OQSL anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The analysis of OQSL under the noise requires the CRC methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here we only discuss the robustness of the evolution time for the states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Recall that the states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A12) can be written into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A13) in the basis {|E0⟩ , |E1⟩ , · · · , |E2n−1⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' With- out the external field, the degeneracy of ground states and the highest energy levels are both two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the mean- time, due to the fact that σz j (for any j) and Jz are both diagonal in this basis, we are allowed to denote Jz = diag(A, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' , G) with A and G 2-dimensional diagonal ma- trices, and σz j = diag(Cj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' , Dj) with Cj and Dj 2- dimensional diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Utilizing these notations, the master equation for global dephasing [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A62)] re- duces to the evolution of the block ξ as follows ∂tξt = i(Emax − Emin)ξt + γgAξtG − γg 2 � ξtG2 + A2ξt � , (A64) where ξt is the evolved block at time t, and the one for local dephasing [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A63)] reduces to ∂tξt = i(Emax − Emin)ξt + � j γl,j (CjξtDj − ξt) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A65) As long as the specific forms of A, G, Cj, and Dj are known, the dynamics can be easily solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Next, we show the calculations of these blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It is not difficult to see that σz j is easy to be expressed in the basis {|↑⟩ , |↓⟩}⊗n, and the specific forms of σz j (diago- nal values) for different values of j are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 6(a), where ⃗1k (−⃗1k) represents a k-dimensional vector with all entries 1 (−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' To find the expressions of Cj and Dj, we need to know the entry positions of minimum and maximum energies for the Hamiltonian − � j σz j σz j+1 and extract the values of σz j in the same positions to recon- struct Cj and Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The expression of −σz j σz j+1 in the basis {|↑⟩ , |↓⟩}⊗n for different values of j are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 6(b).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Schematic for the search of entry positions of the minimum and maximum energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (a) The diagonal entry distribution for σz j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (b) The diagonal entry distribution for −σz j σz j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [(c),(d)] The second blocks for σz j and −σz j σz j+1 for the search of the maximum energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In this diagram, searching the entry positions of the min- imum and maximum energies is equivalent to searching a column with the most number of −1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It can be seen that the entries of −σz j σz j+1 for all values of j are symmetric, indicating that the entire diagram can be di- vided into four blocks, where the first and fourth (second and third) blocks are mirror symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The positions with respect to the minimum energy are easy to locate since only the first and last entries of −σz j σz j+1 are always −1 for all values of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, their summation (summa- tion of the column in dashed-red boxes) would also be the minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the meantime, the first and last entries of σz j are always 1 and −1 for all values of j, indicating that Cj = σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Moreover, due to the fact that Jz is half of the summation of all σz j , the entry positions in Jz that correspond to the minimum energy are also the first and last entries, which means A = diag(n/2, −n/2) = nσz/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For the sake of finding the entry positions of the max- imum energy, we need to locate the position where the entry is always 1 for any value of j, namely, a column in the diagram where all entries are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It is obvious that it can only exist in the second and third blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Due to the symmetry, we only need to consider the second block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 6(d), a significant feature in this block is that the overlap between the positions of ⃗1 in the jth and (j + 1)th lines halves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' More specifically to say, compared to the position of ⃗1 in the jth line, only the left (right) half in the same position keeps being 1 in the (j + 1)th line if j is odd (even).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For example, in the first line (j = 1) all entries are 1, and hence the length of ⃗1 is 2n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the second line (j = 2), only the left half keeps being one, and the length of ⃗1 becomes 2n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Similarly, in the third line (j = 3) only the right half keeps being 1 compared to the position of ⃗1 in the sec- ond line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Utilizing this feature, one can find that when n is even, the 1 3(2n − 1)th and 1 3(2n + 2)th entries keep being 1 in the (n − 2)th line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Notice that the entry num- ber here starts from the beginning of all diagonal entries of −σz j σz j+1, not the beginning of the second block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' And in the (n − 1)th line, the 1 3(2n + 2)th entry is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case of open boundary condition, this is the last line and the position is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case of the periodic boundary condition, one more line of −σz nσz 1 needs to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Luckily, this position of −σz nσz 1 is also 1 when n is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, the maximum energy is at the 1 3(2n + 2)th entry under both boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Due to the symmetry, the 1 3(2n+1 + 1)th entry, which is in the third block, is also maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Now we locate the values of 1 3(2n + 2)th and 1 3(2n+1 + n-2n-176 2+1114 1)th entries in σz j , which is irrelevant to the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The block of entries in σz j with respect to the second block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 6(b) is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 6(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As shown in this diagram, the 1 3(2n + 2)th entry is 1 for an odd j and −1 for an even j, namely, it is (−1)j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Similarly, one can find that the 1 3(2n+1+1)th entry is (−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, Dj = (−1)j+1σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the meantime, both 1 3(2n +2)th and 1 3(2n+1+1)th entries are zero in Jz when n is even, which means G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In summary, we have found that A = nσz/2, G = 0, Cj = σz, and Dj = (−1)j+1σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Utilizing these expres- sions, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A64) and (A65) can be further written into ∂tξt = � i(Emax − Emin) − n2γg 8 � ξt, (A66) and ∂tξt = i(Emax − Emin)ξt + � j γl,j � (−1)j+1σzξtσz − ξt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A67) Equation (A66) can be easily solved as ξt = e � i(Emax−Emin)− n2γg 8 � ξ, (A68) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A67) can be solved as [ξt]00(11) =ei(Emax−Emin)−�n j=1 γl,j[1+(−1)j][ξ]00(11), [ξt]01(10) =ei(Emax−Emin)−�n j=1 γl,j[1−(−1)j][ξ]01(10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here [·]ab represents the abth entry (a, b = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Next we calculate cos(θ(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A61) can be expressed by cos(θ(t)) = Re � Tr(ξξ† t ) � � Re (Tr(ξξ†)) Re � Tr(ξtξ† t ) �, (A69) where Re(·) represents the real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case of global dephasing [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A66)], the expression above reduces to cos(θ(t)) = cos ((Emax − Emin)t) , (A70) which is irrelevant to the decay rate γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the evo- lution time to reach the target for the optimal states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A13) is indeed robust to the global dephasing in both periodic and open boundary conditions, indicating that their difference is also robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case of local dephasing, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A69) can be ex- pressed by cos(θ(t)) = cos ((Emax − Emin)t) ς1 + ς2e−2tγall √ς1 + ς2 √ς1 + ς2e−4tγall , where ς1 = |[ξt]00|2 + |[ξt]11|2, ς2 = |[ξt]01|2 + |[ξt]10|2, and γall = �n j=1 γl,j(−1)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='If the values of all decay ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='AC2nicjVHLSsQwFD1T3+NrVFy5KQ6Cq6HVQd0IohuXCs4YEXSmNFgpy1pKkiZjTtx6w+41Q8S/0D/wptYwQeiKW1Pzr3nJPfeMI1kpj3vueIMDA4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='Nj4yOVcnJqemazOz7SzJFRctnkSJ6oQsE5GMRUtLHYlOqgTrhZE4DC92TPzwUqhMJvGBvkrFcY+dxbIrOdNEndTmg4Nzodlm0FWMF6v9otkPUnlSq3sNzy73J/BLUEe59pLaEwKcIgFHjh4EYmjCERgyeo7gw0NK3DEK4hQhaeMCfVRJm1OWoAxG7AV9z2h3VLIx7Y1nZtWcTonoVaR0sUSahPIUYXOa+O5dTbsb96F9TR3u6J/WHr1iNU4J/Yv3Ufmf3WmFo0uNmwNkmpKLWOq46VLbrtibu5+qkqTQ0qcwacUV4S5VX702bWazNZuests/MVmGtb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='seZmb49Xckgbsfx/nT9BeafhrDX+/Wd/aLkc9igUsYpnmuY4t7GIPLfIucI8HPDqBc+3cOLfvqU6l1Mzhy3Lu3gBVY5gX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='⇥ = 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='4⇡ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The variety of the gap between the maximum and minimum values of the evolution time to reach the target Θ among 100 random states with random values of {γl,j} ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The insets present the ratios of 10000 states at different evolution time to reach the target Θ = 3π/4 for periodic (red dots) and open (green pentagrams) boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' n = 10 in all plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' rates {γl,j} are very close, for example γl,j ≈ γ for any j, then γall ≈ 0 and cos(θ(t)) still approximates to cos ((Emax − Emin)t), which is also irrelevant to the de- cay rates, and thus in this case the evolution time, as well as the time difference, are also robust to the local dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case that the values of {γl,j} are not close, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A69) is indeed dependent on the decay rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' However, since ς1 + ς2e−2tγall is always positive at finite time, the evolution time is still irrelevant to γall for the target Θ = π/2 and hence robust to the local dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For a general target, we have tested 100 random states in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' (A13) with random values of {γl,j} ∈ (0, 1) for each target in the case of n = 10, and the gap between the maximum and minimum values of the evolution time for these 100 states are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It can be seen that the robustness is quite good when the target is no larger than π/2, and it is indeed compromised when Θ is larger than π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Even for those targets with large gaps, the evolution time for different states could concentrate on some specific values, namely, the distribution of states in the gap has a sharp peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' For example, the insets of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 7 show the distributions of 10000 states for periodic (red dots) and open (green pentagrams) boundary con- ditions in the case of Θ = 3π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It can be seen that the distributions for both periodic and open boundary conditions have a sharp peak at the minimum values, in- dicating that the evolution time is still relatively robust for most states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 15 Appendix B: Learning the OQSL in Landau-Zener model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Verification of the validity of CRC methodology Here we present the process of learning the OQSL in the Landau-Zener model and show the validity of the CRC methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The Hamiltonian of this model is H = ∆σx + vtσz, (B1) where ∆ and v are two time-independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' σx and σz are the Pauli matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The OQSL in this model has been thoroughly discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' [18], in which the set S is obtained via the brute-force search among around one million pure states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The reason why only pure states are considered here is due to the fact that unitary evo- lution does not affect the purity and in the Bloch rep- resentation all states in the same direction can/cannot reach the target simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The dynamics is solved via QuTiP [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The full evolution is truncated at vt = 10, namely, the state is treated to not be in S if it cannot reach the target within the truncated time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The Bloch vector of the initial state is parameterized by ⃗r = (sin α cos φ, sin α sin φ, cos α)T with α ∈ [0, π] and φ ∈ [0, 2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the step of classification, a multilayer neural net- work with two inputs (α and φ) and one output (1 or 0) is created with a hyperbolic tangent function as the ac- tivation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The output result 1/0 represents that the input initial state can/cannot realize the given tar- get, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Supervised learning is performed via Scikit-learn [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The network contains five to six hidden layers each with about 200 to 250 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The Cross- Entropy loss function [53] is used as the loss function, and Adam [54] is applied in the updates of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The test set contains all the initial states (around one million states) used in the brute-force search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The per- formance of training for different values of ∆ are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The first, second, and third rows represent the learned S for ∆ = 0, ∆ = 1, and ∆ = 2 (in the units of √v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The solid blue and dashed red lines repre- sent the boundaries between S and its complementary set obtained via supervised learning and brute-force search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Different numbers of the training set, including 15000, 22500, and 30000, have also been tested and compared, as shown in the first (15000), second (22500), and third (30000) columns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 8(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The percentage numbers in the plots are the scores of learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', the correctness of the network’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It can be seen that the perfor- mance of 15000 training data is better than the others in the case of ∆ = 0, and 22500 training data present the best performance in the cases of ∆ = 1 and ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' One should notice that all the parameters of the network are manually tuned case by case, and the slight difference in the performance may not be fully due to the difference in the training data number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' In the case of 22500 train- ing data, the correctness is around 99%, indicating that about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='99 million states are correctly classified into S and its complementary set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Therefore, the neural net- work indeed works for the classification in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The second step is the regression process, in which basically the same neural network is created but with rectified linear unit function as the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The loss function is taken as the square error loss func- tion [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The training data are sorted by the evolution time to reach the target from smallest to largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Sim- ilar to the classification process, all the states in S are used to test the performance of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Notice that S here is the exact reachable state set obtained via the brute-force search since we need to check the validity of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The performance of regression is presented for different values of ∆ and training data number in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' All the plots in this figure are semi-logarithmic (x axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The first, second, and third rows represent the results for ∆ = 0, ∆ = 1, and ∆ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The first, second, and third columns represent the results for 15000, 22500, and 30000 training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The number in the plots are the mean square errors of learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=', 1 m �m i=1 � t(i) pre−t(i) ext �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Here t(i) pre and t(i) ext are the predicted time obtained via learning and exact time obtained via brute-force search for the ith state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The order of states in the figure is sorted by the evolution time obtained in the brute-force search from smallest to largest, and the learned time is plotted using the same order of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Notice that these states are not exactly the same for different values of ∆ due to the dependence of S on ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' It can be seen that the performance of learning (solid blue lines) is good for all values of ∆, especially when the training data num- ber is 22500 and 30000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Basically the mean square errors of learning in these two cases for all values of ∆ are in the scale of 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, the network also works for the regression in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As a matter of fact, in practice the reachable state set used in the regression process is the one obtained in the classification process (denoted by Slearn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Hence, al- though it is reasonable to use the true S to check the validity of the regression, Slearn has to be applied to test if the OQSL obtained from CRC methodology is rea- sonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The performance of regression with respect to Slearn for 22500 training data is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 8(c) for dif- ferent values of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The results for the other two training data numbers are not shown here due to their similar- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' Since the training set chosen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 8(b) is also a subset of Slearn, we can directly use it as the training set in this case and the trained network is then the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' The states in the plots are sorted by the evolution time to reach the target from smallest to largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' As shown in this figure, the trend of learned time basically coincides with the exact time in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' 8(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content=' One should notice that in fact these two lines cannot be compared directly as the 16 (a) (b) training data number: 15000 training data number: 22500 training data number: 30000 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='75×10-5 4.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='00% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='29% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='02% Predicted time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='2552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='7310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} +page_content='3859 True time 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HtAyT4oBgHgl3EQfrvl-/content/2301.00566v1.pdf'} 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based on its sign. In general, it is also +possible to have 𝜑0 junctions, where also cosine terms need +to be included in the Fourier expansion; however, no sign of +such 𝜑0 effects were found in any of our simulations. While +the first harmonic is ideal for locating 0-𝜋 transitions, we also +calculated the critical current 𝐼𝑐 ≡ max𝛿𝜑 |𝐼(𝛿𝜑)| for some +interesting junctions as it is more experimentally accessible. +Methodology. The fermionic operators at each site 𝑖 can be +grouped into Nambu vectors ˆ𝑐𝑖 ≡ (𝑐𝑖↑, 𝑐𝑖↓, 𝑐† +𝑖↑, 𝑐† +𝑖↓), which may +in turn be collected into a 4𝑁-element vector ˇ𝑐 ≡ ( ˆ𝑐1, . . . , ˆ𝑐𝑁 ) +containing every fermionic operator on the lattice. The Hamilto- +nian operator can then be expressed via a 4𝑁 ×4𝑁 Hamiltonian +matrix: H = 𝐸0 + 1 +2 ˇ𝑐† ˇ𝐻 ˇ𝑐. The most common approach to solv- +ing the BdG equations consists of diagonalizing ˇ𝐻, and then +expressing physical observables of interest in terms of its eigen- +vectors and eigenvalues. However, an alternative approach has +gained momentum over the last decade: The Kernel Polynomial +Method [20–22]. Instead of diagonalizing the Hamiltonian, +one calculates a Green function matrix from the Hamiltonian +matrix, which can be done efficiently and accurately using a +series expansion in Chebyshev polynomials. Many physical +observables of interest can then be directly extracted from the +elements of this Green function. +There are many variants of the Chebyshev methods outlined +above. We here use the Fermi operator expansion method [23], +which for the case of the BdG Hamiltonian is explained in +detail in Ref. [24]. The starting point is then the Fermi matrix +ˇ𝐹 ≡ 𝑓 ( ˇ𝐻), where 𝑓 (𝜖) = [1 + exp(𝜖/𝑇)]−1 is the Fermi– +Dirac distribution at temperature 𝑇. The function 𝑓 should be +interpreted in terms of its Taylor expansion when applied to +the matrix ˇ𝐻. Using the kernel polynomial method, we can +expand the Fermi matrix in Chebyshev polynomials as +ˇ𝐹 = 1 +2 𝑓0 ˇ𝐼 + +𝑀−1 +∑︁ +𝑚=1 +𝑓𝑚𝑔𝑚 ˇ𝑇𝑚, +(3) +where 𝑓𝑚 are the Chebyshev moments of the Fermi–Dirac +distribution [20, 24], 𝑔𝑚 are the Jackson kernel coefficients [20], +ˇ𝑇𝑚 ≡ 𝑇𝑚( ˇ𝐻) are Chebyshev matrix polynomials [20], and ˇ𝐼 is +the identity matrix. Note that the above assumes that ˇ𝐻 has been +normalized such that all its eigenvalues have magnitudes below +unity; this is in practice easily achieved by scaling ˇ𝐻 by its 1- +norm. The Chebyshev polynomials are calculated via the usual +recursion relation ˇ𝑇0 = ˇ𝐼, ˇ𝑇1 = ˇ𝐻, ˇ𝑇𝑚 = 2 ˇ𝐻 ˇ𝑇𝑚−1 − ˇ𝑇𝑚−2 [20]. +Calculating { ˇ𝑇𝑚} is the computationally limiting part of our +calculation, but was significantly sped up using sparse matrices +with fully-parallelized block-wise matrix multiplication. All +simulations presented here were performed using 𝑀 = 4000 +Chebyshev moments. We found that this provides negligible +truncation error for typical junctions, which is consistent with +the findings for LDOS calculations in e.g. Ref. [21]. The +calculations were performed at a temperature 𝑇 = 𝑇𝑐/20. +The procedure above provides us with a 4𝑁 × 4𝑁 Fermi +matrix ˇ𝐹. This can be deconstructed into 4×4 blocks in Nambu +space, ˇ𝐹 = [ ˆ𝐹𝑖 𝑗]. Following a similar approach as Ref. [24], it +can then be shown that the elements of these matrices are: +ˆ𝐹𝑖 𝑗 = +������� +� +⟨𝑐† +𝑗↑𝑐𝑖↑⟩ ⟨𝑐† +𝑗↓𝑐𝑖↑⟩ ⟨𝑐 𝑗↑𝑐𝑖↑⟩ ⟨𝑐 𝑗↓𝑐𝑖↑⟩ +⟨𝑐† +𝑗↑𝑐𝑖↓⟩ ⟨𝑐† +𝑗↓𝑐𝑖↓⟩ ⟨𝑐 𝑗↑𝑐𝑖↓⟩ ⟨𝑐 𝑗↓𝑐𝑖↓⟩ +⟨𝑐† +𝑗↑𝑐† +𝑖↑⟩ ⟨𝑐† +𝑗↓𝑐† +𝑖↑⟩ ⟨𝑐 𝑗↑𝑐† +𝑖↑⟩ ⟨𝑐 𝑗↓𝑐† +𝑖↑⟩ +⟨𝑐† +𝑗↑𝑐† +𝑖↓⟩ ⟨𝑐† +𝑗↓𝑐† +𝑖↓⟩ ⟨𝑐 𝑗↑𝑐† +𝑖↓⟩ ⟨𝑐 𝑗↓𝑐† +𝑖↓⟩ +������� +� +. +(4) + +3 +0 +20 +40 +Altermagnet length 퐿/푎 +0.00 +0.25 +0.50 +0.75 +1.00 +First harmonic 퐼1(퐿)/퐼1(0) +(a) 푚 = 0.5Δ +0 +20 +40 +Altermagnet length 퐿/푎 +(b) 푚 = 1.5Δ +0 +10 +20 +30 +Altermagnet length 퐿/푎 +(c) 푚 = 0.5푡 +0 +5 +10 +15 +Altermagnet length 퐿/푎 +(d) 푚 = 0.9푡 +Straight junction +Diagonal junction +FIG. 2. First harmonic 𝐼1 of the Josephson supercurrent 𝐼(𝛿𝜑) as function of the altermagnet length 𝐿 for the junctions in Fig. 1. To simplify +the comparison between the two different geometries, each curve was normalized to the amplitude 𝐼1(0) in the absence of the altermagnetic +interlayer. As indicated above the plots, different panels and curves correspond to different altermagnetic order parameters and junction types, +respectively. The insets zoom in on the regions in the golden boxes, in order to highlight the 0–𝜋 oscillations for large junction lengths. +This implies that any physical observable which can be calcu- +lated from two-point finite-temperature correlation functions +on the lattice can be calculated directly from the Fermi matrix. +We evaluate the charge current inside the normal metal +spacer. Charge conservation ensures that the current is constant +anywhere along the junction in a stationary system. We compute +the charge current by summing over bond currents. The bond +current between two sites 𝑖 and 𝑗 can be written [18] +𝐽𝑖 𝑗 = 𝑖𝑒 +∑︁ +𝜎 +� +𝑡𝑖 𝑗 ⟨𝑐† +𝑖𝜎𝑐 𝑗 𝜎⟩ − 𝑡 𝑗𝑖 ⟨𝑐† +𝑗 𝜎𝑐𝑖𝜎⟩ +� +, +(5) +where 𝑒 < 0 is the electron charge. By comparison with Eq. (4), +we see that the bond current 𝐽𝑖 𝑗 can be trivially calculated from +appropriate traces of ˆ𝐹𝑖 𝑗 and ˆ𝐹𝑗𝑖. The bond current along the +junction direction 𝒏 is then simply 𝐽𝑖 𝑗 (𝜹𝑖 𝑗 · 𝒏), where 𝜹𝑖 𝑗 is +a unit vector that points from site 𝑖 towards site 𝑗. The total +current 𝐼 flowing through the junction is found by integrating +this over a cross section of the junction. +Results and discussion. The main results of our numerical +simulations are in Fig. 2. First, we observe that 0-𝜋 oscillations +are possible in both straight and diagonal junctions. This +finding is interesting since such oscillations are typically found +in Josephson junctions with magnetic interlayers, whereas +altermagnets have zero magnetization. Moreover, 0-𝜋 oscil- +lations do not appear in Rashba spin-orbit coupled junctions +either, which have a different spin-momentum coupling (odd- +in-momentum) compared to altermagnets. Second, we see +that the 0-𝜋 oscillations behave qualitatively differently from +ferromagnetic Josephson junctions: the latter typically has an +exponential decay with superimposed oscillations, whereas in +the altermagnet case there is an initial large decay followed by +oscillations with a much weaker damping. This result is most +striking in Fig. 2(b), where we find a pure decay at 𝐿 < 8𝑎 +followed by nearly pure oscillation at 𝐿 > 10𝑎. +Physically, the oscillations in straight junctions can be un- +derstood as follows. Conventional superconductivity consists +of singlet Cooper pairs |↑↓⟩ − |↓↑⟩. As the Cooper pairs leak +into the altermagnet along the 𝑥 axis, spin-up electrons have +a hopping amplitude 𝑡 + 𝑚 while spin-down electrons have a +hopping amplitude 𝑡 − 𝑚 (see Fig. 1). This “speed difference” +causes position-dependent phase differences between spin-up +and spin-down electrons. Such spin-dependent phase shifts +are well-known to cause 0-𝜋 oscillations from previous studies +on ferromagnetic Josephson junctions. For diagonal junctions, +however, the most direct path between the superconductors +consists of an equal number of hops along the 𝑥 and 𝑦 axes. +Since the electrons experience opposite spin-dependent phase +shifts in these two cases, their effects appear to partially cancel. +Figure 2 shows that the initial decay in 𝐼1(𝐿) is in general +accelerated as 𝑚 is increased. However, 0-𝜋 oscillations are +found over a much wider parameter range for straight than +diagonal junctions, consistent with the discussion above. For +example, for small altermagnetic order parameters 𝑚 = 0.05𝑡 +[Fig. 2(a)], the first 0-𝜋 oscillation occurs already at 𝐿 = 15𝑎 +for the straight junction, but not until 𝐿 = 35𝑎 for the diagonal +junction. On the other hand, for large altermagnetic order +parameters 𝑚 = 0.5𝑡 [Fig. 2(c)], the first 0-𝜋 oscillation occurs +simultaneously, but sustained 0-𝜋 oscillations for a large range +of junction lengths is found only for straight junctions. It is +only for intermediate values 𝑚 = 0.15𝑡 [Fig. 2(b)] that we find +qualitatively similar results in straight and diagonal junctions. +Finally, for very large values 𝑚 = 0.9𝑡, the supercurrent decays +extremely fast for both straight and diagonal junctions, limiting +the number of visible oscillations for both junction types. +For very large altermagnetic order parameters 𝑚 → 𝑡, spin- +down electrons become nearly immobile along the 𝑥 axis. In this +limit, spin-zero Cooper pairs clearly cannot propagate through +the altermagnet, and the Josephson effect vanishes. +This +explains the extremely sharp decay in Fig. 2(d). Interestingly, +if an altermagnet with 𝑚 → 𝑡 could be realized experimentally, + +4 +0 +10 +20 +30 +40 +Altermagnet length 퐿 +10−3 +10−2 +10−1 +100 +Critical current 퐼푐 (퐿)/퐼푐 (0) +Straight junction +Diagonal junction +FIG. 3. Critical current 𝐼𝑐 as a function of the altermagnet length 𝐿 +for 𝑚 = 0.5Δ and 𝑇 = 0.05𝑇𝑐 [cf. Fig. 2(a)]. +this might also serve as a new kind of filter for spin-triplet +Cooper pairs. Specifically, we would expect |↑↑⟩ pairs to only +move along the 𝑥 axis, |↓↓⟩ pairs to only move along the 𝑦 axis, +and any |↑↓⟩ ∓ |↓↑⟩ pairs to decay. This may thus provide a non- +destructive way to separate the different equal-spin-triplet pairs +generated in superconducting spintronics while eliminating any +remaining spin-zero pairs. +In Fig. 3, we show the critical current, rather than just the first +harmonic, for one of the junctions considered (𝑚 = 0.5Δ). In +Fig. 2(a) we saw that this choice of 𝑚 produces 0-𝜋 oscillations +at 𝐿 = 15𝑎 and 𝐿 = 21𝑎 for straight but not diagonal junctions, +which causes significant 𝐼𝑐 suppression. However, due to the +presence of higher harmonics, it is difficult from the 𝐼𝑐 curve +alone in Fig. 3 to observe that there are two such 0-𝜋 oscillations +in this area. This becomes even more challenging for higher +values of 𝑚 (not shown), where qualitatively similar oscillations +are observed, but the shorter oscillation period makes it more +difficult to determine the exact number of zero crossings. We +also see that both junctions display a 0-𝜋 oscillation for 𝐿 ≈ 35𝑎, +which for the diagonal junction is the first 0-𝜋 oscillation. +Figure 4 shows the critical current 𝐼𝑐 vs. temperature 𝑇, +which is one experimental signature of 0-𝜋 transitions in Joseph- +son junctions [10]. For these calculations, we picked an alter- +magnet length 𝐿 = 12𝑎 which according to Fig. 2(b) is close to +a 0-𝜋 transition for straight but not diagonal junctions. For the +straight junction, we find a non-monotonic critical current that +dips sharply at 𝑇 = 0.6𝑇𝑐. For both 𝑇 = 0.5𝑇𝑐 and 𝑇 = 0.7𝑇𝑐, +the current-phase relation is completely dominated by the first +harmonic 𝐼1. However, it changes sign between these two +points, so the dip is a signature of a 0-𝜋 transition as a function +of temperature. This is in contrast to the diagonal junction, +where we find no 0-𝜋 transition for these parameters. +Conclusion. In summary, we have demonstrated that the +Josephson effect through altermagnets exhibits different proper- +ties than in the two conventional classes of magnetic materials, +ferromagnets and antiferromagnets. Despite the absence of +a net magnetization, altermagnets induce 0-𝜋 oscillations in +the Josephson effect. We have also shown that the decay and +oscillation period of the Josephson current strongly depend +on the crystallographic orientation of the altermagnet relative +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Temperature 푇/푇푐 +10−4 +10−3 +10−2 +10−1 +100 +Critical current 퐼푐 (푇)/퐼푐 (0) +Straight junction +Diagonal junction +FIG. 4. +Critical current 𝐼𝑐 as a function of temperature 𝑇 for +𝑚 = 1.5Δ and 𝐿 = 12𝑎 [cf. Fig. 2(b)]. +the superconductors. The Josephson effect can therefore be +used both to distinguish the altermagnet from conventional +(anti)ferromagnetism and additionally offers a way to tune the +supercurrent via flow direction anisotropy. +Acknowledgments. This work was supported by the Research +Council of Norway through Grant No. 323766 and its Centres +of Excellence funding scheme Grant No. 262633 “QuSpin.” +The simulations were performed on resources provided by +Sigma2—the National Infrastructure for High Performance +Computing and Data Storage in Norway. +[1] A. Hirohata, K. 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PhD thesis (KTH, 2022). + diff --git a/NNE2T4oBgHgl3EQfBgZ1/content/tmp_files/load_file.txt b/NNE2T4oBgHgl3EQfBgZ1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bf1418f68670d113630ba4ac0e66642863583c5 --- /dev/null +++ b/NNE2T4oBgHgl3EQfBgZ1/content/tmp_files/load_file.txt @@ -0,0 +1,459 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf,len=458 +page_content='Josephson effect in altermagnets Jabir Ali Ouassou, Arne Brataas, and Jacob Linder Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway The ability of magnetic materials to modify superconducting systems is an active research area for possible applications in thermoelectricity, quantum sensing, and spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We consider the fundamental properties of the Josephson effect in a third class of magnetic materials beyond ferromagnets and antiferromagnets: altermagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We show that despite having no net magnetization, altermagnets induce 0-𝜋 oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The decay length and oscillation period of the Josephson coupling are qualitatively different from ferromagnetic junctions and depend on the crystallographic orientation of the altermagnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The Josephson effect in altermagnets thus serves a dual purpose: it acts as a signature that distinguishes altermagnetism from conventional (anti)ferromagnetism and offers a way to tune the supercurrent via flow direction anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Spin splitting of quasiparticle bands in con- densed matter systems is a crucial functional property of materi- als explored in spintronics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Recent works detail mechanisms for such splitting distinct from ferromagnetic and relativisti- cally spin-orbit coupled systems [2, 5], originally envisioned in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' One such mechanism consists of a spin-lattice cou- pling due to an internal periodic magnetic field in the material [2], which ultimately leads to a sizeable momentum-dependent spin splitting in the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' It is important to note that such splitting occurs even when disregarding conventional atomic spin-orbit coupling, which has a relativistic origin, the latter thus being much weaker than the new mechanism considered in recent works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The materials displaying this type of magnetic properties are known as altermagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' They have a large 𝒌- dependent spin splitting of the bands, which is even in powers of 𝒌, and that persists in the absence of relativistic spin-orbit couplings [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Ab initio calculations have identified several possible material candidates that can host an altermagnetic state, including metals like RuO2 and Mn5Si3 [3–5] as well as semiconductors/insulators like MnF2 and La2CuO4 [2, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Through the proximity effect, metallic materials which are not superconducting can inherit the two fundamental char- acteristics of superconductors: the Meissner effect [8] and dissipationless charge flow [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' When magnetic materials become superconducting via the proximity effect, both the Meissner effect and the dissipationless transport change in qualitatively new ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For instance, the Josephson coupling through ferromagnetic materials displays an effect known as 0- 𝜋 oscillations [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This means that the ground-state phase difference between the superconductors alternates between 0 and 𝜋, depending on the junction parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' As a result, the supercurrent vanishes at certain lengths and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Such 𝜋 junctions can be used for qubits [12, 13], and have also been generalized to 𝜙0 junctions [14–16] where the system acts as a quantum phase battery supplying an arbitrary phase between 0 and 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In this work, we consider Josephson junctions with altermag- netic interlayers (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Surprisingly, we find that despite the absence of any magnetization in altermagnets, the supercur- rent displays 0-𝜋 oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The behavior is different from both ferromagnetic and antiferromagnetic Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In the latter scenario, the 𝜋-state occurs in the very specific Superconductor Normal metal Altermagnet t–m t–m t+m t+m y x (a) (b) (d) t+m t+m t–m t–m (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (a–b) Illustration of the Josephson junctions considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Each circle represents one atom in a 2D square lattice in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In an experiment, there is likely a lattice mismatch between the different regions at the interfaces, which can reduce the supercurrent amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (a) “Straight junctions” are aligned with the crystallographic axes, and thus have the interface normals 𝒏 = 𝒆𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (b) “Diagonal junctions” have 45◦ misalignment relative to the lattice, so 𝒏 = (𝒆𝑥 − 𝒆𝑦)/ √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (c–d) Illustration of the altermagnetic order parameter 𝒎𝑖 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (c) Spin- up electrons have increased hopping amplitudes 𝑡 → 𝑡 + 𝑚 along the 𝑥 axis, and decreased hopping amplitudes 𝑡 → 𝑡 − 𝑚 along the 𝑦 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (d) For spin-down electrons, the situation is exactly reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' case of a junction with exactly an odd number of atoms [17], so that a net magnetic moment exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In addition, we find that both the decay and oscillation period of the supercurrent in the altermagnetic case exhibits anisotropy with respect to the crystallographic orientation of the interface relative the superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' These unique characteristics of the Josephson current in altermagnets can be used as a tool to identify the altermagnetic state among the list of candidate materials that have recently been identified through ab initio calculations [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 1, we consider two kinds of Joseph- son junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Both are created from a 2D square lattice with arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='03603v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='supr-con] 9 Jan 2023 2 lattice constant 𝑎, but have different junction orientations rela- tive to the crystallographic axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' At the ends of each junction is a 20𝑎 × 20𝑎 BCS superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The two superconductors that form each Josephson junction have a variable phase dif- ference 𝛿𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Next to the superconductors are thin normal-metal spacers of lengths 3𝑎 (straight junctions) or 3 √ 2𝑎 (diagonal junctions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Finally, the center of each junction is an altermag- net of varying length 𝐿 ∈ [0, 40𝑎].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 1, we plot an intermediate junction length 𝐿 = 20𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' To model the proposed physical setup we employ the Bogoliubov–de Gennes (BdG) method [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Our starting point is a mean-field tight-binding Hamiltonian that includes altermagnetism and conventional superconductivity: H = 𝐸0 − ∑︁ 𝑖𝜎 𝜇𝑖𝑐† 𝑖𝜎𝑐𝑖𝜎 − ∑︁ 𝑖 (Δ𝑖𝑐† 𝑖↓𝑐† 𝑖↑ + Δ∗ 𝑖 𝑐𝑖↑𝑐𝑖↓) − ∑︁ ⟨𝑖, 𝑗⟩𝜎 𝑡𝑖 𝑗𝑐† 𝑖𝜎𝑐 𝑗 𝜎 − ∑︁ ⟨𝑖, 𝑗⟩𝜎𝜎′ (𝒎𝑖 𝑗 · 𝝈)𝜎𝜎′𝑐† 𝑖𝜎𝑐 𝑗 𝜎′, (1) where 𝑐𝑖𝜎 and 𝑐† 𝑖𝜎 are the usual electronic annihilation and creation operators, and 𝝈 = (𝜎1, 𝜎2, 𝜎3) is the Pauli vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 𝐸0 describes a constant contribution which is not important for the non-selfconsistent calculations below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We choose constant nearest-neighbor hopping amplitudes 𝑡𝑖 𝑗 ≡ 𝑡 and chemical potentials 𝜇𝑖 = −𝑡/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In the two superconductors, we set Δ𝑖 = Δ𝑒±𝑖 𝛿𝜑/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The gap was calculated using the interpolation formula Δ(𝑇) ≈ Δ(0) tanh � 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='74 √︁ 𝑇𝑐/𝑇 − 1 � , where we chose a zero-temperature gap Δ(0) = 𝑡/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The critical temperature was determined using the BCS ratio Δ(0)/𝑇𝑐 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In the altermagnet, we set 𝒎𝑖 𝑗 = +𝑚𝒆𝑧 for nearest-neighbor hopping along the 𝑥 axis and 𝒎𝑖 𝑗 = −𝑚𝒆𝑧 for hopping along the 𝑦 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This corresponds to a low-energy effective Hamiltonian 𝑚𝑘𝑥𝑘𝑦𝜎𝑧 or 𝑚(𝑘2 𝑥 − 𝑘2 𝑦)𝜎𝑧, depending on the crystallographic orientation of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This Hamiltonian differs from both the momentum-independent spin-splitting 𝑚𝜎𝑧 of a ferromag- net and a Rashba-type spin-orbit coupling 𝑚𝑘𝑥(𝑦)𝜎𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The model above has three parameters that were varied be- tween simulations: The altermagnet length 𝐿 ∈ [0, 40𝑎], the magnitude of its order parameter 𝑚 ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5Δ, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5Δ, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5𝑡, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='9𝑡}, and the phase difference 𝛿𝜑 ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='02𝜋, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For each com- bination of these parameters, we calculated the Josephson supercurrent 𝐼 flowing along the junction using the method- ology described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The current-phase relation 𝐼(𝛿𝜑) for each junction was then fit to Fourier sine series, 𝐼(𝛿𝜑) = ∑︁ 𝑛>0 𝐼𝑛 sin(𝑛 𝛿𝜑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (2) The amplitude of the first harmonic 𝐼1 was extracted from these fits, and used to judge whether the Josephson junction is in a 0-state or 𝜋-state based on its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In general, it is also possible to have 𝜑0 junctions, where also cosine terms need to be included in the Fourier expansion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' however, no sign of such 𝜑0 effects were found in any of our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' While the first harmonic is ideal for locating 0-𝜋 transitions, we also calculated the critical current 𝐼𝑐 ≡ max𝛿𝜑 |𝐼(𝛿𝜑)| for some interesting junctions as it is more experimentally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The fermionic operators at each site 𝑖 can be grouped into Nambu vectors ˆ𝑐𝑖 ≡ (𝑐𝑖↑, 𝑐𝑖↓, 𝑐† 𝑖↑, 𝑐† 𝑖↓), which may in turn be collected into a 4𝑁-element vector ˇ𝑐 ≡ ( ˆ𝑐1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' , ˆ𝑐𝑁 ) containing every fermionic operator on the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The Hamilto- nian operator can then be expressed via a 4𝑁 ×4𝑁 Hamiltonian matrix: H = 𝐸0 + 1 2 ˇ𝑐† ˇ𝐻 ˇ𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The most common approach to solv- ing the BdG equations consists of diagonalizing ˇ𝐻, and then expressing physical observables of interest in terms of its eigen- vectors and eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' However, an alternative approach has gained momentum over the last decade: The Kernel Polynomial Method [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Instead of diagonalizing the Hamiltonian, one calculates a Green function matrix from the Hamiltonian matrix, which can be done efficiently and accurately using a series expansion in Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Many physical observables of interest can then be directly extracted from the elements of this Green function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' There are many variants of the Chebyshev methods outlined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We here use the Fermi operator expansion method [23], which for the case of the BdG Hamiltonian is explained in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The starting point is then the Fermi matrix ˇ𝐹 ≡ 𝑓 ( ˇ𝐻), where 𝑓 (𝜖) = [1 + exp(𝜖/𝑇)]−1 is the Fermi– Dirac distribution at temperature 𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The function 𝑓 should be interpreted in terms of its Taylor expansion when applied to the matrix ˇ𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Using the kernel polynomial method, we can expand the Fermi matrix in Chebyshev polynomials as ˇ𝐹 = 1 2 𝑓0 ˇ𝐼 + 𝑀−1 ∑︁ 𝑚=1 𝑓𝑚𝑔𝑚 ˇ𝑇𝑚, (3) where 𝑓𝑚 are the Chebyshev moments of the Fermi–Dirac distribution [20, 24], 𝑔𝑚 are the Jackson kernel coefficients [20], ˇ𝑇𝑚 ≡ 𝑇𝑚( ˇ𝐻) are Chebyshev matrix polynomials [20], and ˇ𝐼 is the identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Note that the above assumes that ˇ𝐻 has been normalized such that all its eigenvalues have magnitudes below unity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' this is in practice easily achieved by scaling ˇ𝐻 by its 1- norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The Chebyshev polynomials are calculated via the usual recursion relation ˇ𝑇0 = ˇ𝐼, ˇ𝑇1 = ˇ𝐻, ˇ𝑇𝑚 = 2 ˇ𝐻 ˇ𝑇𝑚−1 − ˇ𝑇𝑚−2 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Calculating { ˇ𝑇𝑚} is the computationally limiting part of our calculation, but was significantly sped up using sparse matrices with fully-parallelized block-wise matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' All simulations presented here were performed using 𝑀 = 4000 Chebyshev moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We found that this provides negligible truncation error for typical junctions, which is consistent with the findings for LDOS calculations in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The calculations were performed at a temperature 𝑇 = 𝑇𝑐/20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The procedure above provides us with a 4𝑁 × 4𝑁 Fermi matrix ˇ𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This can be deconstructed into 4×4 blocks in Nambu space, ˇ𝐹 = [ ˆ𝐹𝑖 𝑗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Following a similar approach as Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [24], it can then be shown that the elements of these matrices are: ˆ𝐹𝑖 𝑗 = ������� � ⟨𝑐† 𝑗↑𝑐𝑖↑⟩ ⟨𝑐† 𝑗↓𝑐𝑖↑⟩ ⟨𝑐 𝑗↑𝑐𝑖↑⟩ ⟨𝑐 𝑗↓𝑐𝑖↑⟩ ⟨𝑐† 𝑗↑𝑐𝑖↓⟩ ⟨𝑐† 𝑗↓𝑐𝑖↓⟩ ⟨𝑐 𝑗↑𝑐𝑖↓⟩ ⟨𝑐 𝑗↓𝑐𝑖↓⟩ ⟨𝑐† 𝑗↑𝑐† 𝑖↑⟩ ⟨𝑐† 𝑗↓𝑐† 𝑖↑⟩ ⟨𝑐 𝑗↑𝑐† 𝑖↑⟩ ⟨𝑐 𝑗↓𝑐† 𝑖↑⟩ ⟨𝑐† 𝑗↑𝑐† 𝑖↓⟩ ⟨𝑐† 𝑗↓𝑐† 𝑖↓⟩ ⟨𝑐 𝑗↑𝑐† 𝑖↓⟩ ⟨𝑐 𝑗↓𝑐† 𝑖↓⟩ ������� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (4) 3 0 20 40 Altermagnet length 퐿/푎 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='00 First harmonic 퐼1(퐿)/퐼1(0) (a) 푚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5Δ 0 20 40 Altermagnet length 퐿/푎 (b) 푚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5Δ 0 10 20 30 Altermagnet length 퐿/푎 (c) 푚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5푡 0 5 10 15 Altermagnet length 퐿/푎 (d) 푚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='9푡 Straight junction Diagonal junction FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' First harmonic 𝐼1 of the Josephson supercurrent 𝐼(𝛿𝜑) as function of the altermagnet length 𝐿 for the junctions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' To simplify the comparison between the two different geometries, each curve was normalized to the amplitude 𝐼1(0) in the absence of the altermagnetic interlayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' As indicated above the plots, different panels and curves correspond to different altermagnetic order parameters and junction types, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The insets zoom in on the regions in the golden boxes, in order to highlight the 0–𝜋 oscillations for large junction lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This implies that any physical observable which can be calcu- lated from two-point finite-temperature correlation functions on the lattice can be calculated directly from the Fermi matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We evaluate the charge current inside the normal metal spacer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Charge conservation ensures that the current is constant anywhere along the junction in a stationary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We compute the charge current by summing over bond currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The bond current between two sites 𝑖 and 𝑗 can be written [18] 𝐽𝑖 𝑗 = 𝑖𝑒 ∑︁ 𝜎 � 𝑡𝑖 𝑗 ⟨𝑐† 𝑖𝜎𝑐 𝑗 𝜎⟩ − 𝑡 𝑗𝑖 ⟨𝑐† 𝑗 𝜎𝑐𝑖𝜎⟩ � , (5) where 𝑒 < 0 is the electron charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' By comparison with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' (4), we see that the bond current 𝐽𝑖 𝑗 can be trivially calculated from appropriate traces of ˆ𝐹𝑖 𝑗 and ˆ𝐹𝑗𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The bond current along the junction direction 𝒏 is then simply 𝐽𝑖 𝑗 (𝜹𝑖 𝑗 · 𝒏), where 𝜹𝑖 𝑗 is a unit vector that points from site 𝑖 towards site 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The total current 𝐼 flowing through the junction is found by integrating this over a cross section of the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Results and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The main results of our numerical simulations are in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' First, we observe that 0-𝜋 oscillations are possible in both straight and diagonal junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This finding is interesting since such oscillations are typically found in Josephson junctions with magnetic interlayers, whereas altermagnets have zero magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Moreover, 0-𝜋 oscil- lations do not appear in Rashba spin-orbit coupled junctions either, which have a different spin-momentum coupling (odd- in-momentum) compared to altermagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Second, we see that the 0-𝜋 oscillations behave qualitatively differently from ferromagnetic Josephson junctions: the latter typically has an exponential decay with superimposed oscillations, whereas in the altermagnet case there is an initial large decay followed by oscillations with a much weaker damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This result is most striking in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(b), where we find a pure decay at 𝐿 < 8𝑎 followed by nearly pure oscillation at 𝐿 > 10𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Physically, the oscillations in straight junctions can be un- derstood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Conventional superconductivity consists of singlet Cooper pairs |↑↓⟩ − |↓↑⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' As the Cooper pairs leak into the altermagnet along the 𝑥 axis, spin-up electrons have a hopping amplitude 𝑡 + 𝑚 while spin-down electrons have a hopping amplitude 𝑡 − 𝑚 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This “speed difference” causes position-dependent phase differences between spin-up and spin-down electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Such spin-dependent phase shifts are well-known to cause 0-𝜋 oscillations from previous studies on ferromagnetic Josephson junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For diagonal junctions, however, the most direct path between the superconductors consists of an equal number of hops along the 𝑥 and 𝑦 axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Since the electrons experience opposite spin-dependent phase shifts in these two cases, their effects appear to partially cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Figure 2 shows that the initial decay in 𝐼1(𝐿) is in general accelerated as 𝑚 is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' However, 0-𝜋 oscillations are found over a much wider parameter range for straight than diagonal junctions, consistent with the discussion above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For example, for small altermagnetic order parameters 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='05𝑡 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(a)], the first 0-𝜋 oscillation occurs already at 𝐿 = 15𝑎 for the straight junction, but not until 𝐿 = 35𝑎 for the diagonal junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' On the other hand, for large altermagnetic order parameters 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5𝑡 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(c)], the first 0-𝜋 oscillation occurs simultaneously, but sustained 0-𝜋 oscillations for a large range of junction lengths is found only for straight junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' It is only for intermediate values 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='15𝑡 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(b)] that we find qualitatively similar results in straight and diagonal junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Finally, for very large values 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='9𝑡, the supercurrent decays extremely fast for both straight and diagonal junctions, limiting the number of visible oscillations for both junction types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For very large altermagnetic order parameters 𝑚 → 𝑡, spin- down electrons become nearly immobile along the 𝑥 axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In this limit, spin-zero Cooper pairs clearly cannot propagate through the altermagnet, and the Josephson effect vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This explains the extremely sharp decay in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Interestingly, if an altermagnet with 𝑚 → 𝑡 could be realized experimentally, 4 0 10 20 30 40 Altermagnet length 퐿 10−3 10−2 10−1 100 Critical current 퐼푐 (퐿)/퐼푐 (0) Straight junction Diagonal junction FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Critical current 𝐼𝑐 as a function of the altermagnet length 𝐿 for 𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5Δ and 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='05𝑇𝑐 [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' this might also serve as a new kind of filter for spin-triplet Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Specifically, we would expect |↑↑⟩ pairs to only move along the 𝑥 axis, |↓↓⟩ pairs to only move along the 𝑦 axis, and any |↑↓⟩ ∓ |↓↑⟩ pairs to decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This may thus provide a non- destructive way to separate the different equal-spin-triplet pairs generated in superconducting spintronics while eliminating any remaining spin-zero pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 3, we show the critical current, rather than just the first harmonic, for one of the junctions considered (𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5Δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(a) we saw that this choice of 𝑚 produces 0-𝜋 oscillations at 𝐿 = 15𝑎 and 𝐿 = 21𝑎 for straight but not diagonal junctions, which causes significant 𝐼𝑐 suppression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' However, due to the presence of higher harmonics, it is difficult from the 𝐼𝑐 curve alone in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 3 to observe that there are two such 0-𝜋 oscillations in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This becomes even more challenging for higher values of 𝑚 (not shown), where qualitatively similar oscillations are observed, but the shorter oscillation period makes it more difficult to determine the exact number of zero crossings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We also see that both junctions display a 0-𝜋 oscillation for 𝐿 ≈ 35𝑎, which for the diagonal junction is the first 0-𝜋 oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Figure 4 shows the critical current 𝐼𝑐 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' temperature 𝑇, which is one experimental signature of 0-𝜋 transitions in Joseph- son junctions [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For these calculations, we picked an alter- magnet length 𝐿 = 12𝑎 which according to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(b) is close to a 0-𝜋 transition for straight but not diagonal junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For the straight junction, we find a non-monotonic critical current that dips sharply at 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='6𝑇𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' For both 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5𝑇𝑐 and 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='7𝑇𝑐, the current-phase relation is completely dominated by the first harmonic 𝐼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' However, it changes sign between these two points, so the dip is a signature of a 0-𝜋 transition as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This is in contrast to the diagonal junction, where we find no 0-𝜋 transition for these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' In summary, we have demonstrated that the Josephson effect through altermagnets exhibits different proper- ties than in the two conventional classes of magnetic materials, ferromagnets and antiferromagnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Despite the absence of a net magnetization, altermagnets induce 0-𝜋 oscillations in the Josephson effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' We have also shown that the decay and oscillation period of the Josephson current strongly depend on the crystallographic orientation of the altermagnet relative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='0 Temperature 푇/푇푐 10−4 10−3 10−2 10−1 100 Critical current 퐼푐 (푇)/퐼푐 (0) Straight junction Diagonal junction FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Critical current 𝐼𝑐 as a function of temperature 𝑇 for 𝑚 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='5Δ and 𝐿 = 12𝑎 [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' the superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' The Josephson effect can therefore be used both to distinguish the altermagnet from conventional (anti)ferromagnetism and additionally offers a way to tune the supercurrent via flow direction anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' This work was supported by the Research Council of Norway through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 323766 and its Centres of Excellence funding scheme Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 262633 “QuSpin.” The simulations were performed on resources provided by Sigma2—the National Infrastructure for High Performance Computing and Data Storage in Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Hirohata, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Yamada, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Nakatani, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Prejbeanu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Dieny, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Pirro, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Hillebrands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 509, 166711 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content='-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Yuan, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Wang, J.' metadata={'source': 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J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Sinova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Goennenwein, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Jung- wirth, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Smejkal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} 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Muñoz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 18, 33250 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Pekar and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Rashba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Zh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Eksp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Teor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Fiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' 47, 1927 (1964).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' [8] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Clarke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} +page_content=' A 308, 447 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/NNE2T4oBgHgl3EQfBgZ1/content/2301.03603v1.pdf'} diff --git a/O9AyT4oBgHgl3EQftflG/content/tmp_files/2301.00595v1.pdf.txt b/O9AyT4oBgHgl3EQftflG/content/tmp_files/2301.00595v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..586f1e7bb98c59358f5593913482159c23ee2348 --- /dev/null +++ b/O9AyT4oBgHgl3EQftflG/content/tmp_files/2301.00595v1.pdf.txt @@ -0,0 +1,2096 @@ +Chains of Autoreplicative Random Forests for +missing value imputation in high-dimensional +datasets +Ekaterina Antonenko1,2 and Jesse Read1 +1 LIX, École Polytechnique, Institut Polytechnique de Paris, France +2 Digitalent lab (Moteur Intelligence Artificielle), Paris, France +{ekaterina.antonenko,jesse.read}@polytechnique.edu +Abstract. Missing values are a common problem in data science and +machine learning. Removing instances with missing values can adversely +affect the quality of further data analysis. This is exacerbated when there +are relatively many more features than instances, and thus the propor- +tion of affected instances is high. Such a scenario is common in many +important domains, for example, single nucleotide polymorphism (SNP) +datasets provide a large number of features over a genome for a relatively +small number of individuals. To preserve as much information as possi- +ble prior to modeling, a rigorous imputation scheme is acutely needed. +While Denoising Autoencoders is a state-of-the-art method for imputa- +tion in high-dimensional data, they still require enough complete cases +to be trained on which is often not available in real-world problems. In +this paper, we consider missing value imputation as a multi-label classifi- +cation problem and propose Chains of Autoreplicative Random Forests. +Using multi-label Random Forests instead of neural networks works well +for low-sampled data as there are fewer parameters to optimize. Ex- +periments on several SNP datasets show that our algorithm effectively +imputes missing values based only on information from the dataset and +exhibits better performance than standard algorithms that do not require +any additional information. In this paper, the algorithm is implemented +specifically for SNP data, but it can easily be adapted for other cases of +missing value imputation. +Keywords: Missing value imputation · Multi-label classification · High- +dimensional data. +1 +Introduction +Missing values are a common problem and an important issue in the domain +of data science and machine learning. Most off-the-shelf statistical and machine +learning methods cannot handle missing values, and such values must be im- +puted, or the whole instance or row removed, before the actual data analysis. +When many values are missing, considering only instances with complete infor- +mation can lead to a loss of necessary information and can yield a very poor or +even empty dataset. +arXiv:2301.00595v1 [cs.LG] 2 Jan 2023 + +2 +E. Antonenko and J. Read. +A special challenge is missing values occurring in several or even most of the +samples and/or features of the training set, and when there are sufficiently more +features than samples (p ≫ N), which means that removing samples amplifies +the imbalance. Single Nucleotide Polymorphism (SNP) is an example of high- +dimensional and low-sampled categorical data where missing values are very +common and can affect a large number of the features. Other examples of data +with such characteristics include gene expression arrays, classification problems +in astronomy, tool development for finance data, and weather prediction [12]. +Multiple Imputation with Chained Equations (MICE) [4] remains a state- +of-the-art approach in the imputation domain, very powerful and flexible, but +requires proper parameter tuning and a deep understanding of the data. We +have not seen evidence of MICE usage for high-dimensional data. We propose +possible MICE parameters to make computation time feasible but do not obtain +promising results. Denoising Autoencoders have proved to work well for the +missing value imputation [5], but they require enough complete cases for the +training phase, which is not always the case in real-world data and, in particular, +high-dimensional data. +In this paper, we consider missing value imputation of categorical features +as a multi-label classification problem, and thus exploiting multi-label models +such as Random Forests [3] becomes possible. We present an algorithm that ef- +ficiently imputes missing values when data has complete cases to be trained on +and can be adapted for high-dimensional data when having fully complete cases +is very unlikely. We explore the idea of a multi-label prediction cascade, using a +methodology similar to that of, e.g., Classifier Chains [15] in the multi-label clas- +sification literature, where already processed targets are stacked as new features +for further targets. We use Chains of Autoreplicative Random Forests (ChARF) +such that each Random Forest processes one window of consequent features and +incorporates information from already imputed previous windows. We treat win- +dows as data that can be given to an Autoencoder, however, noticing that we +do not explicitly need a hidden-layer representation, we use multi-label Random +Forests instead of neural network architecture. To the best of our knowledge, +using simpler multi-label models as Autoencoders without explicit encoding has +not been widely studied in the literature. We argue that such an approach can +be useful for similar purposes, at least for imputation (similarly to Denoising +Autoencoders), and has its advantages, such as an ability to efficiently process +data of small sample size. +We study imputation for SNP data as it exhibits all the aspects we are inter- +ested in tackling: high dimensional data (such that p ≫ N), the possibility of a +significant proportion of missing values, and no reference panel but at least some +local correlations in the feature space. At the same time, our approach can be +adapted to any data exhibiting these characteristics. Although SNP data is an +instance of categorical data, our model can easily be modified to work with con- +tinuous data by using regressor models, which have already been investigated in +the context of ‘regressor chains’, e.g., [1]. For high-dimensional and low-sampled +datasets, the ChARF method shows to be very competitive w.r.t. both impu- + +Chains of Autoreplicative Random Forests +3 +tation quality and time complexity. At the same time, even for low-dimensional +data we can adapt this approach with personalized splitting for blocks depending +on missing patterns. +The rest of the paper is organized as follows. After summarizing background +and related work in Section 2, we present our method in Section 3. The results +and their discussion as well as complexity analysis are in Section 4. In Section 5, +we draw conclusions and describe future work. +2 +Related work +Traditionally, missing data is categorized into three types: Missing Completely +At Random (MCAR, the absence occurs entirely independently from feature +values), Missing At Random (MAR, the absence depends only on the observed +feature values), and Missing Not At Random (MNAR, the absence depends on +both observed and the unobserved feature values) [18]. +Within this work, we consider high-dimensional data with categorical fea- +tures. An example of such a situation is Single Nucleotide Polymorphism (SNP) +data, which presents a range of genetic differences between the individuals. Typ- +ically the associations between traits/diseases are studied. A standard coding for +values in SNP datasets is 0, 1, and 2 for variants AA, Aa, and aa respectively, +where allele A corresponds to the prevalent variant in the population and allele +a to the minor one. Due to linkage disequilibrium [7], neighboring features can +correlate to each other, and taking such dependencies into account is helpful +for missing value imputation. At the same time, some long-distance correla- +tions (across the genome) are also possible, though rare. A typical SNP dataset +contains a number of features (positions on the genome) greatly exceeding the +number of samples (individuals in the population of the study). SNP datasets +are prone to a missing values problem due to a variety of reasons, such as de- +viations from the Hardy-Weinberg equilibrium, an abundance of rare variants, +and missing features in combining different datasets in meta-studies [7]. Within +this study, we assume that missing data is missing completely at random as it +depends on external factors rather than observed/unobserved values. Removing +features or samples containing missing values may be inefficient as this implies +loss of important information and impoverishment of the data. +For SNP data, imputation methods can be split into reference-based and +reference-free methods. Reference-based techniques require a reference panel +based on whole-genome sequencing samples and show the advantage of using +large datasets with complete data as well as additional information such as +linkage patterns, mutations, and recombination hotspots [7]. The main limit- +ing factors for such methods are the size and sequencing coverage of reference +panels, as well as the conformity of ethnicity in the reference panel and data con- +taining missing values to impute. While reference-based methods are considered +a first-choice approach to impute missing values in SNP data, the correspond- +ing reference panels are not always available, especially for non-human species. +Moreover, these methods require similarity of populations in the references and + +4 +E. Antonenko and J. Read. +data to impute. These facts necessitate the study and development of methods +that are independent of the reference panels and impute missing values exploiting +only statistical inferences from the data itself. +The existing missing value imputation methods range from simple replace- +ment with mean, mode, or median statistics [13] to more sophisticated tech- +niques such as k-Nearest Neighbours (kNN) [19], Singular Value Decomposition +(SVD) [21], Random Forests [20], and logistic regression. Recently developed +deep learning techniques have also been applied for imputation, e.g. Denoising +Autoencoders [5] method. Below we present the listed methods in more detail. +Mode [13]. For each feature, a mode of non-missing values, i.e. the most +frequent value, is estimated, and the missing values are imputed with this mode. +k-Nearest Neighbors (kNN) [19]. The imputation procedure is based +on the weighted k-Nearest Neighbors algorithm. The algorithm looks for the k +samples that are most similar to the one whose missing values need to be replaced +and uses these k neighbors to impute the missing values. For experiments, we +used knncatimpute function implemented in scrime R package. +Singular Value Decomposition (SVD) [21]. This method calculates the +k most significant eigenvectors and then imputes the missing values using a low- +rank SVD approximation estimated by an Expectation-Maximization algorithm. +For experiments, we used IterativeSVD function implemented in fancyimpute [17] +python package. +Multivariate Imputation by Chained Equations (MICE) +[4]. The +imputation process is organized into several cycles of prediction, on one cycle +each variable is regressed on the other variables (all or subset). Initially, MICE +imputes missing values with samples from features distributions and then carries +out a number of imputation iterations. The MICE method is very flexible w.r.t. +base model, i.e. any per feature estimator is possible. +MissForest [20]. MissForest also works in an iterative manner by predicting +missing values by Random Forests trained on the observed features. MissForest +builds a new Random Forest for each feature and iteration, while we propose to +use a much smaller number of forests with a small number of features each, and +this allow to significantly alleviate the computation (see Subsection 4.1) +The MICE and missForest methods are commonly used for different types +of data and, in particular, clinical data, and can be considered state-of-the-art +for missing value imputation, but we have not found big evidence of using these +methods for high-dimensional datasets, as they become very costly with the rise +of the number of features. In our empirical study we try to adapt the MICE +method for SNP data, but do not obtain promising results (see Section 4). +Denoising Autoencoders (DAE) [23,5]. Neural networks reproducing in- +put X as output are called Autoencoders [2]. Classical Autoencoders imple- +mented within neural networks architecture consist of Encoder and Decoder +structures as illustrated in Fig. 1a. While the inner structure of hidden layers +can be very different, the typical common property is having a narrow middle +layer H to restrict the model to learning only important information from the +data. Optimizing hidden layers implies a search for some inner patterns in the + +Chains of Autoreplicative Random Forests +5 +(a) Classic Autoencoder +(b) Denoising Autoencoder +(c) Autoencoder without an explicit encoding +Fig. 1: An illustration of (a) Classic Autoencoder (with hidden representation H), (b) +Denoising Autoencoder (input is corrupted with noise or missing values as +˙X before +encoding), and (c) Autoencoder without an explicit encoding (as we use in our work). +In all cases, the goal is to minimize the difference between X and its reconstruction Z. +data. Denoising Autoencoders [23] receive data ˙X corrupted by noise or missing +values as input and complete data X as output during the training phase when +they learn to remove noise or impute missing values (see Fig. 1b). Denoising Au- +toencoders have been successfully applied to address the missing data problems +in various fields [23]. In [5] the authors suggest Sparse Convolutional Denoising +Autoencoders (SCDA) to impute missing values in SNP datasets. Sparsity is +required due to high dimensionality and insufficient number of samples to train +on, and convolution layers are used as neighboring features have a bigger chance +to explain each other, at least in SNP data. The main limitation of the SCDA +method is that they require training data of complete cases, which is usually very +limited in SNP data. For this reason, we don’t include the SCDA method in an +experimental setting where all or almost all cases are affected by missingness. +Although apparently largely overlooked in the literature, we have noticed +that any other model designed for the multi-label prediction can be used instead +of a neural network as an Autoencoder. One such example is a combination of +decision trees [25] where the first decision tree is used as an encoder, and the +second one is used in a vice versa manner as a decoder. Meanwhile, this idea can +be simplified even more: if we are not aiming to extract the information about +the inner patterns of the data, the usage of a straightforward model such as a +decision tree or random forest is sufficient. Such an approach can facilitate the +optimization process for the model on data containing a small number of samples, +and at the same time, decision trees and random forests are both efficient and +simple to understand and interpret. To the best of our knowledge, this simple but +efficient idea has not been well studied in the literature. We argue, that however it +deserves attention and can be further investigated. Applying this idea, we suggest + +Loss function +L(X, Z)Loss function +L(X, Z)Loss function +L(X, Z)6 +E. Antonenko and J. Read. +further Autoreplicative Random Forests. While we choose Random Forests as +a well-known and stable multi-label method with good performance, this idea +may be developed by using other multi-label methods, such as e.g. Classifier +Chains [15], Multilabel k Nearest Neighbours [24], Random k-Labelsets [22], +Conditional Dependency Networks [11]. +3 +Method +Our method consists of two main novelties. First, we use multi-label classifiers +(e.g. Random Forests) as imputational Autoreplicative models (Subsection 3.1). +Second, Chains of subsequent windows of Autoreplicative models allow adapting +the idea of Autoencoders to real-world high-dimensional scenarios when there is +no complete data available for training (Subsection 3.2). +3.1 +Autoencoders without an explicit encoding +Our method is inspired by the idea of Denoising Autoencoders for missing value +imputation. We use multi-label predictive models that are not based on neu- +ral networks. With this approach, we want to efficiently process relatively low- +sampled (compared to a number of features) datasets, where complex neural +networks are prone to overfitting and get stuck in optimizing parameters. Fur- +thermore, as discovering the inner structure of the data itself is out of the scope +of this task, we do not explicitly need hidden layers of the neural network. +We would like to point here that any multi-label classification model can be +used for this goal. We will use Random Forest as it proved to be a compet- +itive and robust method. However, an approach of autoreplicative imputation +models deserves better research and possibly may be improved by the usage +of more sophisticated multi-label models. To the best of our knowledge, multi- +label classification models and, e.g., Random Forests have not been used before +as autoreplicative models reproducing input as output. +The training process is illustrated in Fig. 1c. First, we select complete cases +of the entire dataset or of a features subset (more on that in the following +subsection) X, obtain dataset +˙X by manually corrupting them with missing +values (uniformly distributed, following the proportion of missing values in the +original dataset) , and train an Autoreplicative forest to reproduce Z ∼ X from +˙X, i.e. fill missing values by minimizing loss function between Z and X. Then +the fitted model can be used to replace actual missing values. +3.2 +Ensemble of chains of Autoencoders +Autoencoders are considered a baseline method for missing value imputation. +However, they require complete data for training. It is often difficult or impos- +sible to obtain such datasets in real-world problems when missing values can +be abundant. This is especially the case for high-dimensional data: with a large +number of features, it is likely not feasible to select a reasonable number of rows + +Chains of Autoreplicative Random Forests +7 +without missing values, even for a small ratio of missingness. As a consequence, +we need to adapt our approach for the high-dimensional datasets, when training +data may be not available. For this goal, we split the whole set of consequent +features into windows of size ∆, such that for the features within one window it +is possible to select a training subset of reasonable size with full information. We +fit the model on the selected subset and then predict values to impute missing +ones in the remaining subset. +In the case of SNP data, it makes sense to select windows of consequetive +features, as they are more likely to provide information about each there. This +effect is due to linkage disequilibrium, i.e. close neighbor positions in the genomes +are more likely to be inherited from the same ancestors. This window approach +may serve for other types of ordered data, such as e.g. gene expression arrays, +time series, images, and sound fragments. +Fig. 2: Average complete training size (i.e. rows without missing values) according to +window size ∆. Missing values are simulated for the MCAR scenario with a uniform +distribution. Dashed black lines show examples of possible window sizes for τ = 0.4. +Fig. 2 shows the average size of available training data in simulation with +uniformly distributed missing values. As it can be seen, it decreases dramatically +with the growth of window size. To increase the method’s power to catch and +use dependencies between the features, we suggest chains of imputation models, +similar to the Classifier Chains methodology [15], i.e. stacking of already pro- +cessed features as new features for the consequent estimators (see Fig. 3). To +keep the complexity of the algorithm feasible and reduce computation time, we +do not incorporate all previous windows but select only ν last ones. +The basic intuition behind using windows of consequent features is that short +chromosome segments can be inherited from a distant common ancestor [7] and +thus shared between some individuals. For this reason, we select one forward and +one backward chain, as well as several (up to 3) random chains, to incorporate +possible long-term interactions. Selection of previously imputed windows can be +generalized as, for example, sampling from a normal distribution with a mean + +Average % of complete rows per window +1% missing values +g0 +5% missing values +10% missing values +Size of training data = +20% missing values +30% missing values +8 +OE +0 +0 +5 +10 +15 +20 +25 +OE +35 +40 +45 +window size8 +E. Antonenko and J. Read. +Fig. 3: Model processes windows in a chain, incorporating windows with already im- +puted values as additional features. At one step, we split the window of size ∆ into +training part with complete data and testing part with missing values. After fitting on +training data corrupted with missing values, we impute missing values in testing part. +equal to the current window number (Fig. 4a) or other kinds of distributions for +different kinds of data. In the ensemble of chains, we average predictions (i.e. +take a major vote) for each missing value. +To support the hypothesis that neighboring features have a higher chance to +explain each other, in Fig. 4b-4f we include experiments for using all neighboring +(strategy A) or only two distant (strategy B) windows on distance ν. We can +see that including very close neighbors significantly increases the quality of im- +putation, while with including distant neighbors the improvement may present +(this fact corresponds to possible long-term correlations), but is very unstable +and cannot be guaranteed. +Table 1: Example window sizes ∆ according to desired training samples, via Eq. 1 +Size of training data +% of missing data 1% +5% +10% +20% +30% +20% of original data +160 +31 +15 +7 +4 +30% of original data +120 +23 +11 +5 +3 +50% of original data +69 +14 +7 +3 +2 +Our method is summarised as pseudocode in Alg.1. We compare performance +of the models with hyperparameters ∆ and ν in Section 4. We estimate theo- + +Corrupting with +missing values +Training +PredictingChains of Autoreplicative Random Forests +9 +(a) Probabilities to take already predicted +windows into chain at position 50; 100 win- +dows; 10 previous windows for each chain; 3 +chains: forward, backward, and random. +(b) Two strategies to include distant win- +dows into analysis +(c) Window size ∆ = 10, strategy A +(d) Window size ∆ = 5, strategy A +(e) Window size ∆ = 10, strategy B +(f) Window size ∆ = 5, strategy B +Fig. 4: Including the 2ν closest neighbor windows as additional features (strategy A) +significantly increases the accuracy while including only 2 windows on distance ν (strat- +egy B) has occasional and unstable improvement. + +6'0 +ccuracy +1: +1% missing values +5% missing values +0.6 +10% missing values +20% missing values +30% missing values +0 +5 +10 +15 +20 +35 +40 +45 +50 +Distance from the selected window0.9 +0.B +0.7 +0.6 +1% missing values +5% missing values +0.5 +10% missing values +20% missing values +0.4 +30% missing values +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +O +Distance from the selected windownormaldistribution +0.04 +ChARFstrategy +0.03 +0.02 +0.01 +0.00 +0 +10 +20 +30 +40 +50 +60 +70 +80 +06 +100△=310 +6'0 +ccuracy +1% missing values +5% missing values +10% missing values +0.6 +20% missing values +30% missing values +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Distance from the selected window10 +0.9 +Accuracy +0.B +0.7 +0.6 +1% missing values +5% missing values +0.5 +10% missing values +20% missing values +0.4 +30% missing values +0 +10 +30 +50 +60 +70 +80 +90 +Distance from the selected window10 +E. Antonenko and J. Read. +retically the maximum size ∆ of one window according to the desired size of +training data τ. As τ ∼ (1 − f)∆, then +∆(τ) ∼ log1−f τ = +ln τ +ln(1 − f), +τ ∈ (0, 1), +(1) +where f is a fraction of missing values and τ is a desirable threshold for a ratio +of complete rows in the training subset. The empirical results of the simula- +tion (Fig. 2) correspond to this estimation. We see that with the growth of +window size the size of training data decreases dramatically. As a consequence, +the window size should be selected carefully by taking the missing value ratio +into account. We suggest possible window sizes according to the desired size of +training data in Table 1. +Algorithm 1 +1: procedure ChARF(XN×p, window size ∆, # of previous steps ν, # of chains K) +2: +Split features into ∆-wide windows +▷ Last window has size +p (mod ∆) +3: +Generate K permutations of (1, 2, ..., n = ⌈ p +∆⌉)} +4: +for each permutation {σ(1), ..., σ(n)} do +5: +for each window Xσ(i) do +6: +Xext ← Xσ(i) +� Xσ(i−1) +� ... � Xσ(i−ν) +▷ Stack last ν processed +windows as additional +features +7: +Xtrain ← Xcomplete +ext +▷ Select complete cases +for training +8: +˜ +Xtrain ← Xtrain corrupted with missing values ▷ Uniformly +dis- +tributed, % of m.v. +calculated from Xσ(i) +9: +Xtest ← Xmissing +ext +10: +Fit model on ( ˜ +Xtrain, Xtrain) +11: +Xpred ← predictions of fitted model on Xtest +12: +replace missing values in Xtest with corresponding values from Xpred +13: +Take major vote for all K predictions per missing value +4 +Results and discussion +We evaluate the performance of our method by imputation accuracy, i.e. per- +centage of correctly imputed values out of missing ones. We test Chains of au- +toreplicative Random Forests (of 10 trees each) on several high-dimensional SNP +datasets (p ≫ N), briefly summarized in Table 2. For the SNP data, we test only +the MCAR scenario, as missing values are likely to happen because of external +reasons rather than depending on missing or observed data. We simulate the +missing values by masking true values in the data under a uniform distribution, + +Chains of Autoreplicative Random Forests +11 +with the proportion of missing values 1%, 5%, 10%, 20%, and 30%. The proce- +dure is repeated 5 times to produce independent incomplete datasets. Average +accuracy is shown. The empirical study has shown a significant improvement, +when the features are one-hot encoded (paired t-test statistics 3.442, df=29, +p-value=0.0018, see Fig. 5). +Table 2: Datasets used in experiments, p features, N samples. +Name +p +N +Maize [14] +44,729 +247 +Eucalyptus [10] +33,398 +970 +Colorado Beetle [6] +34,186 +188 +Arabica Coffee [8] +4,666 +596 +Wheat (Zuchtwert study) [16] +9,763 +388 +Coffea Canephora [9] +45,748 +119 +Fig. 5: One-Hot encoding may significantly improve the imputation power of ChARF +method in SNP datasets. +For ChARF, we first evaluate hyperparameters: window size ∆ ∈ [3, 5, 8, 10, 15], +and number of previous windows in the chain to take as new features ν ∈ +[0, 1, 3, 5, 10]. To reduce the computation time, we first search for the best values +of ∆ and ν on reduced datasets (first 1000 features) and then use these values +for computation on the entire datasets. The grid-search results are presented in +Fig. 6. As expected (from estimation in Subsection 3.2), from Fig. 6 we see that +the most effective window size decreases with the growth of a number of missing +values (since a bigger part of instances gets corrupted). +For the MICE method, with the default settings, each estimator considers all +other variables, which makes the total complexity at least quadratic and thus +requires huge computational and time resources in the case of high-dimensional + +Imputation accuracy +LD +0.9 +80 +without One-Hot +0.B +8 +00 +Q +0.7 +0.6 +0.5 +0.5 +0.6 +o.B +60 +with One-Hot encoding12 +E. Antonenko and J. Read. +1% +5% +10% +20% +30% +Maize +Eucalyptus +C. Beetle +A. Coffee +Wheat +C. Canephora +Fig. 6: Accuracy of imputation for SNP datasets for different ratios of missing values +(indicated in column headers). 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The original R +package suggests a pre-processing quickpred function, which selects the predictive +features based on pairwise correlation, but in this case, quadratic complexity is +required in this step. With the intuition that the neighboring features in SNP +data have the highest chance to explain each other, for the experiments we select +windows of 10 neighbors for each feature. Such an approach is computationally +feasible, but the imputation still leaves some missing values in the data (around +10-20%). The possible explanation is the collinearity between features [4]. This +approach worked for smaller SNP datasets (Arabica Coffee and Wheat), but for +the other ones, the computations still failed. +Table 3: Accuracy. An asterisk (∗) indicates optimal hyperparameters estimated via +internal validation. For knnimpute we selected the best of k ∈ {3, 5, 10, 20, 50} (shown +in brackets) in a similar way. Likewise for rank ∈ {10, 20, 50, 100, 200, 300, 500} for SVD +method. The missForest method was run for first 100 features only as it is not possible +to run it for a whole dataset. Best accuracy per column in bold. All results rounded +to 3 dp. +0.01 +0.05 +0.1 +0.2 +0.3 +0.01 +0.05 +0.1 +0.2 +0.3 +Maize +Eucalyptus +∆ = 15∗ ∆ = 15∗ ∆ = 10∗ +∆ = 5∗ +∆ = 5∗ +∆ = 10∗ +∆ = 5∗ +∆ = 5∗ +∆ = 3∗ +∆ = 3∗ +ChARF +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 5∗ +ν = 10∗ +ν = 10∗ ν = 10∗ ν = 10∗ +0.952 +0.935 +0.916 +0.882 +0.845 +0.970 +0.950 +0.926 +0.866 +0.810 +kNN (5/10) +0.803 +0.802 +0.801 +0.798 +0.794 +0.851 +0.849 +0.847 +0.843 +0.839 +mode +0.727 +0.727 +0.726 +0.727 +0.726 +0.725 +0.732 +0.731 +0.730 +0.729 +SVD (50/500) +0.647 +0.648 +0.645 +0.643 +0.636 +0.788 +0.788 +0.788 +0.785 +0.780 +MICE +– +– +– +– +– +– +– +– +– +– +missForest +0.662 +0.650 +0.622 +0.593 +0.580 +0.684 +0.673 +0.626 +0.564 +0.521 +Colorado Beetle +Arabica Coffee +∆ = 10∗ ∆ = 10∗ +∆ = 5∗ +∆ = 5∗ +∆ = 3∗ +∆ = 15∗ ∆ = 10∗ +∆ = 5∗ +∆ = 3∗ +∆ = 3∗ +ChARF +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 3∗ +ν = 3∗ +ν = 5∗ +ν = 10∗ +ν = 3∗ +0.835 +0.824 +0.818 +0.805 +0.792 +0.897 +0.886 +0.878 +0.866 +0.854 +kNN (50/10) +0.765 +0.763 +0.765 +0.765 +0.764 +0.867 +0.866 +0.866 +0.865 +0.864 +mode +0.761 +0.760 +0.762 +0.761 +0.761 +0.807 +0.804 +0.805 +0.805 +0.804 +SVD (50/100) +0.740 +0.737 +0.737 +0.735 +0.734 +0.693 +0.694 +0.696 +0.692 +0.690 +MICE +– +– +– +– +– +0.757 +0.741 +0.724 +0.689 +0.664 +missForest +0.352 +0.349 +0.361 +0.326 +0.335 +0.497 +0.480 +0.533 +0.541 +0.586 +Wheat +Coffea Canephora +∆ = 8∗ +∆ = 5∗ +∆ = 5∗ +∆ = 3∗ +∆ = 3∗ +∆ = 10∗ ∆ = 10∗ +∆ = 5∗ +∆ = 5∗ +∆ = 3∗ +ChARF +ν = 10∗ +ν = 10∗ +ν = 10∗ +ν = 10∗ ν = 10∗ +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 1∗ +ν = 1∗ +0.821 +0.808 +0.795 +0.777 +0.762 +0.799 +0.781 +0.761 +0.731 +0.717 +kNN (10/10) +0.823 +0.819 +0.818 +0.815 +0.811 +0.737 +0.739 +0.737 +0.734 +0.731 +mode +0.729 +0.727 +0.729 +0.729 +0.727 +0.691 +0.693 +0.692 +0.692 +0.691 +SVD (200/50) +0.622 +0.618 +0.609 +0.600 +0.594 +0.456 +0.453 +0.450 +0.449 +0.450 +MICE +0.641 +0.635 +0.621 +0.585 +0.545 +– +– +– +– +– +missForest +0.614 +0.736 +0.746 +0.756 +0.755 +0.377 +0.449 +0.442 +0.395 +0.383 +In most cases, the experiments show better or competitive performance w.r.t. +benchmark methods (Table 3). At the same time, we see that with the rise of +the missing value ratio the accuracy of imputation diminishes. This is explained +by the very small size of training data even on small windows for a big number +of missing values. However, for a moderate missing value ratio, our method +consistently outperforms its alternatives. + +14 +E. Antonenko and J. Read. +4.1 +Time complexity analysis +The complexity of ChARF is O(pN log N) w.r.t. the number of features for a +single tree and for an ensemble is O( p +∆ ·∆N log N) ∼ O(pN log N) (fixed number +of chains and ensemble members). +We expect that the kNN and SVD methods’ time complexity is linear w.r.t. +a number of features. For the MICE method time, the complexity depends on +the base per feature estimator. In the simulation, we use a decision tree and +random forest as base estimators. For a single decision tree, the complexity +is O(pN log N), and thus total complexity is quadratic w.r.t. the number of +features. Random forests consist of individual decision trees, but it is possible to +select a number of features per tree. The standard choice is √p features per tree, +which makes the total complexity O(p√p) and is already substantially slower +than linear complexity for a big number of features p. The same estimation +holds for the MissForest method. However, we can reach linear complexity if +we put a number of features per tree equal to come constant, which we try in +the previous subsection. These theoretic estimations are well supported in the +simulation study, see Fig. 7. We empirically compare the time complexity of +the imputation methods on subsets of the Eucalyptus dataset under the MCAR +scenario with 10% missing values. The subsets are selected as first ps features of +the original dataset, 10 < ps < 500. +Fig. 7: Empirical results on time complexity for imputation methods. +5 +Conclusions and future work +We propose a new approach: tackling missing value imputation as a multi-label +predictive problem. First, we suggest Autoreplicative Random Forests as a sim- +pler alternative for Denoising Autoencoders. With this simple but efficient idea, +we can obtain competitive results, which may be further improved by more so- +phisticated multi-label models. This method does require complete cases for the +training phase, though we suppose that in real-world low-dimensional data this + +RF +80 +MICE(dt) +MICE(rf) +kNN +60 +SVD +40 +20 +0 +0 +100 +200 +300 +400 +500Chains of Autoreplicative Random Forests +15 +scenario is still quite realistic. Besides, it is possible to tune this method by +splitting the data into blocks of complete and missing features. +For high-dimensional datasets, when having complete data in all features +is very unlikely, we propose Chains of Autoreplicative Random Forests. This +method splits data into windows of consequent features and imputes missing +values window by window while incorporating information from already pro- +cessed features. We test this approach on SNP datasets and demonstrate a very +competitive predictive power. Our method requires neither reference panels nor +complete data for training and thus can be used in a variety of real-world sce- +narios when imputation of missing data is required. Our approach consists of +two main novelties: it is model agnostic (we used using Random Forests in ex- +periments) in regards to the Autoreplicator; essentially an Autoencoder with +no explicit encoding; and operates on windows of data. Our approach proved +competitive, and is promising for further investigation. +In future work, we are going to improve algorithm performance on datasets +with a big number of missing values and make it more stable w.r.t. high missing +value ratios. As preliminary experiments show that this approach works for the +MAR scenario as well, we will further analyze the performance of the ChARF +method for other patterns of missingness. +We will look at allowing multiple hypotheses and their distribution per a +single missing value. This would allow a greater chance of capturing the true +mode and incorporating this uncertainty to further data analysis. +Acknowledgments +We would like to thank Ander Carreño, University of the Basque Country, for +fruitful discussions and three anonymous reviewers for their editorial comments. +References +1. Antonenko, E., Read, J.: Multi-modal ensembles of regressor chains for multi- +output prediction. 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Zhang, M.L., Zhou, Z.H.: Ml-knn: A lazy learning approach to multi-label learning. +Pattern recognition 40(7), 2038–2048 (2007) +25. İrsoy, O., Alpaydin, E.: Autoencoder trees. In: Holmes, G., Liu, T.Y. (eds.) Asian +Conference on Machine Learning. Proceedings of Machine Learning Research, +vol. 45, pp. 378–390. PMLR, Hong Kong (20–22 Nov 2016) + diff --git a/O9AyT4oBgHgl3EQftflG/content/tmp_files/load_file.txt b/O9AyT4oBgHgl3EQftflG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cecf0ea1622e04f71f724ea536fefc2e564480a0 --- /dev/null +++ b/O9AyT4oBgHgl3EQftflG/content/tmp_files/load_file.txt @@ -0,0 +1,1456 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf,len=1455 +page_content='Chains of Autoreplicative Random Forests for missing value imputation in high-dimensional datasets Ekaterina Antonenko1,2 and Jesse Read1 1 LIX, École Polytechnique, Institut Polytechnique de Paris, France 2 Digitalent lab (Moteur Intelligence Artificielle), Paris, France {ekaterina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='antonenko,jesse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='read}@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Missing values are a common problem in data science and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Removing instances with missing values can adversely affect the quality of further data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This is exacerbated when there are relatively many more features than instances, and thus the propor- tion of affected instances is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Such a scenario is common in many important domains, for example, single nucleotide polymorphism (SNP) datasets provide a large number of features over a genome for a relatively small number of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To preserve as much information as possi- ble prior to modeling, a rigorous imputation scheme is acutely needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' While Denoising Autoencoders is a state-of-the-art method for imputa- tion in high-dimensional data, they still require enough complete cases to be trained on which is often not available in real-world problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In this paper, we consider missing value imputation as a multi-label classifi- cation problem and propose Chains of Autoreplicative Random Forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Using multi-label Random Forests instead of neural networks works well for low-sampled data as there are fewer parameters to optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Ex- periments on several SNP datasets show that our algorithm effectively imputes missing values based only on information from the dataset and exhibits better performance than standard algorithms that do not require any additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In this paper, the algorithm is implemented specifically for SNP data, but it can easily be adapted for other cases of missing value imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Keywords: Missing value imputation · Multi-label classification · High- dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 1 Introduction Missing values are a common problem and an important issue in the domain of data science and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Most off-the-shelf statistical and machine learning methods cannot handle missing values, and such values must be im- puted, or the whole instance or row removed, before the actual data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' When many values are missing, considering only instances with complete infor- mation can lead to a loss of necessary information and can yield a very poor or even empty dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='00595v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='LG] 2 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' A special challenge is missing values occurring in several or even most of the samples and/or features of the training set, and when there are sufficiently more features than samples (p ≫ N), which means that removing samples amplifies the imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Single Nucleotide Polymorphism (SNP) is an example of high- dimensional and low-sampled categorical data where missing values are very common and can affect a large number of the features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Other examples of data with such characteristics include gene expression arrays, classification problems in astronomy, tool development for finance data, and weather prediction [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Multiple Imputation with Chained Equations (MICE) [4] remains a state- of-the-art approach in the imputation domain, very powerful and flexible, but requires proper parameter tuning and a deep understanding of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We have not seen evidence of MICE usage for high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We propose possible MICE parameters to make computation time feasible but do not obtain promising results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Denoising Autoencoders have proved to work well for the missing value imputation [5], but they require enough complete cases for the training phase, which is not always the case in real-world data and, in particular, high-dimensional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In this paper, we consider missing value imputation of categorical features as a multi-label classification problem, and thus exploiting multi-label models such as Random Forests [3] becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We present an algorithm that ef- ficiently imputes missing values when data has complete cases to be trained on and can be adapted for high-dimensional data when having fully complete cases is very unlikely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We explore the idea of a multi-label prediction cascade, using a methodology similar to that of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=', Classifier Chains [15] in the multi-label clas- sification literature, where already processed targets are stacked as new features for further targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We use Chains of Autoreplicative Random Forests (ChARF) such that each Random Forest processes one window of consequent features and incorporates information from already imputed previous windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We treat win- dows as data that can be given to an Autoencoder, however, noticing that we do not explicitly need a hidden-layer representation, we use multi-label Random Forests instead of neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To the best of our knowledge, using simpler multi-label models as Autoencoders without explicit encoding has not been widely studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We argue that such an approach can be useful for similar purposes, at least for imputation (similarly to Denoising Autoencoders), and has its advantages, such as an ability to efficiently process data of small sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We study imputation for SNP data as it exhibits all the aspects we are inter- ested in tackling: high dimensional data (such that p ≫ N), the possibility of a significant proportion of missing values, and no reference panel but at least some local correlations in the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' At the same time, our approach can be adapted to any data exhibiting these characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Although SNP data is an instance of categorical data, our model can easily be modified to work with con- tinuous data by using regressor models, which have already been investigated in the context of ‘regressor chains’, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=', [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For high-dimensional and low-sampled datasets, the ChARF method shows to be very competitive w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' both impu- Chains of Autoreplicative Random Forests 3 tation quality and time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' At the same time, even for low-dimensional data we can adapt this approach with personalized splitting for blocks depending on missing patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' After summarizing background and related work in Section 2, we present our method in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The results and their discussion as well as complexity analysis are in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In Section 5, we draw conclusions and describe future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 2 Related work Traditionally, missing data is categorized into three types: Missing Completely At Random (MCAR, the absence occurs entirely independently from feature values), Missing At Random (MAR, the absence depends only on the observed feature values), and Missing Not At Random (MNAR, the absence depends on both observed and the unobserved feature values) [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Within this work, we consider high-dimensional data with categorical fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' An example of such a situation is Single Nucleotide Polymorphism (SNP) data, which presents a range of genetic differences between the individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Typ- ically the associations between traits/diseases are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' A standard coding for values in SNP datasets is 0, 1, and 2 for variants AA, Aa, and aa respectively, where allele A corresponds to the prevalent variant in the population and allele a to the minor one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Due to linkage disequilibrium [7], neighboring features can correlate to each other, and taking such dependencies into account is helpful for missing value imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' At the same time, some long-distance correla- tions (across the genome) are also possible, though rare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' A typical SNP dataset contains a number of features (positions on the genome) greatly exceeding the number of samples (individuals in the population of the study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' SNP datasets are prone to a missing values problem due to a variety of reasons, such as de- viations from the Hardy-Weinberg equilibrium, an abundance of rare variants, and missing features in combining different datasets in meta-studies [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Within this study, we assume that missing data is missing completely at random as it depends on external factors rather than observed/unobserved values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Removing features or samples containing missing values may be inefficient as this implies loss of important information and impoverishment of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For SNP data, imputation methods can be split into reference-based and reference-free methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Reference-based techniques require a reference panel based on whole-genome sequencing samples and show the advantage of using large datasets with complete data as well as additional information such as linkage patterns, mutations, and recombination hotspots [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The main limit- ing factors for such methods are the size and sequencing coverage of reference panels, as well as the conformity of ethnicity in the reference panel and data con- taining missing values to impute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' While reference-based methods are considered a first-choice approach to impute missing values in SNP data, the correspond- ing reference panels are not always available, especially for non-human species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Moreover, these methods require similarity of populations in the references and 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' data to impute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' These facts necessitate the study and development of methods that are independent of the reference panels and impute missing values exploiting only statistical inferences from the data itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The existing missing value imputation methods range from simple replace- ment with mean, mode, or median statistics [13] to more sophisticated tech- niques such as k-Nearest Neighbours (kNN) [19], Singular Value Decomposition (SVD) [21], Random Forests [20], and logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Recently developed deep learning techniques have also been applied for imputation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Denoising Autoencoders [5] method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Below we present the listed methods in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Mode [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For each feature, a mode of non-missing values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' the most frequent value, is estimated, and the missing values are imputed with this mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' k-Nearest Neighbors (kNN) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The imputation procedure is based on the weighted k-Nearest Neighbors algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The algorithm looks for the k samples that are most similar to the one whose missing values need to be replaced and uses these k neighbors to impute the missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For experiments, we used knncatimpute function implemented in scrime R package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Singular Value Decomposition (SVD) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This method calculates the k most significant eigenvectors and then imputes the missing values using a low- rank SVD approximation estimated by an Expectation-Maximization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For experiments, we used IterativeSVD function implemented in fancyimpute [17] python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Multivariate Imputation by Chained Equations (MICE) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The imputation process is organized into several cycles of prediction, on one cycle each variable is regressed on the other variables (all or subset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Initially, MICE imputes missing values with samples from features distributions and then carries out a number of imputation iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The MICE method is very flexible w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' base model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' any per feature estimator is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' MissForest [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' MissForest also works in an iterative manner by predicting missing values by Random Forests trained on the observed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' MissForest builds a new Random Forest for each feature and iteration, while we propose to use a much smaller number of forests with a small number of features each, and this allow to significantly alleviate the computation (see Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='1) The MICE and missForest methods are commonly used for different types of data and, in particular, clinical data, and can be considered state-of-the-art for missing value imputation, but we have not found big evidence of using these methods for high-dimensional datasets, as they become very costly with the rise of the number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In our empirical study we try to adapt the MICE method for SNP data, but do not obtain promising results (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Denoising Autoencoders (DAE) [23,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Neural networks reproducing in- put X as output are called Autoencoders [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Classical Autoencoders imple- mented within neural networks architecture consist of Encoder and Decoder structures as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' While the inner structure of hidden layers can be very different, the typical common property is having a narrow middle layer H to restrict the model to learning only important information from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Optimizing hidden layers implies a search for some inner patterns in the Chains of Autoreplicative Random Forests 5 (a) Classic Autoencoder (b) Denoising Autoencoder (c) Autoencoder without an explicit encoding Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 1: An illustration of (a) Classic Autoencoder (with hidden representation H), (b) Denoising Autoencoder (input is corrupted with noise or missing values as ˙X before encoding), and (c) Autoencoder without an explicit encoding (as we use in our work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In all cases, the goal is to minimize the difference between X and its reconstruction Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Denoising Autoencoders [23] receive data ˙X corrupted by noise or missing values as input and complete data X as output during the training phase when they learn to remove noise or impute missing values (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Denoising Au- toencoders have been successfully applied to address the missing data problems in various fields [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In [5] the authors suggest Sparse Convolutional Denoising Autoencoders (SCDA) to impute missing values in SNP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Sparsity is required due to high dimensionality and insufficient number of samples to train on, and convolution layers are used as neighboring features have a bigger chance to explain each other, at least in SNP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The main limitation of the SCDA method is that they require training data of complete cases, which is usually very limited in SNP data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For this reason, we don’t include the SCDA method in an experimental setting where all or almost all cases are affected by missingness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Although apparently largely overlooked in the literature, we have noticed that any other model designed for the multi-label prediction can be used instead of a neural network as an Autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' One such example is a combination of decision trees [25] where the first decision tree is used as an encoder, and the second one is used in a vice versa manner as a decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Meanwhile, this idea can be simplified even more: if we are not aiming to extract the information about the inner patterns of the data, the usage of a straightforward model such as a decision tree or random forest is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Such an approach can facilitate the optimization process for the model on data containing a small number of samples, and at the same time, decision trees and random forests are both efficient and simple to understand and interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To the best of our knowledge, this simple but efficient idea has not been well studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We argue, that however it deserves attention and can be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Applying this idea, we suggest Loss function L(X, Z)Loss function L(X, Z)Loss function L(X, Z)6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' further Autoreplicative Random Forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' While we choose Random Forests as a well-known and stable multi-label method with good performance, this idea may be developed by using other multi-label methods, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Classifier Chains [15], Multilabel k Nearest Neighbours [24], Random k-Labelsets [22], Conditional Dependency Networks [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 3 Method Our method consists of two main novelties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' First, we use multi-label classifiers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Random Forests) as imputational Autoreplicative models (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Second, Chains of subsequent windows of Autoreplicative models allow adapting the idea of Autoencoders to real-world high-dimensional scenarios when there is no complete data available for training (Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='1 Autoencoders without an explicit encoding Our method is inspired by the idea of Denoising Autoencoders for missing value imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We use multi-label predictive models that are not based on neu- ral networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' With this approach, we want to efficiently process relatively low- sampled (compared to a number of features) datasets, where complex neural networks are prone to overfitting and get stuck in optimizing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Fur- thermore, as discovering the inner structure of the data itself is out of the scope of this task, we do not explicitly need hidden layers of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We would like to point here that any multi-label classification model can be used for this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We will use Random Forest as it proved to be a compet- itive and robust method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' However, an approach of autoreplicative imputation models deserves better research and possibly may be improved by the usage of more sophisticated multi-label models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To the best of our knowledge, multi- label classification models and, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=', Random Forests have not been used before as autoreplicative models reproducing input as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The training process is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' First, we select complete cases of the entire dataset or of a features subset (more on that in the following subsection) X, obtain dataset ˙X by manually corrupting them with missing values (uniformly distributed, following the proportion of missing values in the original dataset) , and train an Autoreplicative forest to reproduce Z ∼ X from ˙X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' fill missing values by minimizing loss function between Z and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Then the fitted model can be used to replace actual missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='2 Ensemble of chains of Autoencoders Autoencoders are considered a baseline method for missing value imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' However, they require complete data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' It is often difficult or impos- sible to obtain such datasets in real-world problems when missing values can be abundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This is especially the case for high-dimensional data: with a large number of features, it is likely not feasible to select a reasonable number of rows Chains of Autoreplicative Random Forests 7 without missing values, even for a small ratio of missingness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' As a consequence, we need to adapt our approach for the high-dimensional datasets, when training data may be not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For this goal, we split the whole set of consequent features into windows of size ∆, such that for the features within one window it is possible to select a training subset of reasonable size with full information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We fit the model on the selected subset and then predict values to impute missing ones in the remaining subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In the case of SNP data, it makes sense to select windows of consequetive features, as they are more likely to provide information about each there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This effect is due to linkage disequilibrium, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' close neighbor positions in the genomes are more likely to be inherited from the same ancestors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This window approach may serve for other types of ordered data, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' gene expression arrays, time series, images, and sound fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 2: Average complete training size (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' rows without missing values) according to window size ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Missing values are simulated for the MCAR scenario with a uniform distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Dashed black lines show examples of possible window sizes for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 2 shows the average size of available training data in simulation with uniformly distributed missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' As it can be seen, it decreases dramatically with the growth of window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To increase the method’s power to catch and use dependencies between the features, we suggest chains of imputation models, similar to the Classifier Chains methodology [15], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' stacking of already pro- cessed features as new features for the consequent estimators (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To keep the complexity of the algorithm feasible and reduce computation time, we do not incorporate all previous windows but select only ν last ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The basic intuition behind using windows of consequent features is that short chromosome segments can be inherited from a distant common ancestor [7] and thus shared between some individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For this reason, we select one forward and one backward chain, as well as several (up to 3) random chains, to incorporate possible long-term interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Selection of previously imputed windows can be generalized as, for example, sampling from a normal distribution with a mean Average % of complete rows per window 1% missing values g0 5% missing values 10% missing values Size of training data = 20% missing values 30% missing values 8 OE 0 0 5 10 15 20 25 OE 35 40 45 window size8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 3: Model processes windows in a chain, incorporating windows with already im- puted values as additional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' At one step, we split the window of size ∆ into training part with complete data and testing part with missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' After fitting on training data corrupted with missing values, we impute missing values in testing part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' equal to the current window number (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 4a) or other kinds of distributions for different kinds of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In the ensemble of chains, we average predictions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' take a major vote) for each missing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To support the hypothesis that neighboring features have a higher chance to explain each other, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 4b-4f we include experiments for using all neighboring (strategy A) or only two distant (strategy B) windows on distance ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We can see that including very close neighbors significantly increases the quality of im- putation, while with including distant neighbors the improvement may present (this fact corresponds to possible long-term correlations), but is very unstable and cannot be guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Table 1: Example window sizes ∆ according to desired training samples, via Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 1 Size of training data % of missing data 1% 5% 10% 20% 30% 20% of original data 160 31 15 7 4 30% of original data 120 23 11 5 3 50% of original data 69 14 7 3 2 Our method is summarised as pseudocode in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We compare performance of the models with hyperparameters ∆ and ν in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We estimate theo- Corrupting with missing values Training PredictingChains of Autoreplicative Random Forests 9 (a) Probabilities to take already predicted windows into chain at position 50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 100 win- dows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 10 previous windows for each chain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 3 chains: forward, backward, and random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' (b) Two strategies to include distant win- dows into analysis (c) Window size ∆ = 10, strategy A (d) Window size ∆ = 5, strategy A (e) Window size ∆ = 10, strategy B (f) Window size ∆ = 5, strategy B Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 4: Including the 2ν closest neighbor windows as additional features (strategy A) significantly increases the accuracy while including only 2 windows on distance ν (strat- egy B) has occasional and unstable improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=" 6'0 ccuracy 1: 1% missing values 5% missing values 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='6 10% missing values 20% missing values 30% missing values 0 5 10 15 20 35 40 45 50 Distance from the selected window0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='6 1% missing values 5% missing values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='5 10% missing values 20% missing values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='4 30% missing values 0 10 20 30 40 50 60 70 80 90 O Distance from the selected windownormaldistribution 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='04 ChARFstrategy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content="00 0 10 20 30 40 50 60 70 80 06 100△=310 6'0 ccuracy 1% missing values 5% missing values 10% missing values 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='6 20% missing values 30% missing values 0 5 10 15 20 25 30 35 40 45 50 Distance from the selected window10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='9 Accuracy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='6 1% missing values 5% missing values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='5 10% missing values 20% missing values 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='4 30% missing values 0 10 30 50 60 70 80 90 Distance from the selected window10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' retically the maximum size ∆ of one window according to the desired size of training data τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' As τ ∼ (1 − f)∆, then ∆(τ) ∼ log1−f τ = ln τ ln(1 − f), τ ∈ (0, 1), (1) where f is a fraction of missing values and τ is a desirable threshold for a ratio of complete rows in the training subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The empirical results of the simula- tion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 2) correspond to this estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We see that with the growth of window size the size of training data decreases dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' As a consequence, the window size should be selected carefully by taking the missing value ratio into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We suggest possible window sizes according to the desired size of training data in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Algorithm 1 1: procedure ChARF(XN×p, window size ∆, # of previous steps ν, # of chains K) 2: Split features into ∆-wide windows ▷ Last window has size p (mod ∆) 3: Generate K permutations of (1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=', n = ⌈ p ∆⌉)} 4: for each permutation {σ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=', σ(n)} do 5: for each window Xσ(i) do 6: Xext ← Xσ(i) � Xσ(i−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' � Xσ(i−ν) ▷ Stack last ν processed windows as additional features 7: Xtrain ← Xcomplete ext ▷ Select complete cases for training 8: ˜ Xtrain ← Xtrain corrupted with missing values ▷ Uniformly dis- tributed, % of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' calculated from Xσ(i) 9: Xtest ← Xmissing ext 10: Fit model on ( ˜ Xtrain, Xtrain) 11: Xpred ← predictions of fitted model on Xtest 12: replace missing values in Xtest with corresponding values from Xpred 13: Take major vote for all K predictions per missing value 4 Results and discussion We evaluate the performance of our method by imputation accuracy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' per- centage of correctly imputed values out of missing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We test Chains of au- toreplicative Random Forests (of 10 trees each) on several high-dimensional SNP datasets (p ≫ N), briefly summarized in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For the SNP data, we test only the MCAR scenario, as missing values are likely to happen because of external reasons rather than depending on missing or observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We simulate the missing values by masking true values in the data under a uniform distribution, Chains of Autoreplicative Random Forests 11 with the proportion of missing values 1%, 5%, 10%, 20%, and 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The proce- dure is repeated 5 times to produce independent incomplete datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Average accuracy is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The empirical study has shown a significant improvement, when the features are one-hot encoded (paired t-test statistics 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='442, df=29, p-value=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='0018, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Table 2: Datasets used in experiments, p features, N samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Name p N Maize [14] 44,729 247 Eucalyptus [10] 33,398 970 Colorado Beetle [6] 34,186 188 Arabica Coffee [8] 4,666 596 Wheat (Zuchtwert study) [16] 9,763 388 Coffea Canephora [9] 45,748 119 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 5: One-Hot encoding may significantly improve the imputation power of ChARF method in SNP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For ChARF, we first evaluate hyperparameters: window size ∆ ∈ [3, 5, 8, 10, 15], and number of previous windows in the chain to take as new features ν ∈ [0, 1, 3, 5, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' To reduce the computation time, we first search for the best values of ∆ and ν on reduced datasets (first 1000 features) and then use these values for computation on the entire datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The grid-search results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' As expected (from estimation in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='2), from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 6 we see that the most effective window size decreases with the growth of a number of missing values (since a bigger part of instances gets corrupted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For the MICE method, with the default settings, each estimator considers all other variables, which makes the total complexity at least quadratic and thus requires huge computational and time resources in the case of high-dimensional Imputation accuracy LD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='9 80 without One-Hot 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='B 8 00 Q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='6 o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='B 60 with One-Hot encoding12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 1% 5% 10% 20% 30% Maize Eucalyptus C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Beetle A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Coffee Wheat C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Canephora Fig.' metadata={'source': 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the predictive features based on pairwise correlation, but in this case, quadratic complexity is required in this step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' With the intuition that the neighboring features in SNP data have the highest chance to explain each other, for the experiments we select windows of 10 neighbors for each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Such an approach is computationally feasible, but the imputation still leaves some missing values in the data (around 10-20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The possible explanation is the collinearity between features [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This approach worked for smaller SNP datasets (Arabica Coffee and Wheat), but for the other ones, the computations still failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Table 3: Accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' An asterisk (∗) indicates optimal hyperparameters estimated via internal validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For knnimpute we selected the best of k ∈ {3, 5, 10, 20, 50} (shown in brackets) in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Likewise for rank ∈ {10, 20, 50, 100, 200, 300, 500} for SVD method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The missForest method was run for first 100 features only as it is not possible to run it for a whole dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Best accuracy per column in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' All results rounded to 3 dp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='01 0.' metadata={'source': 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+page_content=' benchmark methods (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' At the same time, we see that with the rise of the missing value ratio the accuracy of imputation diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This is explained by the very small size of training data even on small windows for a big number of missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' However, for a moderate missing value ratio, our method consistently outperforms its alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='1 Time complexity analysis The complexity of ChARF is O(pN log N) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' the number of features for a single tree and for an ensemble is O( p ∆ ·∆N log N) ∼ O(pN log N) (fixed number of chains and ensemble members).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We expect that the kNN and SVD methods’ time complexity is linear w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' a number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For the MICE method time, the complexity depends on the base per feature estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In the simulation, we use a decision tree and random forest as base estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For a single decision tree, the complexity is O(pN log N), and thus total complexity is quadratic w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' the number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Random forests consist of individual decision trees, but it is possible to select a number of features per tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The standard choice is √p features per tree, which makes the total complexity O(p√p) and is already substantially slower than linear complexity for a big number of features p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The same estimation holds for the MissForest method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' However, we can reach linear complexity if we put a number of features per tree equal to come constant, which we try in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' These theoretic estimations are well supported in the simulation study, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We empirically compare the time complexity of the imputation methods on subsets of the Eucalyptus dataset under the MCAR scenario with 10% missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' The subsets are selected as first ps features of the original dataset, 10 < ps < 500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 7: Empirical results on time complexity for imputation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 5 Conclusions and future work We propose a new approach: tackling missing value imputation as a multi-label predictive problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' First, we suggest Autoreplicative Random Forests as a sim- pler alternative for Denoising Autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' With this simple but efficient idea, we can obtain competitive results, which may be further improved by more so- phisticated multi-label models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This method does require complete cases for the training phase, though we suppose that in real-world low-dimensional data this RF 80 MICE(dt) MICE(rf) kNN 60 SVD 40 20 0 0 100 200 300 400 500Chains of Autoreplicative Random Forests 15 scenario is still quite realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Besides, it is possible to tune this method by splitting the data into blocks of complete and missing features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' For high-dimensional datasets, when having complete data in all features is very unlikely, we propose Chains of Autoreplicative Random Forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This method splits data into windows of consequent features and imputes missing values window by window while incorporating information from already pro- cessed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We test this approach on SNP datasets and demonstrate a very competitive predictive power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Our method requires neither reference panels nor complete data for training and thus can be used in a variety of real-world sce- narios when imputation of missing data is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Our approach consists of two main novelties: it is model agnostic (we used using Random Forests in ex- periments) in regards to the Autoreplicator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' essentially an Autoencoder with no explicit encoding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' and operates on windows of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Our approach proved competitive, and is promising for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' In future work, we are going to improve algorithm performance on datasets with a big number of missing values and make it more stable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' high missing value ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' As preliminary experiments show that this approach works for the MAR scenario as well, we will further analyze the performance of the ChARF method for other patterns of missingness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' We will look at allowing multiple hypotheses and their distribution per a single missing value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' This would allow a greater chance of capturing the true mode and incorporating this uncertainty to further data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Acknowledgments We would like to thank Ander Carreño, University of the Basque Country, for fruitful discussions and three anonymous reviewers for their editorial comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Antonenko, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=', Read, J.' 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on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' Proceedings of Machine Learning Research, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 45, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' 378–390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} +page_content=' PMLR, Hong Kong (20–22 Nov 2016)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9AyT4oBgHgl3EQftflG/content/2301.00595v1.pdf'} diff --git a/PdFRT4oBgHgl3EQf5zjP/content/tmp_files/2301.13674v1.pdf.txt b/PdFRT4oBgHgl3EQf5zjP/content/tmp_files/2301.13674v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..993dc6692eefe6b12010012b7cd3d7040b67acec --- /dev/null +++ b/PdFRT4oBgHgl3EQf5zjP/content/tmp_files/2301.13674v1.pdf.txt @@ -0,0 +1,886 @@ +Springer Nature 2021 LATEX template +Improved distinct bone segmentation in +upper-body CT through multi-resolution +networks. +Eva Schnider1*, Julia Wolleb1, Antal Huck1, Mireille +Toranelli2, Georg Rauter1, Magdalena M¨uller-Gerbl2 +and Philippe C. Cattin1 +1*Department of Biomedical Engineering, University of Basel, +Hegenheimermattweg 167B, Allschwil, 4123, Switzerland. +2Department of Biomedicine, Musculoskeletal Research, +University of Basel, Basel, Switzerland. +*Corresponding author(s). E-mail(s): eva.schnider@unibas.ch; +Abstract +Purpose: Automated distinct bone segmentation from CT scans is +widely used in planning and navigation workflows. U-Net variants are +known to provide excellent results in supervised semantic segmenta- +tion. However, in distinct bone segmentation from upper body CTs a +large field of view and a computationally taxing 3D architecture are +required. This leads to low-resolution results lacking detail or localisation +errors due to missing spatial context when using high-resolution inputs. +Methods: We propose to solve this problem by using end-to-end train- +able segmentation networks that combine several 3D U-Nets working +at different resolutions. Our approach, which extends and generalizes +HookNet and MRN, captures spatial information at a lower resolution +and skips the encoded information to the target network, which operates +on smaller high-resolution inputs. We evaluated our proposed archi- +tecture against single resolution networks and performed an ablation +study on information concatenation and the number of context networks. +Results: Our proposed best network achieves a median DSC of 0.86 +taken over all 125 segmented bone classes and reduces the confusion +among similar-looking bones in different locations. These results out- +perform our previously published 3D U-Net baseline results on the +task and distinct-bone segmentation results reported by other groups. +1 +arXiv:2301.13674v1 [eess.IV] 31 Jan 2023 + +Springer Nature 2021 LATEX template +2 +Improved distinct bone segmentation using multi-resolution networks +Conclusion: The presented multi-resolution 3D U-Nets address current +shortcomings in bone segmentation from upper-body CT scans by allow- +ing for capturing a larger field of view while avoiding the cubic growth of +the input pixels and intermediate computations that quickly outgrow the +computational capacities in 3D. The approach thus improves the accu- +racy and efficiency of distinct bone segmentation from upper-body CT. +Keywords: Multi-resolution, Distinct Bone Segmentation, Deep Learning +1 Introduction +Segmentation of bones is used in bone disease diagnosis, in image-based assess- +ment of fracture risks [1], bone-density [2], for planning and navigation of +interventions [3], and for post-treatment assessment. +Bone tissue segmentation from CT has been shown to work well using slice- +wise 2D CNN-based segmentation algorithms [4–6]. The tasks and solutions +become more varied when moving from bone-tissue segmentation to distinct +bone segmentation (our task) where we distinguish individual bones. Vertebrae +segmentation has gained much attention, with many of the algorithms using +multi-stage approaches and leveraging the sequential structure of the spine [7]. +Rib segmentation has been tackled by [8], who use a point cloud approach tar- +geted at leveraging their dataset’s spatial sparsity. Carpal bone segmentation +is performed from X-rays of hands that were placed on a flat surface [9]. +Simultaneous segmentation of distinct bones of multiple groups is still rela- +tively little studied. [10] segment 62 different bones from upper-body CT using +an atlas-based approach and kinematic joint models. [11] use a multi-stage +approach with a localisation network, shape models, and a segmentation net- +work to segment 49 distinct bones of the upper body. Segmentation of bones +of different groups in one shot can be used as a starting point for more fine- +grained atlas segmentations [10], or as a guide for a follow-up inner organ +segmentation [12]. Segmenting multiple structures at once can also be benefi- +cial for the segmentation accuracy, [13] found their network trained on multiple +bone classes to outperform the one-class networks. +The region of interest in upper-body or full-body CT scans is typically +larger than the possible input sizes of 3D convolutional neural networks +(CNNs). As a result, the input needs to be sampled as patches, restricting the +input field of view to the patch size. This problem exacerbates with the devel- +opment of CT scanners that produce ever more highly resolved images. While +a higher resolution allows for capturing more fine-grained details, it covers +smaller body areas within a fixed-size input patch. +In order to extend the field of view, larger input patches can be sampled. +Using bigger patches, i.e. more input pixels does not increase the number of +trainable parameters in a fully connected network, but it does increase the +number of necessary intermediate computations. Doubling the patch size in + +Springer Nature 2021 LATEX template +Improved distinct bone segmentation using multi-resolution networks +3 + + +CT input +context 3D U-Net +target 3D U-Net +down-sampled CT, +extended field of few +ℒcontext +ℒtarget +Fig. +1 Task overview: We segment 125 distinct bones from upper-body CT scans +using SneakyNet, a multi-encoder-decoder network which incorporates inputs at various +resolutions. The example here features one context network, but multiple are possible. +all three dimensions leads to at least eight times more forward- and back- +ward computations, which are taxing for the generally scarce GPU memory. +Countermeasures fall into two categories. A) keeping the resolution and input +pixel size high, but reducing the computational load elsewhere. Those mea- +sures include reducing the batch size (not to be confused with the patch size), +using a simpler model, or reducing the output size. All of those means poten- +tially hamper training and inference. B) Keeping a large field of view by using +a small patch size of down-sampled inputs. This approach allows for a wider +field of view for a constant input size while losing detail information. +To decide upon the better of the two approaches presented above, the +requirements for the task at hand need to be considered. A suitable network +for our task of complete distinct bone segmentation from upper-body CT scans +(see 1) should provide the following: Its field of view should be sufficiently big +to distinguish similar bones at different body locations, e.g. left from right +humerus or the fourth from the eighth rib while keeping the computational +burden in a feasible area. +The merits of high-resolution inputs – accurate details – and low-resolution +inputs – a larger field of view – can be combined in many ways. Cascaded U- +Nets consist of two or more individual U-Nets that are trained consecutively. A +first model is trained on downsampled input. Its one-hot encoded segmentation +results are then upsampled, potentially cropped and used as additional input +channels for the following model at higher resolution [14]. These approaches all +have the downside of requiring the training and sequential inference of multiple +models. Instead of this, we focus on end-to-end trainable models here. +End-to-end trained multi-resolution architectures have been proposed in +histopathology whole-slide segmentation. For example, MRN [15] combines a + +Springer Nature 2021 LATEX template +4 +Improved distinct bone segmentation using multi-resolution networks +Table 1 Comparison of architectures with different field of view (FOV) of their target +and context network(s). +Config +Target +Context +trainable +input +training time +network +network(s) +param. +pixels +per iteration +FOV +FOV +·107 +·104 +(s) +A 3D U-Net +323 +— +5.8 +3.3 +0.44 +643 +— +26.2 +0.57 +3D U-Net slim∗ +1283 +— +1.5 +209.7 +4.24 +B HookNet +323 +643 +3.7 +6.6 +0.41 +643 +1283 +52.4 +0.72 +C MRN +323 +643 +4.7 +6.6 +0.43 +643 +1283 +52.4 +1.27 +D Sneakynet (ours) +323 +643 +4.9 +6.6 +0.45 +643 − 1283 +5.8 +9.9 +0.70 +643 − 1283 − 2563 +6.2 +13.1 +3.16 +643 +1283 +4.9 +52.4 +1.28 +1283 − 2563 +5.8 +78.6 +3.11 +∗ Operating the full 3D U-Net on patches of size 1283 exceeds the available GPU memory. +2D target U-Net and one context encoder with drop-skip-connections crossing +over at every level. MRN does not contain a context decoder or context loss +and is studied on a binary segmentation problem. Another such architecture +is HookNet [16], which contains both a target and a context 2D U-Net and +two individual losses, but only uses skip connections in the bottleneck layer. +The purpose of our work is to address common segmentation errors that +originate from a lack of global context while using 3D U-Nets for distinct +bone segmentation. We propose to use a multi-resolution approach and present +SneakyNet, an expansion and generalization of the MRN and HookNet archi- +tectures. We compare the segmentation accuracy, complexity, and run-time of +baseline 3D U-Nets with the SneakyNet. We ablate the model components and +find that the use of our generalized architecture improves the results over the +HookNet and MRN variants. We will use our bone segmentation in conjunc- +tion with 3D rendering of anatomical images in augmented- and virtual reality +applications, where segmentations can be used on top or in conjunction with +existing transfer functions [17, 18]. +2 Materials and Methods +To assess the performance of SneakyNet on upper-body distinct bone segmen- +tation, we train it on our in-house upper-body CT dataset, which has been +described in [19]. We make ablation studies on the combination of context and +target information and on the optimal number of context networks. +2.1 SneakyNet Architecture +In general, SneakyNet consists of one target network and one or more context +networks. The target network operates on high-resolution data and eventu- +ally produces the desired segmentation maps. The context networks operate + +Springer Nature 2021 LATEX template +Improved distinct bone segmentation using multi-resolution networks +5 + + +✂ +Central crop, skip connection +3x3x3 Conv, leReLu +2x2x2 maxpool +Skip connection +2x2x2 upsample +context U-Net +target U-Net +level m +level m+1 +Fig. 2 Detailed view of the architecture. Displayed are only two out of five levels of the +U-Nets. Left: the context U-Net working on low-resolution data with a larger field of view. +Right: The U-Net working with the central cropped high-resolution data. After all encoder +convolutions of level m, a cropped copy of the output is skipped to the target decoder at level +m+1. The decoder receives skip connections from its own encoder and the context network. +The intermediate results of the decoder and both skip connections are concatenated along +the channel axis before undergoing further convolutions. +on lower resolution inputs spanning a larger field of view. Information is +propagated from the context networks to the target network using crop-skip +connections presented in Section 2.1.1. We present a detailed visual overview +of the architecture with one context network in Figure 1. +In our previous work [20], we have explored the suitability of different 2D +and 3D network architectures and parameter configurations for upper-body +distinct bone segmentation. We found that there is little leeway in architectural +choices due to the tasks large required field of view and the many classes +that are segmented in parallel. A lean 3D U-Net variant was found to work +best [20]. We use this variant’s architecture for our target and context U-Nets +here. In our baseline computations, where we have only a target network and +omit the context networks, we select the number of channels in order for our +variants and the baselines to have approximately the same number of trainable +parameters, to facilitate comparison. Inputs to the network are required to be +multiples of 2M−1, where M denotes the number of levels of the U-Net. We +use the basic architecture of M = 5 and therefore need multiples of 16 pixels +in every dimension as input. +For the target network we use inputs of size (Sx, Sy, Sz) at full resolution. +For each of the context networks we use that input plus its surrounding area, +which together span a field of view of 2κ · (Sx, Sy, Sz). We display the case +of κ = 1 in Figure 1, but also use context networks with κ = 2 and κ = 3 in +our ablation studies. The context network inputs are down-sampled to reduce +their size to (Sx, Sy, Sz). We perform the down-sampling using (2κ × 2κ × 2κ) +average-pooling with a stride of 2κ. Both target and context network inputs +eventually have a size of (Sx, Sy, Sz), but at different resolutions and fields of +view. +2.1.1 Crop-skip connections +We use crop-skip connections to transfer information from the context to the +target branch. We crop the encoder output at the desired level m such that + +Springer Nature 2021 LATEX template +6 +Improved distinct bone segmentation using multi-resolution networks + + +A +B +Context network +Target network +Loss +Crop-skip connection +C +D +Fig. 3 Schematic of the four network configurations used in our ablation study. A shows a +base U-Net, while B, C, D show different possibilities of how to insert information into the +target network, see also Section 2.1.1 for a written description. +only the centre cube of half the size per dimension remains. This centre cube +is now spatially aligned to the input of the target branch. We concatenate the +centre cube to the next lower level m + 1 of the target decoder to match the +spatial size. We refer to the central cropping and subsequent concatenation +into a lower level of the target branch as crop-skip-connection. A detailed +schematic of the crop-skip connection is depicted in Figure 2. +We explore three network configurations which differ in their number of +crop-skip connections and their use of a context loss, and compare it to a +baseline U-Net. A visual comparison of the architectures is given in Figure 3 +and the parameters are provided in Table 1. +• A – Baseline: 3D U-Net with optimal configuration found for the task [20]. +• B – HookNet: One context network with a single crop-skip connection +is added to the target network. The crop-skip connection enters the target +network at its bottleneck layer. This configuration is used in [16]. +• C – MRN: Crop-skip connections connect the context encoder and the +target decoder at every level. There is neither a context decoder nor a context +loss function. This configuration was used in [15]. +• D – proposed SneakyNet: Crop-skip connections connect all levels of the +context and target networks. The context network has a decoder with its +own loss function. +2.2 Training +Our dataset is split into 11 scans for training, 2 for validation and 3 for testing. +We use 5-fold cross-validation, ensuring that every scan appears in precisely +one of the cross-validation folds in the test set. +The loss is composed of an unweighted combination of the target network’s +loss and the losses of the K context networks. For both networks, we use the +sum of the cross-entropy loss LX-Ent and Dice-Loss LDSC [21]. As in [20], we +sum the Dice-Loss for every class separately and normalize by the number of +classes. We optimized the network weights using the Adam optimizer with an +initial learning rate of 0.001. We trained our networks for 100000 iterations +until convergence was observed. + +Springer Nature 2021 LATEX template +Improved distinct bone segmentation using multi-resolution networks +7 +humerus l +humerus r +radius l +radius r +ulna l +ulna r +femur l +femur r +Predicted label +humerus l +humerus r +radius l +radius r +ulna l +ulna r +femur l +femur r +True label +0.98 +0 +0 +0 +0 +0 +0.021 +0 +0 +0.99 +0 +0 +0 +0 +0 +0.005 +0.016 +0 +0.98 +0 +0 +0 +0 +0 +0 +0 +0 +0.99 +0 +0.0052 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0.0074 0.99 +ours: D 64-128 +humerus l +humerus r +radius l +radius r +ulna l +ulna r +femur l +femur r +Predicted label +humerus l +humerus r +radius l +radius r +ulna l +ulna r +femur l +femur r +True label +0.75 +0 +0 +0 +0 +0 +0.21 +0.027 +0.014 +0.87 +0 +0.0066 +0 +0 +0 +0.11 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0.0096 +0 +0.99 +0 +0 +0 +0 +0 +0 +0 +0 +0.99 +0 +0 +0 +0 +0 +0 +0 +0 +0.98 +0 +0.01 +0 +0 +0 +0 +0 +0 +0.99 +0.013 +0 +0 +0 +0 +0 +0 +0.039 +0.96 +baseline 3D U-Net +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +0.4 +0.6 +0.8 +Fig. 4 Confusion matrix among the long bones of the arms and legs. With our method, +there is considerably less confusion between the left and right sides of the body and between +arm and leg bones. +Our input images are padded by (S −Starget)/2 all-around using edge value +padding. The padding step ensures that we can sample high-resolution patch +centres right to the image’s border. +We implemented and trained our networks using Tensorflow Keras 2.5. All +training and inference were conducted on NVidia Quadro RTX 6000 GPUs of +24 GB RAM size. +2.3 Evaluation +We evaluate the performance of our models using a class-wise Dice Score Coef- +ficient (DSC). To indicate the performance over all classes, we give the median +and the 16 and 84 quantiles (1σ) over all classes c. To not give a distorted +impression of the distribution, we exclude classes where no true positives of +c have been detected and therefore DSCc = 0. We present the percentage of +classes included as ’non-zero DSC’ in Table 2 and Table 3 to make up for the +omission. +3 Results and Discussion +Our experiments show how automated distinct bone segmentation can be +improved using a multi-resolution approach. We evaluate our results on mul- +tiple target resolutions with different numbers of context networks and field +of view sizes and perform an ablation study to determine the most beneficial +way to combine context and target network information. +We evaluated some of the most common errors when using a baseline seg- +mentation method. We found that the missing context information leads to +similar-looking bones in different body regions being mistaken for one another. +In the confusion matrix presented in Figure 4, we observe that when using + +Springer Nature 2021 LATEX template +8 +Improved distinct bone segmentation using multi-resolution networks +Table 2 Ablation results in DSC for different model configurations. +Target patch size +32 +64 +non-zero +non-zero +DSC +Median +σ +−σ +DSC +Median +σ +−σ +DSC +A 3D U-Net +0.64 ++0.19 +-0.34 +94.5% +0.83 ++0.09 +-0.27 +94.5% +B HookNet +0.66 ++0.17 +-0.34 +94.1% +0.85 ++0.09 +-0.32 +95.3% +C MRN +0.69 ++0.16 +-0.37 +95.1% +0.84 ++0.09 +-0.31 +96.0% +D SneakyNet (ours) +0.75 ++0.14 +-0.33 +95.3% +0.86 ++0.08 +-0.28 +96.7% +a baseline 3D U-Net, humerus pixels were predicted as femur, and the left +and right humerus were confused for one another (right confusion matrix). +When using context information, these errors are reduced almost entirely (left +confusion matrix). +We performed an ablation study to see how different strategies of combining +the context and target information within the network perform. In Table 2 +we present the quantitative results. For both target patch sizes, 32 and 64, all +strategies (B-D) improve upon the baseline 3D U-Net (A). The observed effect +is substantially bigger when using the smaller target patch size of 323, where +the median DCS rises from 0.64 to 0.75. The DSC still increases from 0.83 to +0.86 median DSC on the bigger target patches. +The combination of skip connections at every level and a context loss func- +tion in our proposed architecture increases the accuracy further, as compared +to the HookNet [16] and the MRN [15]. +In Table 3 we ablate the influence of different numbers of context networks +and input patch sizes. Qualitative results are depicted in Figure 5. Compar- +ing the baseline 3D U-Nets with the SneakyNet results, we see that adding +context networks to very small target patches of 323 pixels almost reaches +the performance of our baseline networks operating on 643 patches. Going +up, the SneakyNet operating on patch size 643 even outperforms the baseline +3D U-Net of patchsize 1283. We recall, that we had to reduce the number of +channels in the baseline 1283 network, due to memory restraints. Our ablation +results suggests, that the addition of context networks are more valuable in +adding performance when reaching memory limits. When considering the dif- +ferent FOV of the context networks, we observe the best results when including +context FOVs of 1283. This covers roughly half of the L-R and A-P dimen- +sions of the scans and seems to contain the necessary information to correctly +locate bones, see e.g. the purple lumbar vertebra in Figure 5, which is correctly +located in cases where the context FOV reaches 1283. +We provide a comparison to other results published on distinct bone seg- +mentation in Table 4. While a direct comparison is difficult due to different +datasets, our results compare favourably to both the convolutional neural +networks and shape model approach by [11], and to the hierarchical atlas +segmentation by [10]. + +Springer Nature 2021 LATEX template +Improved distinct bone segmentation using multi-resolution networks +9 + + +64 +64-128 +32-64-128 +32-64 +64-128-256 +32 +32-64-128-256 +256 +128 +64 +32 +128 +64 +32 +64 +32 +32 +256 +128 +64 +128 +64 +64 +Fig. 5 Qualitative prediction results from our ablation study comparing different numbers +of context networks at various resolutions. The first four results from the left were obtained +using a target patch size of 32px per dimension (turquoise), and the remaining three scans +with target patch sizes of 64px per dimension (light blue). The grey areas indicate the field +of view of the context networks. The sizes of the squares are proportional to the prediction +sizes. +Table 3 Ablation results for the number of context networks in the SneakyNet +architecture (D). Zero context networks corresponds to the baseline 3D U-Nets (A) with +different input patch sizes. +Config +target FOV +context FOV(s) +DSC +per dim. +per dim. +Median +σ +−σ +non-zero DSC +A +32 +— +0.64 ++0.19 +-0.34 +94.5% +D +32 +64 +0.75 ++0.14 +-0.33 +95.3% +D +32 +64-128 +0.79 ++0.11 +-0.33 +94.4% +D +32 +64-128-256 +0.79 ++0.11 +-0.33 +95.9% +A +64 +— +0.83 ++0.09 +-0.27 +95.6% +D +64 +128 +0.86 ++0.08 +-0.28 +96.7% +D +64 +128-256 +0.85 ++0.09 +-0.28 +96.1% +A +128 +— +0.82 ++0.11 +-0.30 +94.3% +4 Conclusion +This works presents improvements in distinct bone segmentation from upper- +body CT. The proposed multi-resolution networks use additional inputs at a +lower resolution but with a larger field of view to provide the necessary context +information to assign the proper bone classes. We compared three different +ways of combining the context and target information and evaluated the results +using zero to three context networks. Using context networks improves the +segmentation results on all target patch sizes. + +Springer Nature 2021 LATEX template +10 +Improved distinct bone segmentation using multi-resolution networks +Table 4 Comparison of our best-performing SneakyNet (D, target patch size of 643 and +one context network with a FOV of 1283 pixels) to other work on distinct bone +segmentation from upper-body CT. Results are in DSC. +ours (median) +[11] (median) +[10] (mean) +L3 +0.85 +0.85 +0.91 +Sacrum +0.92 +0.88 +Right 7th rib +0.78 +0.84 +Clavicula +0.94 +0.87 +Right femur +0.96 +0.92 +Pelvic bones +0.94 +0.86 +Hamate +0.81 +Inference time per scan (min) +∼ 3 +∼ 20 +Scans in training dataset (#) +11 +100 +19 +Classes (#) +125 +49 +62 +Acknowledgments. +This work was financially supported by the Werner +Siemens Foundation through the MIRACLE project. We thank Azhar Zam for +valuable discussions that helped shape this work. +Declarations +• Funding: This work was financially supported by the Werner Siemens +Foundation through the MIRACLE project. +• Competing interests: None of the authors have competing interests to declare +that are relevant to the content of this article. +• Ethics approval: This research study was conducted retrospectively from CT +data routinely obtained from body donours. No ethical approval is required. +• Consent to participate: Informed consent was obtained from all individual +body donours included in the study. +• Consent for publication: Body donours signed informed consent regarding +publications using their data. +• Availability of data and materials: The upper-body CT dataset is not +publicly available. An anonymized version can be shared on request. +• Code availability: Our code is shared at: https://gitlab.com/cian.unibas.ch/ +sneakynet +References +[1] Deng, Y., Wang, L., Zhao, C., Tang, S., Cheng, X., Deng, H.-W., Zhou, +W.: A deep learning-based approach to automatic proximal femur seg- +mentation in quantitative ct images. Medical & Biological Engineering & +Computing, 1–13 (2022) +[2] Uemura, K., Otake, Y., Takao, M., Makino, H., Soufi, M., Iwasa, M., +Sugano, N., Sato, Y.: Development of an open-source measurement system + +Springer Nature 2021 LATEX template +Improved distinct bone segmentation using multi-resolution networks +11 +to assess the areal bone mineral density of the proximal femur from clinical +ct images. Archives of Osteoporosis 17(1), 1–11 (2022) +[3] Su, Z., Liu, Z., Wang, M., Li, S., Lin, L., Yuan, Z., Pang, S., Feng, Q., +Chen, T., Lu, H.: Three-dimensional reconstruction of kambin’s triangle +based on automated magnetic resonance image segmentation. 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In: International Workshop on +Machine Learning in Medical Imaging, pp. 40–49 (2020). Springer +[21] Milletari, F., Navab, N., Ahmadi, S.-A.: V-net: Fully convolutional neu- +ral networks for volumetric medical image segmentation. In: 2016 Fourth +International Conference on 3D Vision (3DV), pp. 565–571 (2016). IEEE + diff --git a/PdFRT4oBgHgl3EQf5zjP/content/tmp_files/load_file.txt b/PdFRT4oBgHgl3EQf5zjP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eacb982fe17b4aacb644d441c0552952df9fe517 --- /dev/null +++ b/PdFRT4oBgHgl3EQf5zjP/content/tmp_files/load_file.txt @@ -0,0 +1,526 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf,len=525 +page_content='Springer Nature 2021 LATEX template Improved distinct bone segmentation in upper-body CT through multi-resolution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Eva Schnider1*, Julia Wolleb1, Antal Huck1, Mireille Toranelli2, Georg Rauter1, Magdalena M¨uller-Gerbl2 and Philippe C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Cattin1 1*Department of Biomedical Engineering, University of Basel, Hegenheimermattweg 167B, Allschwil, 4123, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2Department of Biomedicine, Musculoskeletal Research, University of Basel, Basel, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' E-mail(s): eva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='schnider@unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='ch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Abstract Purpose: Automated distinct bone segmentation from CT scans is widely used in planning and navigation workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' U-Net variants are known to provide excellent results in supervised semantic segmenta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' However, in distinct bone segmentation from upper body CTs a large field of view and a computationally taxing 3D architecture are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This leads to low-resolution results lacking detail or localisation errors due to missing spatial context when using high-resolution inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Methods: We propose to solve this problem by using end-to-end train- able segmentation networks that combine several 3D U-Nets working at different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Our approach, which extends and generalizes HookNet and MRN, captures spatial information at a lower resolution and skips the encoded information to the target network, which operates on smaller high-resolution inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We evaluated our proposed archi- tecture against single resolution networks and performed an ablation study on information concatenation and the number of context networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Results: Our proposed best network achieves a median DSC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='86 taken over all 125 segmented bone classes and reduces the confusion among similar-looking bones in different locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' These results out- perform our previously published 3D U-Net baseline results on the task and distinct-bone segmentation results reported by other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='13674v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='IV] 31 Jan 2023 Springer Nature 2021 LATEX template 2 Improved distinct bone segmentation using multi-resolution networks Conclusion: The presented multi-resolution 3D U-Nets address current shortcomings in bone segmentation from upper-body CT scans by allow- ing for capturing a larger field of view while avoiding the cubic growth of the input pixels and intermediate computations that quickly outgrow the computational capacities in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The approach thus improves the accu- racy and efficiency of distinct bone segmentation from upper-body CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Keywords: Multi-resolution, Distinct Bone Segmentation, Deep Learning 1 Introduction Segmentation of bones is used in bone disease diagnosis, in image-based assess- ment of fracture risks [1], bone-density [2], for planning and navigation of interventions [3], and for post-treatment assessment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Bone tissue segmentation from CT has been shown to work well using slice- wise 2D CNN-based segmentation algorithms [4–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The tasks and solutions become more varied when moving from bone-tissue segmentation to distinct bone segmentation (our task) where we distinguish individual bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Vertebrae segmentation has gained much attention, with many of the algorithms using multi-stage approaches and leveraging the sequential structure of the spine [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Rib segmentation has been tackled by [8], who use a point cloud approach tar- geted at leveraging their dataset’s spatial sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Carpal bone segmentation is performed from X-rays of hands that were placed on a flat surface [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Simultaneous segmentation of distinct bones of multiple groups is still rela- tively little studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' [10] segment 62 different bones from upper-body CT using an atlas-based approach and kinematic joint models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' [11] use a multi-stage approach with a localisation network, shape models, and a segmentation net- work to segment 49 distinct bones of the upper body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Segmentation of bones of different groups in one shot can be used as a starting point for more fine- grained atlas segmentations [10], or as a guide for a follow-up inner organ segmentation [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Segmenting multiple structures at once can also be benefi- cial for the segmentation accuracy, [13] found their network trained on multiple bone classes to outperform the one-class networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The region of interest in upper-body or full-body CT scans is typically larger than the possible input sizes of 3D convolutional neural networks (CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' As a result, the input needs to be sampled as patches, restricting the input field of view to the patch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This problem exacerbates with the devel- opment of CT scanners that produce ever more highly resolved images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' While a higher resolution allows for capturing more fine-grained details, it covers smaller body areas within a fixed-size input patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' In order to extend the field of view, larger input patches can be sampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Using bigger patches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' more input pixels does not increase the number of trainable parameters in a fully connected network, but it does increase the number of necessary intermediate computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Doubling the patch size in Springer Nature 2021 LATEX template Improved distinct bone segmentation using multi-resolution networks 3 CT input context 3D U-Net target 3D U-Net down-sampled CT, extended field of few ℒcontext ℒtarget Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 1 Task overview: We segment 125 distinct bones from upper-body CT scans using SneakyNet, a multi-encoder-decoder network which incorporates inputs at various resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The example here features one context network, but multiple are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' all three dimensions leads to at least eight times more forward- and back- ward computations, which are taxing for the generally scarce GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Countermeasures fall into two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A) keeping the resolution and input pixel size high, but reducing the computational load elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Those mea- sures include reducing the batch size (not to be confused with the patch size), using a simpler model, or reducing the output size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' All of those means poten- tially hamper training and inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' B) Keeping a large field of view by using a small patch size of down-sampled inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This approach allows for a wider field of view for a constant input size while losing detail information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' To decide upon the better of the two approaches presented above, the requirements for the task at hand need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A suitable network for our task of complete distinct bone segmentation from upper-body CT scans (see 1) should provide the following: Its field of view should be sufficiently big to distinguish similar bones at different body locations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' left from right humerus or the fourth from the eighth rib while keeping the computational burden in a feasible area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The merits of high-resolution inputs – accurate details – and low-resolution inputs – a larger field of view – can be combined in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Cascaded U- Nets consist of two or more individual U-Nets that are trained consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A first model is trained on downsampled input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Its one-hot encoded segmentation results are then upsampled, potentially cropped and used as additional input channels for the following model at higher resolution [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' These approaches all have the downside of requiring the training and sequential inference of multiple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Instead of this, we focus on end-to-end trainable models here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' End-to-end trained multi-resolution architectures have been proposed in histopathology whole-slide segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' For example, MRN [15] combines a Springer Nature 2021 LATEX template 4 Improved distinct bone segmentation using multi-resolution networks Table 1 Comparison of architectures with different field of view (FOV) of their target and context network(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Config Target Context trainable input training time network network(s) param.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' pixels per iteration FOV FOV 107 104 (s) A 3D U-Net 323 — 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='44 643 — 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='57 3D U-Net slim∗ 1283 — 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='5 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='24 B HookNet 323 643 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='41 643 1283 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='72 C MRN 323 643 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='43 643 1283 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='27 D Sneakynet (ours) 323 643 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='45 643 − 1283 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='70 643 − 1283 − 2563 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='16 643 1283 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='28 1283 − 2563 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='11 ∗ Operating the full 3D U-Net on patches of size 1283 exceeds the available GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2D target U-Net and one context encoder with drop-skip-connections crossing over at every level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' MRN does not contain a context decoder or context loss and is studied on a binary segmentation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Another such architecture is HookNet [16], which contains both a target and a context 2D U-Net and two individual losses, but only uses skip connections in the bottleneck layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The purpose of our work is to address common segmentation errors that originate from a lack of global context while using 3D U-Nets for distinct bone segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We propose to use a multi-resolution approach and present SneakyNet, an expansion and generalization of the MRN and HookNet archi- tectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We compare the segmentation accuracy, complexity, and run-time of baseline 3D U-Nets with the SneakyNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We ablate the model components and find that the use of our generalized architecture improves the results over the HookNet and MRN variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We will use our bone segmentation in conjunc- tion with 3D rendering of anatomical images in augmented- and virtual reality applications, where segmentations can be used on top or in conjunction with existing transfer functions [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2 Materials and Methods To assess the performance of SneakyNet on upper-body distinct bone segmen- tation, we train it on our in-house upper-body CT dataset, which has been described in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We make ablation studies on the combination of context and target information and on the optimal number of context networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1 SneakyNet Architecture In general, SneakyNet consists of one target network and one or more context networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The target network operates on high-resolution data and eventu- ally produces the desired segmentation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The context networks operate Springer Nature 2021 LATEX template Improved distinct bone segmentation using multi-resolution networks 5 ✂ Central crop, skip connection 3x3x3 Conv, leReLu 2x2x2 maxpool Skip connection 2x2x2 upsample context U-Net target U-Net level m level m+1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2 Detailed view of the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Displayed are only two out of five levels of the U-Nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Left: the context U-Net working on low-resolution data with a larger field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Right: The U-Net working with the central cropped high-resolution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' After all encoder convolutions of level m, a cropped copy of the output is skipped to the target decoder at level m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The decoder receives skip connections from its own encoder and the context network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The intermediate results of the decoder and both skip connections are concatenated along the channel axis before undergoing further convolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' on lower resolution inputs spanning a larger field of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Information is propagated from the context networks to the target network using crop-skip connections presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We present a detailed visual overview of the architecture with one context network in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' In our previous work [20], we have explored the suitability of different 2D and 3D network architectures and parameter configurations for upper-body distinct bone segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We found that there is little leeway in architectural choices due to the tasks large required field of view and the many classes that are segmented in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A lean 3D U-Net variant was found to work best [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We use this variant’s architecture for our target and context U-Nets here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' In our baseline computations, where we have only a target network and omit the context networks, we select the number of channels in order for our variants and the baselines to have approximately the same number of trainable parameters, to facilitate comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Inputs to the network are required to be multiples of 2M−1, where M denotes the number of levels of the U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We use the basic architecture of M = 5 and therefore need multiples of 16 pixels in every dimension as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' For the target network we use inputs of size (Sx, Sy, Sz) at full resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' For each of the context networks we use that input plus its surrounding area, which together span a field of view of 2κ · (Sx, Sy, Sz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We display the case of κ = 1 in Figure 1, but also use context networks with κ = 2 and κ = 3 in our ablation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The context network inputs are down-sampled to reduce their size to (Sx, Sy, Sz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We perform the down-sampling using (2κ × 2κ × 2κ) average-pooling with a stride of 2κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Both target and context network inputs eventually have a size of (Sx, Sy, Sz), but at different resolutions and fields of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1 Crop-skip connections We use crop-skip connections to transfer information from the context to the target branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We crop the encoder output at the desired level m such that Springer Nature 2021 LATEX template 6 Improved distinct bone segmentation using multi-resolution networks A B Context network Target network Loss Crop-skip connection C D Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 3 Schematic of the four network configurations used in our ablation study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A shows a base U-Net, while B, C, D show different possibilities of how to insert information into the target network, see also Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1 for a written description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' only the centre cube of half the size per dimension remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This centre cube is now spatially aligned to the input of the target branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We concatenate the centre cube to the next lower level m + 1 of the target decoder to match the spatial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We refer to the central cropping and subsequent concatenation into a lower level of the target branch as crop-skip-connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A detailed schematic of the crop-skip connection is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We explore three network configurations which differ in their number of crop-skip connections and their use of a context loss, and compare it to a baseline U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A visual comparison of the architectures is given in Figure 3 and the parameters are provided in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' A – Baseline: 3D U-Net with optimal configuration found for the task [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' B – HookNet: One context network with a single crop-skip connection is added to the target network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The crop-skip connection enters the target network at its bottleneck layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This configuration is used in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' C – MRN: Crop-skip connections connect the context encoder and the target decoder at every level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' There is neither a context decoder nor a context loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This configuration was used in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' D – proposed SneakyNet: Crop-skip connections connect all levels of the context and target networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The context network has a decoder with its own loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='2 Training Our dataset is split into 11 scans for training, 2 for validation and 3 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We use 5-fold cross-validation, ensuring that every scan appears in precisely one of the cross-validation folds in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The loss is composed of an unweighted combination of the target network’s loss and the losses of the K context networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' For both networks, we use the sum of the cross-entropy loss LX-Ent and Dice-Loss LDSC [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' As in [20], we sum the Dice-Loss for every class separately and normalize by the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We optimized the network weights using the Adam optimizer with an initial learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We trained our networks for 100000 iterations until convergence was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Improved distinct bone segmentation using multi-resolution networks 7 humerus l humerus r radius l radius r ulna l ulna r femur l femur r Predicted label humerus l humerus r radius l radius r ulna l ulna r femur l femur r True label 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='98 0 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='021 0 0 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 4 Confusion matrix among the long bones of the arms and legs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' With our method, there is considerably less confusion between the left and right sides of the body and between arm and leg bones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Our input images are padded by (S −Starget)/2 all-around using edge value padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The padding step ensures that we can sample high-resolution patch centres right to the image’s border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We implemented and trained our networks using Tensorflow Keras 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' All training and inference were conducted on NVidia Quadro RTX 6000 GPUs of 24 GB RAM size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='3 Evaluation We evaluate the performance of our models using a class-wise Dice Score Coef- ficient (DSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' To indicate the performance over all classes, we give the median and the 16 and 84 quantiles (1σ) over all classes c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' To not give a distorted impression of the distribution, we exclude classes where no true positives of c have been detected and therefore DSCc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We present the percentage of classes included as ’non-zero DSC’ in Table 2 and Table 3 to make up for the omission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 3 Results and Discussion Our experiments show how automated distinct bone segmentation can be improved using a multi-resolution approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We evaluate our results on mul- tiple target resolutions with different numbers of context networks and field of view sizes and perform an ablation study to determine the most beneficial way to combine context and target network information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We evaluated some of the most common errors when using a baseline seg- mentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We found that the missing context information leads to similar-looking bones in different body regions being mistaken for one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' In the confusion matrix presented in Figure 4, we observe that when using Springer Nature 2021 LATEX template 8 Improved distinct bone segmentation using multi-resolution networks Table 2 Ablation results in DSC for different model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Target patch size 32 64 non-zero non-zero DSC Median σ −σ DSC Median σ −σ DSC A 3D U-Net 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='64 +0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='28 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='7% a baseline 3D U-Net, humerus pixels were predicted as femur, and the left and right humerus were confused for one another (right confusion matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' When using context information, these errors are reduced almost entirely (left confusion matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We performed an ablation study to see how different strategies of combining the context and target information within the network perform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' In Table 2 we present the quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' For both target patch sizes, 32 and 64, all strategies (B-D) improve upon the baseline 3D U-Net (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The observed effect is substantially bigger when using the smaller target patch size of 323, where the median DCS rises from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='64 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The DSC still increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='83 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='86 median DSC on the bigger target patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The combination of skip connections at every level and a context loss func- tion in our proposed architecture increases the accuracy further, as compared to the HookNet [16] and the MRN [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' In Table 3 we ablate the influence of different numbers of context networks and input patch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Qualitative results are depicted in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Compar- ing the baseline 3D U-Nets with the SneakyNet results, we see that adding context networks to very small target patches of 323 pixels almost reaches the performance of our baseline networks operating on 643 patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Going up, the SneakyNet operating on patch size 643 even outperforms the baseline 3D U-Net of patchsize 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We recall, that we had to reduce the number of channels in the baseline 1283 network, due to memory restraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Our ablation results suggests, that the addition of context networks are more valuable in adding performance when reaching memory limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' When considering the dif- ferent FOV of the context networks, we observe the best results when including context FOVs of 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This covers roughly half of the L-R and A-P dimen- sions of the scans and seems to contain the necessary information to correctly locate bones, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' the purple lumbar vertebra in Figure 5, which is correctly located in cases where the context FOV reaches 1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We provide a comparison to other results published on distinct bone seg- mentation in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' While a direct comparison is difficult due to different datasets, our results compare favourably to both the convolutional neural networks and shape model approach by [11], and to the hierarchical atlas segmentation by [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Springer Nature 2021 LATEX template Improved distinct bone segmentation using multi-resolution networks 9 64 64-128 32-64-128 32-64 64-128-256 32 32-64-128-256 256 128 64 32 128 64 32 64 32 32 256 128 64 128 64 64 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' 5 Qualitative prediction results from our ablation study comparing different numbers of context networks at various resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The first four results from the left were obtained using a target patch size of 32px per dimension (turquoise), and the remaining three scans with target patch sizes of 64px per dimension (light blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The grey areas indicate the field of view of the context networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The sizes of the squares are proportional to the prediction sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Table 3 Ablation results for the number of context networks in the SneakyNet architecture (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Zero context networks corresponds to the baseline 3D U-Nets (A) with different input patch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Config target FOV context FOV(s) DSC per dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' per dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Median σ −σ non-zero DSC A 32 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='64 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='34 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='5% D 32 64 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='4% D 32 64-128-256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='79 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='33 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='9% A 64 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='83 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='27 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='6% D 64 128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='86 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='28 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='7% D 64 128-256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='85 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='28 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='1% A 128 — 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='82 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='30 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='3% 4 Conclusion This works presents improvements in distinct bone segmentation from upper- body CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' The proposed multi-resolution networks use additional inputs at a lower resolution but with a larger field of view to provide the necessary context information to assign the proper bone classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We compared three different ways of combining the context and target information and evaluated the results using zero to three context networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Using context networks improves the segmentation results on all target patch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Springer Nature 2021 LATEX template 10 Improved distinct bone segmentation using multi-resolution networks Table 4 Comparison of our best-performing SneakyNet (D, target patch size of 643 and one context network with a FOV of 1283 pixels) to other work on distinct bone segmentation from upper-body CT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Results are in DSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' ours (median) [11] (median) [10] (mean) L3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='91 Sacrum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='88 Right 7th rib 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='84 Clavicula 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='87 Right femur 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='92 Pelvic bones 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='86 Hamate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='81 Inference time per scan (min) ∼ 3 ∼ 20 Scans in training dataset (#) 11 100 19 Classes (#) 125 49 62 Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' This work was financially supported by the Werner Siemens Foundation through the MIRACLE project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' We thank Azhar Zam for valuable discussions that helped shape this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Declarations Funding: This work was financially supported by the Werner Siemens Foundation through the MIRACLE project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Competing interests: None of the authors have competing interests to declare that are relevant to the content of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Ethics approval: This research study was conducted retrospectively from CT data routinely obtained from body donours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' No ethical approval is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Consent to participate: Informed consent was obtained from all individual body donours included in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Consent for publication: Body donours signed informed consent regarding publications using their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Availability of data and materials: The upper-body CT dataset is not publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' An anonymized version can be shared on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content=' Code availability: Our code is shared at: https://gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='com/cian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='unibas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFRT4oBgHgl3EQf5zjP/content/2301.13674v1.pdf'} +page_content='ch/ sneakynet References [1] Deng, Y.' metadata={'source': 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sha256:3310356a8dc2f9ff122058c397d64206f6c98ef6d2b7e0405d5f93697c29af80 +size 50315 diff --git a/R9E4T4oBgHgl3EQfLAxE/content/tmp_files/2301.04934v1.pdf.txt b/R9E4T4oBgHgl3EQfLAxE/content/tmp_files/2301.04934v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ec9eb9a0bf36f889fd002f28e4f0f5512e3b9f2e --- /dev/null +++ b/R9E4T4oBgHgl3EQfLAxE/content/tmp_files/2301.04934v1.pdf.txt @@ -0,0 +1,3227 @@ +arXiv:2301.04934v1 [math.DG] 12 Jan 2023 +Curvature effect in the spinorial Yamabe problem on +product manifolds +Thomas Bartsch, Tian Xu* +Abstract +Let (M1, g(1)), (M2, g(2)) be closed Riemannian spin manifolds. We study the existence +of solutions of the Spinorial Yamabe problem on the product M1 × M2 equipped with a +family of metrics ε−2g(1) ⊕ g(2), ε > 0. Via variational methods and blow-up techniques, +we prove the existence of solutions which depend only on the factor M1, and which exhibit +a spike layer as ε → 0. Moreover, we locate the asymptotic position of the peak points of +the solutions in terms of the curvature tensor on (M1, g(1)). +MSC 2010: Primary: 53C27; Secondary: 35B40, 35Q40, 35R01, 58E30, 58J60 +Keywords. Spinorial Yamabe equation; strongly indefinite functional; blow-up solutions; +spike layer solutions +1 +Introduction +Let N be an n-dimensional closed spin manifold, n ≥ 2, with Riemannian metric g and a +fixed spin structure σ : PSpin(N) → PSO(N). Denoted by ρ : Spin(n) → End(Sn) the spin +representation, we write S(N) = PSpin(N) ×ρ Sn for the spinor bundle over N and DN +g +: +C∞(N, S(N)) → C∞(N, S(N)) for the (Atiyah-Singer) Dirac operator. +A spinorial analogue of the Yamabe equation can be written as +DN +g ψ = |ψ|n∗−2 +g +ψ, +on (N, g, σ) +(1.1) +where n∗ := +2n +n−1 — in fact, Eq. (1.1) is conformally invariant and is the Euler-Lagrange equa- +tion of a variational problem similar to Yamabe’s problem (see [3]). In case ψ ∈ C1(N, S(N)) is +a non-trivial solution to (1.1), the (generalized) conformal metric ˜g = |ψ| +4 +n−1 +g +g induces a spinor +field ϕ on (N, ˜g, σ) such that +DN +˜g ϕ = ϕ, +|ϕ|˜g ≡ 1 +(1.2) +on N \ ψ−1({0}). As an important geometric application, a solution to Eq. (1.2) on a two- +dimensional manifold N corresponds to the existence of an isometric immersion ( �N, g) → R3 +*Supported by the National Science Foundation of China (NSFC 11601370) and the Alexander von Humboldt +Foundation of Germany +1 + +2 +of the universal covering � +N into the Euclidean 3-space with constant mean curvature (see [3,21] +and references therein for more geometric backgrounds). +A series of works of B. Ammann and his group [3–9] have provided a brief picture of how +variational method is employed to the study of (1.1). From the view point of analysis, as it +was pointed out in [3], standard variational methods do not directly imply the existence of a +solution. This is due to the criticality of the nonlinearity in (1.1). Indeed, the exponent n∗ = +2n +n−1 +is critical in the sense that the Sobolev embedding involved is precisely the one for which the +compactness is lost. Similar to the idea in solving the Yamabe problem, it is possible to find a +criterion which recovers compactness. Here the crucial observation is that a spinorial analogue +of Aubin’s inequality holds (see [7]): +λ+ +min(N, [g], σ) := inf +˜g∈[g] λ+ +1 (˜g)Vol(N, ˜g) +1 +n ≤ λ+ +min(Sn, [gSn], σSn) = n +2ω +1 +nn +(1.3) +where [g] = {f 2g : f ∈ C1(N), f ≥ 0, supp f = N} is the (generalized) conformal class of g +and, for each ˜g ∈ [g], λ+ +1 (˜g) > 0 denotes the smallest positive eigenvalue of the associated Dirac +operator DN +˜g , (Sn, gSn, σSn) is the n-dimensional sphere equipped with its canonical metric gSn +and spin structure σSn, and ωn is the volume of (Sn, gSn). The quantity λ+ +min(N, [g], σ) is known +as the B¨ar-Hijazi-Lott invariant. One of the main results obtained in [3] shows that if inequality +(1.3) is strict then the spinorial Yamabe problem (1.1) has a nontrivial solution. However, the +strict inequality in (1.3) is only verified for some special cases and a general result is still lacking +(cf. [6,9,22]). +The purpose of this paper is to establish existence results for (1.1) on products of compact +spin manifolds without knowing whether the strict inequality in (1.3) holds. Moreover, we are +also interested in the effect of the curvature tensors in our existence results. In particular, given +closed spin manifolds (M1, g(1), σ1) and (M2, g(2), σ2), with fixed spin structures, let us consider +a family of metrics gε on the product N = M1 × M2 defined by gε = ε−2g(1) ⊕ g(2), ε > 0. Let +m1 and m2 be the dimensions of M1 and M2 respectively, we will be interested in solutions of +the spinorial Yamabe equation on the product manifold (N, gε): +DN +gεφ = |φ|n∗−2 +gε +φ +(1.4) +where n∗ = +2n +n−1 and n := dim N = m1+m2. In the sense of separation of variables, we restrict +our study to spinor fields of the form φ = ψ ⊗ϕ ∈ C1(N, S(N)) such that ψ ∈ C1(M1, S(M1)) +and ϕ ∈ C1(M2, S(M2)) are spinor fields on M1 and M2 respectively. +In order to describe our main results, it is useful to recall some notation and definitions in +differential geometry, see for instance [17, 28]. For a closed Riemannian m-manifold (M, g), +let exp : TM → M be the exponential map defined on the tangent bundle TM of M. Since +M is closed, there exists r > 0 such that expξ : Br(0) ⊃ Rm ∼= TξM → Br(ξ) ⊂ M is +a diffeomorphism for any ξ ∈ M. Throughout the paper, Br(0) will denote the ball in Rm +centered at 0 with radius r and, for ξ ∈ M, Br(ξ) will denote the ball in M centered at ξ with +respect to the metric g. For vector fields X, Y, W, Z on M, the Riemannian curvature tensor R +is given by +R(X, Y, W, Z) = g(∇X∇Y W, Z) − g(∇Y ∇XW, Z) − g(∇[X,Y ]W, Z) + +3 +where ∇ is the Riemannian connection. For an orthonormal basis {e1, . . . , em} of TξM, ξ ∈ M, +the Ricci tensor Ric : TξM × TξM → R is given by the trace of R, that is +Ric(X, Y ) = +m +� +j=1 +R(ej, X, Y, ej) +and the scalar curvature and Gaussian curvature will be denoted by Scalg(ξ) and Kg(ξ) re- +spectively. On spin manifolds, since the tangent bundle is embedded in the bundle of Clifford +algebra, vector fields have two different actions on spinors, i.e. the Clifford multiplications and +the covariant derivatives. Here, to distinguish these two actions on a spinor ψ, we denote respec- +tively ∂j ·g ψ the Clifford multiplication of ∂j and ∇∂jψ the covariant derivative, j = 1, . . . , m, +with respect to the background metric g. We also adopt the notation fε ≲ gε for two ε-dependent +functions fε and gε, when there exists a constant C > 0 independent of ε such that fε ≤ Cgε. +1.1 +An illustrative Example +We begin by describing an example when n = 3, where N = Σ × S1 for a closed Riemann +surface Σ and S1 the standard circle. Let g denote the background metric on Σ and dτ denote +the standard metric on S1 with total length 2π, then we are concerned with the product metrics +ε−2g ⊕ dτ, ε > 0. We also equip Σ and S1 with spin structures σΣ and σS1, respectively. In this +setting, we have n∗ = 3 and Eq. (1.4) reads as +DN +gεφ = |φ|gεφ, +φ : N → S(N) +(1.5) +where the spinor bundle S(N) is identified as the tensor product S(N) = S(Σ) ⊗ S(S1). Since +the associated Dirac operator on S1 is simply i d +dτ , and since we are looking for a solution of the +form φ = ψ ⊗ ϕ, we can take ϕ = e−iλτ to be an eigen-spinor on S1 (i.e. i d +dτ ϕ = λϕ) for some +eigenvalue λ ̸= 0. Then φ = ψ ⊗ e−iλτ is a solution to Eq. (1.5) if and only if ψ : Σ → S(Σ) is +a solution to the following reduced equation (see Section 3 for a detailed explanation) +εDΣ +g ψ + λωΣ +C ·g ψ = |ψ|gψ, +on Σ +(1.6) +where ωΣ +C is the chirality operator in the Clifford bundle Cl(TΣ) and “ ·g ” denotes the Clifford +multiplication. +We also introduce the following equation which corresponds to a limiting equation to prob- +lem (1.6) as ε goes to zero: +DgR2ψ + λωC ·gR2 ψ = |ψ|ψ, +ψ : R2 → S(R2) ∼= C2 +(1.7) +where gR2 denotes the standard Euclidean metric and ωC = i∂1 ·gR2 ∂2 is the corresponding +chirality operator with “ ·gR2 ” being the Clifford multiplication and ∂1 = +∂ +∂x1, ∂2 = +∂ +∂x2 are the +canonical base in the tangent bundle TR2. +The blow-up profiles (the so-called concentration phenomenon) appearing in solution se- +quences of Eq. (1.6) (as ε → 0) are described by rescaled solutions of the above limit equation +(1.7). As we will see in Section 4.1, Eq. (1.7) has a variational structure, of strongly indefinite +type. Ground state solutions (i.e. solutions with minimal energy) to Eq. (1.7) can be obtained +via a standard linking arguments, moreover, these solutions decay exponentially at infinity. + +4 +Note that Eq. (1.7) is invariant by translation, we denote B the set of ground state solutions +of (1.7) having maximum modulus at the origin, i.e. |ψ(0)| = maxx∈R2 |ψ(x)| for ψ ∈ B. As +explained in Lemma 4.4, B is compact in W 1,q(R2, S(R2)), q ≥ 2. Now we are ready to state +the results for N = Σ × S1: +Theorem 1.1. There exists ε0 > 0 such that, for any ε ∈ (0, ε0), Eq. (1.6) has a solution +ψε ∈ C1(Σ, S(Σ)). Furthermore, there is a maximum point yε ∈ Σ such that +(1) for a constant c > 0, +|ψε(ξ)|g ≲ exp +� +− c +ε dist(ξ, yε) +� +, +for all ξ ∈ Σ +where dist is the distance induced by the metric g; +(2) as ε → 0, up to a subsequence, the transformed spinor zε(x) ≡ ψε ◦ expyε(εx) converges +uniformly to a ground state solution z0 ∈ B of (1.7) and yε → y0 in (Σ, g) such that +Θ(y0, z0) = +max +(y,ψ)∈Σ×B Θ(y, ψ), +where Θ : Σ × B → R is a functional defined by +Θ(y, ψ) = Kg(y) +36 +� +R2 |ψ|3|x|2dx ++ Kg(y) +12 +Re +� +R2 +� +(x2∇∂1 − x1∇∂2)ψ, (x2∂1 − x1∂2) ·gR2 ψ +� +dx. +(1.8) +The result presented above implies that the concentrating behavior of a solution to (1.5) is +affected by the Gaussian curvature Kg on Σ. Since Σ × B is compact, a maximizer of Θ does +exist. +Remark 1.2. The spin structure of Euclidean spaces is quite explicit and equation (1.7) can be +rewritten in matrix notation as +� +γ1 +∂ +∂x1 ++ γ2 +∂ +∂x2 +� +ψ + λγ3ψ = |ψ|ψ +(1.9) +where γk, k = 1, 2, 3, are the 2 × 2 Pauli matrices +γ1 = +� +0 +i +i +0 +� +, +γ2 = +� +0 +1 +−1 +0 +� +, +γ3 = +� +1 +0 +0 +−1 +� +. +Passing to polar coordinates in R2, i.e. (x1, x2) �→ (r, ϑ), Eq. (1.9) reads as + + + + + + + +e−iϑ� +i ∂ +∂r + 1 +r +∂ +∂ϑ +� +ψ2 = +� +|ψ1|2 + |ψ2|2 ψ1 − λψ1 +eiϑ� +i ∂ +∂r − 1 +r +∂ +∂ϑ +� +ψ1 = +� +|ψ1|2 + |ψ2|2 ψ2 + λψ2 + +5 +where ψ = +�ψ1 +ψ2 +� +∈ C2, and this suggests the following special ansatz (see [18]) +ψ(r, ϑ) = +� +v(r)eiSϑ +iu(r)ei(S+1)ϑ +� +, +r > 0, ϑ ∈ [0, 2π) +(1.10) +with u, v real-valued and S ∈ Z. Plugging such ansatz into the functional Θ, we find that the +second term in (1.8) vanishes, i.e. we get a simpler expression of Θ as +Θ(y, ψ) = πKg(y) +18 +� ∞ +0 +� +u2 + v2� 3 +2r3dr. +This would give a simplified view of the concentration phenomenon in Theorem 1.1, i.e. y0 must +be a global maximum point of the Gaussian curvature Kg on (Σ, g). Unfortunately, we find no +evidence that ground state solutions to the limit equation (1.7) should be in the form (1.10). This +may lead to conjecture that, up to translations and certain group actions (for instance, the multi- +plication by eiω for ω ∈ [0, 2π]) , the ground state solution ψ to Eq. (1.7) is uniquely determined +and takes the form of (1.10) (or may be other symmetric ansatz). We add that solutions of the +form (1.10) to the 2D nonlinear Dirac equation with Kerr-type critical nonlinearity |ψ|2ψ have +been studied in [12, 13]. Furthermore, ground state solutions to the spinorial Yamabe equation +(1.1) on Rn has been recently classified in [14]. +1.2 +Full statement of the results +In order to develop an existence and concentration theory for Eq. (1.4) on general product +spaces N = M1 × M2, we first introduce explicitly the spinor bundle S(N) over N in terms of +the factors M1 and M2, that is +S(N) = +� +(S(M1) ⊕ S(M1)) ⊗ S(M2) +both m1 and m2 are odd, +S(M1) ⊗ S(M2) +else. +In this setting, there is no difficulty to understand that a spinor φ ∈ S(N) has the form φ = ψ⊗ϕ +where ϕ ∈ S(M2) and ψ = ψ1 ⊕ ψ2 ∈ S(M1) ⊕ S(M1) if both m1 and m2 are odd, and +ψ ∈ S(M1) if m1 or m2 is even. +Motivated by the example mentioned previously, we impose the following hypothesis on the +second factor (M2, g(2), σ2): +(H) there is a solution ϕλ with constant length |ϕλ|g(2) = 1 of the Dirac equation DM2 +g(2)ϕ = λϕ, +for some λ ̸= 0. +Under this hypothesis, φ = ψ ⊗ ϕλ is a solution of Eq. (1.4) if and only if ψ is a solution of the +following reduced equation +ε ˜DM1 +g(1)ψ + λωM1 +C +·g(1) ψ = |ψ|n∗−2 +g(1) ψ, +on M1 +(1.11) +where +˜DM1 +g(1) = +�DM1 +g(1) ⊕ −DM1 +g(1) +both m1 and m2 are odd, +DM1 +g(1) +if m1 is even, + +6 +and ωM1 +C ·g(1) denotes the action of the chirality operator with respect to the metric g(1). In case +m1 is odd and m2 is even, one may interchange M1 and M2 to get the above equation. +The corresponding limit equation associated to (1.11) is the following one: +˜DgRm1 ψ + λωC ·gRm1 ψ = |ψ|n∗−2ψ +on Rm1 +(1.12) +where gRm1 is the standard Euclidean metric, +˜DgRm1 = +� +DgRm1 ⊕ −DgRm1 +if both m1 and m2 are odd, +DgRm1 +if m1 is even, +and ωC = i[ m1+1 +2 +]∂1 ·gRm1 · · · ·gRm1 ∂m1 is the corresponding chirality operator in the Clifford +algebra Cl(TRm1) with “ ·gRm1 ” being the Clifford multiplication; ∂1 = +∂ +∂x1, . . . , ∂m1 = +∂ +∂xm1 +are the canonical base in the tangent bundle TRm1. +Eq. (1.12) can be regarded as the Euler-Lagrange equation for the functional +L(ψ) = 1 +2 +� +Rm1 +� ˜DgRm1 ψ + λωC ·gRm1 ψ, ψ +� +dx − 1 +n∗ +� +Rm1 +|ψ|n∗dx +defined for ψ ∈ H1/2(Rm1, S(Rm1)). Since the spectrum of the linear differential operator +˜Dλ = ˜DgRm1 + λωC is given by Spec( ˜Dλ) = +� +−∞, −|λ| +� +∪ +� +|λ|, +∞ +� +, the above functional is +strongly indefinite. Several techniques have been introduced to handle such situations (see for +instance [10, 11, 15, 27, 35] and references therein). Notice that we have set n = m1 + m2 and +n∗ = +2n +n−1, we see the Sobolev embedding +H1/2(Rm1, S(Rm1)) ֒→ Ln∗(Rm1, S(Rm1)) +is locally compact (due to n∗ < m∗ +1 = +2m1 +m1−1). This means that Palais-Smale sequences for the +functional L possess local strong convergence in Ln∗ and thus in H1/2. In the framework of +Concentration-Compactness theory, one obtains the existence of ground state solutions to Eq. +(1.12) via standard variational arguments. +For ease of notation, we still denote B the set of all ground state solutions of (1.12) satis- +fying |ψ(0)| = maxx∈Rm1 |ψ(x)| for ψ ∈ B. We know from Lemma 4.4 that B is compact in +W 1,q(Rm1, S(Rm1)), q ≥ 2. Then our main result reads as +Theorem 1.3. Assume dim M1 = m1 ≥ 2 and (M2, g(2), σ2) satisfies hypothesis (H). Let m1 +be even when n = m1 + m2 is odd. Then there exists ε0 > 0 such that, for any ε ∈ (0, ε0), Eq. +(1.11) has a solution of the form ψε ∈ C1(M1, S(M1)). Furthermore, there is a maximum point +yε ∈ M1 such that +(1) for a constant c > 0, +|ψε(ξ)|g(1) ≲ exp +� +− c +ε dist(1)(ξ, yε) +� +, +for all ξ ∈ M1 +where dist(1) is the distance induced by the metric g(1); + +7 +(2) as ε → 0, up to a subsequence, the transformed spinor zε(x) ≡ ψℓ ◦ expyε(εx) converges +uniformly to a ground state solution z0 ∈ B of (1.12) and yε → y0 in (M1, g(1)) such that +Θ(y0, z0) = +max +(y,ψ)∈M1×B Θ(y, ψ), +where Θ : M1 × B → R is a functional defined by +Θ(y, ψ) = +1 +12n +� +Rm1 +Ricy(x, x)|ψ| +2n +n−1dx ++ 1 +12 +� +j,k +Re +� +Rm1 +Ry(ej, x, x, ek)(∇∂kψ, ∂j ·gRm1 ψ)dx. +The case m1 = m2 = 1, which corresponds to N = S1 × S1, is rather simple and does not +reflect the effect of curvature since the problem is reduced to an ordinary differential equation +on the first circle (see [34]). +Remark 1.4. +a) The hypothesis (H) is rather harmless. This is satisfied by a large class of +manifolds including the circle S1, the m-spheres and many conformally flat manifolds +(see for instance [6,9]). +b) Theorem 1.3 does not treat directly the case of m1 odd and m2 even. As the statements +show, the concentration phenomenon will be obtained on the even dimensional factor. +This is mainly due to the specific spin-representation (see (2.3) below) when n = m1+m2 +is odd. +c) Analogously to Remark 1.2, if we have additionally a symmetric characterization of +ground state solutions to Eq. (1.12) then the functional Θ can be intensively simplified. In +higher dimensions, it is still not very clear how to give a general symmetric characteriza- +tion of the solutions (while the Laplacian commutes with rotations, it is not the case for +Dirac operator). However in 3D Dirac equations, it is known that there is one candidate +symmetric ansatz (see in [20,36]). +Finally we would like to compare problem (1.11) with its counterpart of elliptic type: +− ε2∆gu + u = up−1, +u > 0 +(1.13) +on a smooth closed Riemannian manifold (M, g), with dim M = m ≥ 3 and p ∈ (2, 2m +m−2). +An interesting observation is that our results about (1.11) depend on both curvature tensors +and the ground state solutions of the limit equations. In fact, the point y0 ∈ M1 in our result +locates the blow-up (or concentration) while the ground state solution z0 ∈ B gives the profile +of the blowing-up bubbles. Unlike the well-known results about (1.13) in [16,19,32] etc. where +only scalar curvature enters. The functional Θ in our theorems appear to be complicated and +mysterious, in particular, the second term in Θ is not very clear to us. This is because there is +very little information available for the ground state solutions of the strongly indefinite problems +(1.7) and (1.12). Since the limit equation of (1.13) on the Euclidean space Rm is explicitly +understood, for which there exists a unique positive solution (up to translations) and is radially +symmetric, the results for positive solutions of (1.13) only depends on geometric quantities. It +would be very interesting to characterize those ground state solutions for Dirac equations (1.7) +and (1.12), and then one may have a better understanding for the functional Θ. + +8 +1.3 +Outline of the paper +The proof of our results will be carried out in several steps. First, in Section 2, we recall some +preliminaries and also fix our notation. In particular, we will formally provide the spinor bun- +dles and Dirac operators on product spaces. In Section 3, we explicitly introduce Eq. (1.11) as +the reduced equation of Spinorial Yamabe equation (1.4), and set up the associated variational +framework. The existence result is standard since the nonlinearity in Eq. (1.11) has subcriti- +cal growth (in the sense of Sobolev embedding). In this case the concentration phenomenon +manifests itself in the difficulty of locating the behavior of the solutions when the parameter ε +is small. Here, the key point lies in Corollary 3.8 which provides a refined upper bound esti- +mate for the critical levels in our variational framework. In fact, Corollary 3.8 makes it possible +to compute a asymptotic expansion of the critical levels in terms of ε. Section 4 is devoted +to give the complete proof of our main results. In this section, we first collect basic proper- +ties of the ground state solutions of the limit equation (1.12) since they perform as bubbles in +the concentration phenomenon. The analysis of the concentration phenomenon is quite delicate +and it requires a careful asymptotic expansion of the critical levels. With the help of a well +adopted spinor bundle trivialization (the so-called Bourguignon-Gauduchon trivialization) and +our Corollary 3.8, we establish the critical level expansion in terms of ε in which the effect of +curvature tensors enters. +2 +Spinor bundles and Dirac operators on product spaces +In this section, we collect some basic notations from spin geometry. Instructional material can +be found in [29, Chapter I. 5 and II. 7]. +2.1 +Algebraic preliminaries +Let us denote by {e1, . . . , em} the canonical basis of an oriented Euclidean space V and by +Cℓ(V ) the complex Clifford algebra of V with its multiplication being denoted by “·”. In case +the dimension m of V is even, i.e. m = 2k, the Clifford algebra is isomorphic to the alge- +bra M(2k; C) of all complex 2k × 2k matrices. Hence Cℓ(V ) has precisely one irreducible +module, the spinor module S2k with dimC S2k = 2k. When restricting this representation to +the even subalgebra Cℓ0(V ), the module S2k splits into two irreducible unitary representations +S2k = S+ +2k ⊕ S− +2k, given by the eigensubspaces of the endomorphism ωV +C := ike1 · · · em to the +eigenvalues ±1. In the context, we will call ωV +C the “chirality operator” or the “complex volume +element”. +In case m is odd, that is m = 2k+1, the Clifford algebra Cℓ(V ) is isomorphic to M(2k; C)⊕ +M(2k; C). And thus, we obtain two 2k-dimensional irreducible spinor modules S0 +2k+1 and S1 +2k+1 +if we project the Clifford multiplication onto the first and second component respectively. Simi- +lar to the splitting in even dimensions, the two modules S0 +2k+1 and S1 +2k+1 can be distinguished by +the chirality operator ωV +C := ik+1e1 · · ·em in the sense that on Sj +2k+1 it acts as (−1)j, j = 0, 1. +It will cause no confusion if we simply identify S0 +2k+1 and S1 +2k+1 as the same vector space, that +is S2k+1 = S0 +2k+1 = S1 +2k+1, and equip them with Clifford multiplications of opposite signs. +Let V and W be two oriented Euclidean spaces with dim V = m1 and dim W = m2. We +denote Cℓ(V ) and Cℓ(W) the associated Clifford algebras of V and W respectively. By an + +9 +abuse of notation, we use the same symbol “·” for the Clifford multiplication in Cℓ(V ), Cℓ(W) +and in their representations. As it is well known, the Clifford algebra of the sum of two vector +spaces is the Z2-graded tensor product of the Clifford algebras of the two summands, that is +Cℓ(V ⊕ W) = Cℓ(V )�⊗Cℓ(W) (see [29]). Therefore, we can construct the spinor module of +V ⊕ W from those of V and W as +Sm1+m2 = +� +(Sm1 ⊕ Sm1) ⊗ Sm2 +if both m1 and m2 are odd, +Sm1 ⊗ Sm2 +if m1 is even. +(2.1) +Here, as mentioned before, we simply excluded the case where m1 is odd and m2 is even +because V and W can be interchanged. As for the representation of Clifford multiplications on +Sm1+m2, let ξ ∈ V , ζ ∈ W, ϕ ∈ Sm2 and ψ = ψ1 ⊕ ψ2 ∈ Sm1 ⊕ Sm1 if both m1 and m2 are odd, +and ψ ∈ Sm1 if m1 is even. We set +(ξ ⊕ ζ) · (ψ ⊗ ϕ) = (ξ · ψ) ⊗ ϕ + (ωV +C · ψ) ⊗ (ζ · ϕ), +(2.2) +where, in case both m1 and m2 odd, ξ · ψ = (ξ · ψ1) ⊕ (−ξ · ψ2) and ωV +C · ψ = i(ψ2 ⊕ −ψ1). +With this notation, one easily checks that +(ξ ⊕ ζ) · (ξ ⊕ ζ) · (ψ ⊗ ϕ) = −|ξ ⊕ ζ|2(ψ ⊗ ϕ). +Thus Sm1+m2 is a nontrivial Cℓ(V ⊕ W)-module of (complex) dimension 2[ m1+m2 +2 +]. Moreover, +in case m1 + m2 is even, the splitting of Sm1+m2 into half-spinor modules is given by +S+ +m1+m2 = +� +(ψ ⊕ ψ) ⊗ ϕ : ψ ∈ Sm1, ϕ ∈ Sm2 +� +, +S− +m1+m2 = +� +(ψ ⊕ −ψ) ⊗ ϕ : ψ ∈ Sm1, ϕ ∈ Sm2 +� +for both m1 and m2 odd and +S+ +m1+m2 = (S+ +m1 ⊗ S+ +m2) ⊕ (S− +m1 ⊗ S− +m2), +S− +m1+m2 = (S+ +m1 ⊗ S− +m2) ⊕ (S− +m1 ⊗ S+ +m2) +for both m1 and m2 even. +Next, let us turn to the manifold setting. Let (M1, g(1)) and (M2, g(2)) be two oriented Rie- +mannian manifolds of dimensions m1 and m2, respectively. We henceforth suppose that both +manifolds are equipped with a fixed spin structure (for details about spin structures, we refer +to [21, 29] or to the well written self-contained introduction [25]). This induces a unique spin +structure on the Riemannian product (N = M1 × M2, g = g(1) ⊕ g(2)). Indeed, let πM1 and πM2 +denote the projections on M1 and M2, the tangent bundle of N can be decomposed as +TN = π∗ +M1TM1 ⊕ π∗ +M2TM2. +For simplicity, we omit the projections and write TN = TM1 ⊕ TM2. And such splitting is +orthogonal with respect to g. Hence the frame bundle of N can be reduced to a SO(m1) × +SO(m2)-principal bundle, and this is isomorphic to the product of the frame bundles over M1 +and M2. + +10 +2.2 +The Dirac operator +Fix the spin structures σM1 and σM2, let us consider the Clifford bundles (with Clifford multipli- +cations) (Cl(TM1), ·g(1)), (Cl(TM2), ·g(2)) and spinor bundles S(M1), S(M2) over M1 and M2 +respectively. From the previous considerations in the algebraic settings, we know for the spinor +bundles that +S(N) = +� +(S(M1) ⊕ S(M1)) ⊗ S(M2) +if both m1 and m2 are odd, +S(M1) ⊗ S(M2) +if m1 is even. +For X ∈ TM1, Y ∈ TM2, ϕ ∈ Γ(S(M2)) and ψ = ψ1 ⊕ ψ2 ∈ Γ(S(M1) ⊕ S(M1)) for both m1 +and m2 odd and ψ ∈ Γ(S(M1)) for m1 even, we have +(X ⊕ Y ) ·g (ψ ⊗ ϕ) = (X ·g(1) ψ) ⊗ ϕ + (ωM1 +C +·g(1) ψ) ⊗ (Y ·g(2) ϕ) +(2.3) +where in case m1 and m2 odd we set X ·g(1) ψ = (X ·g(1) ψ1) ⊕ (−X ·g(1) ψ2) and ωM1 +C +·g(1) ψ = +i(ψ2 ⊕ −ψ1). +Let ∇S(M1) and ∇S(M2) be the (lifted) Levi-Civita connections on S(M1) and S(M2). By +∇S(M1)⊗S(M2) = ∇S(M1) ⊗ IdS(M2) + IdS(M1) ⊗ ∇S(M2) +we mean the tensor product connection on S(M1)⊗S(M2). If we take {X1, . . . , Xm1} a locally +positively oriented orthonormal frame of (M1, g(1)), then the Dirac operator on M1 is (locally) +defined by DM1 +g(1) = �m1 +j=1 Xj ·g(1) ∇S(M1) +Xj +. Similarly, if we take {Y1, . . . , Ym2} a locally positively +oriented orthonormal frame of (M2, g(2)), we have DM2 +g(2) = �m2 +j=1 Yj ·g(2) ∇S(M2) +Yj +. Evidently, in +the product setting, {X1 ⊕ 0, . . . , Xm1 ⊕ 0, 0 ⊕ Y1, . . . , 0 ⊕ Ym2} is a local section of the frame +bundle of N. Hence formula (2.3) yields +DN +g := +m1 +� +j=1 +(Xj ⊕ 0) ·g(1) ∇S(M1)⊗S(M2) +Xj⊕0 ++ +m2 +� +j=1 +(0 ⊕ Yj) ·g(2) ∇S(M1)⊗S(M2) +0⊕Yj += ˜DM1 +g(1) ⊗ IdS(M2) + (ωM1 +C +·g(1) IdS(M1)) ⊗ DM2 +g(2) +which defines the Dirac operator on N = M1 × M2, where ˜DM1 +g(1) = DM1 +g(1) ⊕ −DM1 +g(1) if both m1 +and m2 are odd and ˜DM1 +g(1) = DM1 +g(1) if m1 is even. +For the case m1 + m2 even, we have the decomposition S(N) = S(N)+ ⊕ S(N)− and, +moreover, when restrict DN +g on those half-spinor spaces we get DN +g : Γ(S(N)±) → Γ(S(N)∓). +3 +Variational setting +In what follows, we always consider the case N = M1 × M2, m1 = dim M1 ≥ 2 and m2 = +dim M2 ≥ 1. In order to give unified expressions in odd and even cases, we will write simply +S(N) = ˜S(M1) ⊗ S(M2) with +˜S(M1) = +� +S(M1) ⊕ S(M1) +if m1 is odd, +S(M1) +if m1 is even. + +11 +and denote ψ ⊗ ϕ for a spinor field in S(N) when no confusion can arise. +Let us consider the warped metric gε := ε−2g(1) ⊕ g(2), where ε > 0 is a parameter. Accord- +ing to the discussions in the previous section, we know for the Dirac operators that +DN +gε = ˜DM1 +ε−2g(1) ⊗ IdS(M2) + (ωM1 +C +·ε−2g(1) Id˜S(M1)) ⊗ DM2 +g(2) +where ωM1 +C +denotes the chirality operator and ”·ε−2g(1)” denotes the Clifford multiplication on +M1 associated to the conformal metric ε−2g(1) respectively. +Turning to the nonlinear problems, let us denote | · |ε−2g(1) and | · |g(2) the natural hermitian +metrics on S(M1) and S(M2) respectively and | · |gε the induced metric on S(N). Recall the +notation n = m1 + m2 and n∗ = +2n +n−1, we can expand the spinorial Yamabe equation +DN +gεφ = |φ|n∗−2 +gε +φ, +φ = ¯ψ ⊗ ϕ ∈ S(N) +into +( ˜DM1 +ε−2g(1) ¯ψ) ⊗ ϕ + (ωM1 +C +·ε−2g(1) ¯ψ) ⊗ (DM2 +g(2)ϕ) = +� +| ¯ψ|ε−2g(1)|ϕ|g(2) +�n∗−2 ¯ψ ⊗ ϕ. +(3.1) +We will now show how the assumption (H) on (M2, g(2), σM2) enters. In fact, if M2 pos- +sesses a nontrivial eigenspinor ϕM2 of constant length for some λ ̸= 0, then by substituting +¯ψ ⊗ ϕM2 into (3.1) we get the equivalent problem +˜DM1 +ε−2g(1) ¯ψ + λωM1 +C +·ε−2g(1) ¯ψ = +� +| ¯ψ|ε−2g(1) +�n∗−2 ¯ψ +(3.2) +which is sitting on M1. Here, we adopt the convention that λ > 0 since (up to a change of +orientation on M1) the proof for λ < 0 is exactly the same. +Notice that the Dirac operator behaves very nicely under conformal changes (cf. [24, 26]): +Let g0 and g = e2ug0 be two conformal metrics on a Riemannian spin m-manifold M, then +there exists an isomorphism of vector bundles ι : S(M, g0) → S(M, g) which is a fiberwise +isometry such that +DM +g +� +ι(ψ) +� += ι +� +e− m+1 +2 +uDM +g0 +� +e +m−1 +2 +uψ +�� +. +Thus Eq. (3.2) is conformally equivalent to +ε ˜DM1 +g(1)ψ + λωM1 +C +·g(1) ψ = |ψ|n∗−2 +g(1) ψ +on M1 +(3.3) +where ωM1 +C ·g(1) denotes the action of the chirality operator with respect to the metric g(1). +3.1 +The configuration space and the Lagrangian +Plainly, our goal is reduced to find solutions of (3.3) for varying ε > 0. Notice that Eq. (3.3) +is well-defined on M1, and n∗ = +2n +n−1 < +2m1 +m1−1 = m∗ +1. It is not necessary to carry the super- +and sub-scripts in (M1, g(1)), ˜DM1 +g(1) and ωM1 +C +during the proofs, hence in order to simplify the +notation, we consider the following generalized problem +ε ˜Dgψ + aωC ·g ψ = f(|ψ|g)ψ, +ψ : M → ˜S(M) +(3.4) + +12 +on a closed spin m-manifold (M, g, σ), where a > 0 is a constant and f : [0, ∞) → [0, ∞) is +the nonlinearity. Clearly, f(s) = sn∗−2 is our primary concern. Unless otherwise stated, we will +occasionally drop the subscript of | · |g on ˜S(M) for notational convenience. +In the sequel, for q > 1, let us denote Lq := Lq(M, ˜S(M)) with the norm | · |q +q := +� +M | · +|qdvolg. In particular, for q = 2, we have L2 is a Hilbert space with inner product (·, ·)2 = +Re +� +M(·, ·)dvolg. +For each fixed ε > 0, let Aε := ε ˜Dg + aωC·g denote the self-adjoint operator on L2 with +domain D(Aε) = H1 ≡ H1(M, ˜S(M)). It is already known that, on a closed spin manifold +(M, g), the spectrum Spec(Aε) ⊂ (−∞, −a] ∪ [a, +∞) is symmetric about the origin and +consists of an unbounded discrete sequence of eigenvalues (with finite multiplicity for each +eigenvalue), see [34] for details. +Thus, from the classical spectral theory of elliptic self-adjoint operators, we may choose +a complete orthonormal basis ψε +±1, ψε +±2, . . . of L2 consisting of the eigenspinors of Aε, i.e. +Aεψε +±k = λ±k(ε)ψε +±k and the spectrum Spec(Aε) will be denoted as +· · · ≤ λ−2(ε) ≤ λ−1(ε) < 0 < λ1(ε) ≤ λ2(ε) ≤ · · · , +where each eigenvalue appears with its multiplicity. In particular, we have λk(ε) = −λ−k(ε) +and |λ±k(ε)| → +∞ as k → ∞. +Define the unbounded operator |Aε|s : L2 → L2, s ≥ 0, by +|Aε|sψ = +∞ +� +k=−∞ +|λk(ε)|sαkψε +k +where ψ = �∞ +k=−∞ αkψε +k ∈ L2. In this way, we can introduce the domain of |Aε|s in L2 as +H s := +� +ψ = +∞ +� +k=1 +αkψε +k ∈ L2 : +∞ +� +k=−∞ +|λk(ε)|2s|αk|2 < ∞ +� +. +It is worth pointing out that H +1 +2 coincides with the Sobolev space of order 1 +2, that is W +1 +2 ,2(M, ˜S(M)) +(see for instance [1,3]). Moreover, we can equip H := H +1 +2 with the inner product +⟨ψ, ϕ⟩ε := 1 +εm Re +� +M +� +|Aε|1/2ψ, |Aε|1/2ϕ +� +dvolg +(3.5) +and the induced norm ∥ · ∥ε such that (H, ⟨·, ·⟩ε) becomes a Hilbert space. Remark that, in the +above notations, we have emphasized the dependence on the parameter ε because it appears +in the differential operator and its spectrum. The dual space of H will be denoted by H∗ = +W − 1 +2 ,2(M, ˜S(M)). Identifying H with H∗, we will use the same notation ⟨·, ·⟩ε to denote the +norm on H∗. +Recall that we have an (·, ·)2-orthogonal decomposition +L2 = L+ +ε ⊕ L− +ε , +ψ = ψ+ + ψ− +with +L+ +ε := ++∞ +� +k=1 +ker(Aε − λk(ε)) +and +L− +ε := +∞ +� +k=1 +ker(Aε − λ−k(ε)) + +13 +so that Aε is positive definite on L+ +ε and negative definite on L− +ε . This leads to the orthogonal +decomposition of H with respect to the inner product ⟨·, ·⟩ε as +H = H+ +ε ⊕ H− +ε , +H± +ε = H ∩ L± +ε . +On the Banach space Lq, q > 1, we introduce the new norm +|ψ|q,ε = +� 1 +εm +� +M +|ψ|qdvolg +� 1 +q +for each ε > 0. +Then, recall m∗ = +2m +m−1, we have (cf. [34]) +Lemma 3.1. If ε > 0 is small, then for any q ∈ [2, m∗] the embedding IdH : (H, ∥ · ∥ε) ֒→ +(Lq, | · |q,ε) is bounded independent of ε, that is, there exists cq > 0 such that +|ψ|q,ε ≤ cq∥ψ∥ε +for all ψ ∈ H and all ε > 0 small. +In particular, the embedding is compact for q ∈ [2, m∗). +Remark 3.2. The proof of this lemma is a combination of the Lichnerowicz formula +( ˜Dg)2 = ∇∗∇ + 1 +4Scalg, +where Scalg is the scalar curvature of (M, g), and an application of the Calder´on-Lions inter- +polation theorem (see [33]) between H 1 and L2. In fact, for ψ ∈ C∞(M, ˜S(M)), we have +��|Aε|ψ +��2 +2 = +� +M +ε2|∇ψ|2 + +� +a2 + ε2 +4 Scalg +� +|ψ|2dvolg. +(3.6) +It follows that, for large positive ε, the influence of the scalar curvature Scalg enters. In this +situation, (3.6) may no longer be a norm when Scalg possesses certain negative parts. And this +fact will probably affect the embedding constant of H = H +1 +2 ֒→ Lm∗. +With Lemma 3.1, we introduce the following conditions for the nonlinearity f in Eq. (3.4) +which contains the power function f(s) = sp−2 as a special case: +(f1) f(0) = 0, f ∈ C1(0, ∞) and f ′(s) > 0 for s > 0; +(f2) there exists p ∈ (2, m∗), c > 0 such that f ′(s)s ≤ c(1 + sp−2) for s ≥ 0; +(f3) there exists θ > 0 such that f(s) ≤ 1 +θf ′(s)s for s > 0. +Let F(s) be the primitive function of f(s)s, i.e. F(s) = +� s +0 f(t)t dt, it is standard to see that +Eq. (3.4) is the Euler-Lagrange equation of the functional +Lε(ψ) += +1 +εm +� +M +�1 +2(Aεψ, ψ) − F(|ψ|) +� +dvolg += +1 +2 +� +∥ψ+∥2 +ε − ∥ψ−∥2 +ε +� +− 1 +εm +� +M +F(|ψ|)dvolg +(3.7) +defined on H = H+ +ε ⊕ H− +ε . And by Lemma 3.1, we have Lε ∈ C2(H, R). +We emphasize that the relation between F and f is F ′(s) = f(s)s. And by an abuse of nota- +tion, we simply write ψ = ψ+ +ψ− for the orthogonal decomposition of H without mentioning +its dependence on ε. However, one should always keep in mind that, for different values of ε, +this decomposition of a spinor ψ is different. + +14 +3.2 +The reduced action +Let us begin with the compactness of the functional Lε. +Lemma 3.3. For each ε > 0 small, Lε satisfies the (P.S.)c-condition for c ≥ 0, that is, +Lε(ψn) → c +L′ +ε(ψn) → 0 +� +⇒ {ψn} possesses a convergent subsequence in H. +Moreover, ψn → 0 if and only if c = 0. +Proof. Since L′ +ε(ψn) → 0 in H∗, we have +c + o(∥ψn∥ε) = Lε(ψn) − 1 +2L′ +ε(ψn)[ψn] = 1 +εm +� +M +1 +2f(|ψn|)|ψn|2 − F(|ψn|)dvolg. +(3.8) +We also have +o(∥ψn∥ε) = L′ +ε(ψn)[ψ+ +n − ψ− +n ] = ∥ψn∥2 +ε − 1 +εm Re +� +M +f(|ψn|)(ψn, ψ+ +n − ψ− +n )dvolg. +By (f1)-(f3), one checks easily that for arbitrarily small δ > 0 there exists cδ > 0 such that +f(s)s ≤ δs + cδ(f(s)s2) +p−1 +p +and F(s) ≤ +1 +θ+2f(s)s2 for all s ≥ 0. From this, together with +Lemma 3.1, we obtain +∥ψn∥2 +ε +≤ +1 +εm +� +M +f(|ψn|)|ψn| · |ψ+ +n − ψ− +n |dvolg + o(∥ψn∥ε) +≤ +δ|ψn|2,ε|ψ+ +n − ψ− +n |2,ε + cδ +� 1 +εm +� +M +f(|ψn|)|ψn|2dvolg +� p−1 +p |ψ+ +n − ψ− +n |p,ε + o(∥ψn∥ε) +≤ +δc2∥ψn∥2 +ε + Cp,δ +� +c + o(∥ψn∥ε) +� p−1 +p ∥ψn∥ε + o(∥ψn∥ε). +(3.9) +By suitable choosing δ > 0 small, it follows from (3.9) that {ψn} is bounded in H with respect +to the norm ∥ · ∥ε. And due to the compact embedding of H ֒→ Lp, one easily checks that {ψn} +is compact in H. +Finally, we mention that (3.8) and (3.9) together imply: ψn → 0 if and only if c = 0. This +completes the proof. +One may see from (3.7) that the quadratic part in the functional Lε is of strongly indefinite +type, i.e. positive- and negative-definite on infinite dimensional subspaces of H. Hence, in order +to obtain a critical point of Lε, it is now crucial to find a suitable min-max scheme for Lε. +Fortunately, due to our requests on the nonlinearity f (see conditions (f1)-(f3)), we have a very +good geometric behavior of Lε in the following sense: +(i) Since the function F is non-negative, for each fixed u ∈ H+ +ε , the functional +Lε(u + ·) : H− +ε → R, +w �→ Lε(u + w) +is anti-coercive (i.e. Lε(u + w) → −∞ as ∥w∥ε → ∞). + +15 +(ii) Since F ′′(s) = f ′(s)s+f(s) > 0 for s > 0, the quadratic form L′′ +ε(u+w)[·, ·] is negative +definite on H− +ε , in other words, the above functional Lε(u + ·) is strictly concave on H− +ε . +(iii) Since (f3) implies that f(s) ≥ csθ for some constant c > 0 and all s ≥ 1, the function +F has super-quadratic growth at infinity, i.e. F(s) ≥ +c +2+θs2+θ as s → ∞. And hence, for +each fixed u ∈ H+ +ε \ {0}, +Lε(tu + w) → −∞ +as |t| + ∥w∥ε → ∞. +(3.10) +Combining all the above three properties, for each u ∈ H+ +ε \ {0}, we are able to maximize the +functional Lε on R+u ⊕ H− +ε , where R+ = (0, ∞). We remark that u is varying in the space +H+ +ε \ {0}, so we can restrict ourselves to the choice u ∈ S+ +ε := {u ∈ H+ +ε : ∥u∥ε = 1} without +changing the maxima of Lε on R+u ⊕ H− +ε . By (f1)-(f2), one deduces that F(s) ≤ δ +2s2 + Cδsp +for arbitrarily small δ > 0 and hence (by Lemma 3.1) +max +R+u⊕H− +ε +Lε ≥ max +t>0 Lε(tu) ≥ max +t>0 +�1 − δ +2 +t2 − Cδtp� += τ0 > 0. +(3.11) +Therefore, we have found a candidate min-max scheme for Lε: we first maximize the functional +Lε on R+u ⊕ H− +ε and then minimize with respect to u ∈ H+ +ε \ {0}. +By summarizing the above observations, we have the following basic conclusion. +Proposition 3.4. For each ε > 0 small the following holds. +(1) There exists χε ∈ C1(H+ +ε , H− +ε ) such that for u ∈ H+ +ε +w ∈ H− +ε , w ̸= χε(u) ⇒ Lε(u + w) < Lε(u + χε(u)); +that is χε(u) is the unique maximizer of the functional w �→ Lε(u+w) on H− +ε . Moreover, +for u ∈ H+ +ε +∥χε(u)∥2 +ε ≤ 2 +εm +� +M +F(|u|)dvolg +and L′ +ε(u + χε(u))[w] ≡ 0 for all w ∈ H− +ε ; +(2) If {un} is a (P.S.)-sequence for the reduced functional +Iε : H+ +ε → R, +Iε(u) = Lε(u + χε(u)), +then {un + χε(un)} is a (P.S.)-sequence for Lε; +(3) There exists uε ∈ H+ +ε such that I′ +ε(uε) = 0 and +Iε(uε) = µε := inf +γ∈Γε max +t∈[0,1] Iε(γ(t)) > 0, +where Γε = +� +γ ∈ C([0, 1], H+ +ε ) : γ(0) = 0, Iε(γ(1)) < 0 +� +. In particular, ¯ψε = uε + +χε(uε) is a non-trivial critical point of Lε. + +16 +Proof. Similar assertions can be found in [15, Section 2] where a certain abstract theory has +been set up. We only mention here that the existence and uniqueness of the maximizer χε(u) in +assertion (1) follows directly from the anti-coerciveness and strict concavity of the functional +Lε(u+·) on H− +ε . The C1-smoothness of χε is a consequence of the Implicit Function Theorem. +We next provide an upper bound estimate of the critical level µε obtained in the above +proposition. +Lemma 3.5. For every u ∈ H+ +ε \ {0}, the map Iε,u : R → R, Iε,u(t) = Iε(tu), is of class C2 +and satisfies Iε,u(0) = I′ +ε,u(0) = 0 and I′′ +ε,u(0) > 0. Moreover, there holds +I′ +ε,u(t) = 0, t > 0 +=⇒ +I′′ +ε,u(t) < 0. +Proof. First of all, we can use the fact that I′ +ε,u(t) = L′ +ε(tu + χε(tu))[u] to see that Iε,u is of +class C2. By (f1) − (f3) there holds +Iε(tu) ≥ Lε(tu) ≥ (1 − δ)t2 +2 +∥u∥2 +ε − Cp,δ tp∥u∥p +ε +∀u ∈ H+ +ε \ {0} and t > 0, +for any fixed δ > 0 small. Hence Iε(0) = 0 is a strict local minimum in a neighborhood of +0 ∈ H+ +ε . And in particular, we can see that Iε,u(0) = I′ +ε,u(0) = 0 and I′′ +ε,u(0) > 0. Then, to +complete the proof, it is sufficient to show that +I′ +ε(u)[u] = 0, u ̸= 0 +=⇒ +I′′ +ε (u)[u, u] < 0. +(3.12) +For simplicity, let us denote Ψε : H → R by Ψε(ψ) = +1 +εm +� +M F(|ψ|)dvolg and set ψ = u+χε(u) +and w = χ′ +ε(u)[u] − χε(u). By using L′ +ε(u + χε(u))|H− +ε ≡ 0, we see that (3.12) is a direct +consequence of the following computation: +I′′ +ε (u)[u, u] += +L′′ +ε(ψ)[u + χ′ +ε(u)[u], u] = L′′ +ε(ψ)[ψ + w, ψ + w] += +L′′ +ε(ψ)[ψ, ψ] + 2L′′ +ε(ψ)[ψ, w] + L′′ +ε(ψ)[w, w] += +I′ +ε(u)[u] + +� +Ψ′ +ε(ψ)[ψ] − Ψ′′ +ε(ψ)[ψ, ψ] +� ++ 2 +� +Ψ′ +ε(ψ)[w] − Ψ′′ +ε(ψ)[ψ, w] +� +−Ψ′′ +ε(ψ)[w, w] − ∥w∥2 +ε +≤ +I′ +ε(u)[u] − 1 +εm +� +M +f(|ψ|)f ′(|ψ|)|ψ|3 +f(|ψ|) + f ′(|ψ|)|ψ|dvolg − ∥w∥2 +ε. +(3.13) +Since f(s)f ′(s)s ≤ (f(s) + f ′(s)s)2 for all s ≥ 0, and the equality holds iff s = 0, we have +0 < f(|ψ|)f ′(|ψ|)|ψ|3 +f(|ψ|) + f ′(|ψ|)|ψ| ≤ f(|ψ|)|ψ|2 + f ′(|ψ|)|ψ|3 +for ψ ̸= 0. +Therefore, the integration in (3.13) is well-defined due to (f1)-(f3). +The above lemma indicates that, for each u ∈ H+ +ε \ {0}, the function Iε,u(·) has at most one +critical point tε,u ∈ (0, +∞). Due to (3.10), one checks that such tε,u exists for every u. And +then we can define a natural constraint for the reduced functional Iε as +Nε := +� +u ∈ H+ +ε \ {0} : I′ +ε(u)[u] = 0 +� +. + +17 +Lemma 3.5 also implies Nε is a smooth submanifold of codimension 1 in H+ +ε . And conse- +quently, the critical point found in Proposition 3.4 (3) can be characterized by +µε := Lε( ¯ψε) = +inf +u∈H+ +ε \{0} +max +ψ∈Ru⊕H− +ε +Lε(ψ) = +inf +u∈H+ +ε \{0} max +t>0 Iε(tu) = inf +u∈Nε Iε(u). +(3.14) +For later purposes, it is worth to remind that (3.11) implies the existence of some τ0 > 0 +independent of ε such that µε ≥ τ0. +In what follows, we intend to pass to the limit ε → 0 and consider the asymptotic behavior +of the min-max level µε. The idea is to provide an upper bound estimate around an arbitrary +(P.S.)-sequence, so that one may substitute certain test spinors in the functional Lε. +Without loss of generality, we assume that {φε} ⊂ H is an arbitrary sequence such that +c1 ≤ Lε(φε) ≤ c2 +and +∥L′ +ε(φε)∥ε → 0 +(3.15) +as ε → 0 for some constants c1, c2 > 0. Here, we will identify the dual space H∗ with H. +Lemma 3.6. Under (3.15), we have: +(1) ∥φε∥ε is uniformly bounded in ε. +(2) ∥φ− +ε − χε(φ+ +ε )∥ε ≤ O +� +∥L′ +ε(φε)∥ε +� +as ε → 0. +(3) I′ +ε(φ+ +ε ) → 0 as ε → 0 in the dual space of H+ +ε . +Proof. For the boundedness, we recall that Lemma 3.1 implies that the embedding constant for +H ֒→ Lp∗ is independent of ε, and hence the arguments in Lemma 3.3 can be employed. +For (2), let us first set zε = φ+ +ε + χε(φ+ +ε ) and vε = φ− +ε − χε(φ+ +ε ). Then we have vε ∈ H− +ε +and, by the definition of χε, +0 = L′ +ε(zε)[vε] = − +� +χε(φ+ +ε ), vε +� +ε − 1 +εm Re +� +M +f(|zε|)(zε, vε)dvolg. +Since ∥L′ +ε(φε)∥ε → 0 as ε → 0, it follows that +o(∥vε∥ε) = L′ +ε(φε)[vε] = − +� +φ− +ε , vε +� +− 1 +εm Re +� +M +f(|φε|)(φε, vε)dvolg. +And hence, we get +o(∥vε∥ε) = ∥vε∥2 +ε + 1 +εm Re +� +M +f(|φε|)(φε, vε)dvolg − 1 +εm Re +� +M +f(|zε|)(zε, vε)dvolg. (3.16) +Since the map ψ → F(|ψ|) is convex by (f1), we have +1 +εm Re +� +M +f(|φε|)(φε, vε)dvolg − 1 +εm Re +� +M +f(|zε|)(zε, vε)dvolg ≥ 0. +Thus, from (3.16), we can infer that ∥vε∥ε ≤ O +� +∥L′ +ε(φε)∥ε +� +as ε → 0. +In order to check (3) we compute I′ +ε(φ+ +ε ) = L′ +ε +� +φ+ +ε +χε(φ+ +ε ) +� +, which implies that ∥I′ +ε(φ+ +ε )∥ε → +0 as ε → 0 is a direct consequence of (2) and the C2 smoothness of Lε. + +18 +Next, let us introduce the functional Kε : H+ +ε → R by Kε(u) = I′ +ε(u)[u]. Then, it is clear +that Kε is C1 and its derivative is given by the formula +K′ +ε(u)[w] = I′ +ε(u)[w] + I′′ +ε (u)[u, w] +for u, w ∈ H+ +ε . We also have Nε = K−1 +ε (0) \ {0}. Moreover, by (3.13), there holds +K′ +ε(u)[u] ≤ 2Kε(u) − 1 +εm +� +M +f(|ψ|)f ′(|ψ|)|ψ|3 +f(|ψ|) + f ′(|ψ|)|ψ|dvolg, +for u ∈ H+ +ε +where ψ = u + χε(u). By virtue of (f3), one checks easily that +f(s)f′(s)s +f(s)+f′(s)s ≥ +θ +θ+1f(s) for s > 0. +Thus the above estimate implies +K′ +ε(u)[u] ≤ 2Kε(u) − +θ +(θ + 1)εm +� +M +f(|u + χε(u)|)|u + χε(u)|2dvolg, +(3.17) +for u ∈ H+ +ε . +Proposition 3.7. For the sequence {φε} in (3.15), there exists {tε} ⊂ R such that tεφ+ +ε ∈ Nε +and |tε − 1| ≤ O +� +∥I′ +ε(φ+ +ε )∥ε +� +. +Proof. We begin with the observation that F(s) ≤ +1 +θ+2f(s)s2 for all s ≥ 0. Due to the condition +(3.15) and Lemma 3.6 (3), it follows directly that +lim inf +ε→0 +1 +εm +� +M +f +� +|φ+ +ε + χε(φ+ +ε )| +� +|φ+ +ε + χε(φ+ +ε )|2dvolg ≥ c0 +(3.18) +for some constant c0 > 0. Let us set ηε : (0, ∞) → R by ηε(t) = Kε(tφ+ +ε ). One easily checks +that tη′ +ε(t) = K′ +ε(tφ+ +ε )[tφ+ +ε ] for all t > 0. Hence, by (3.17) and Taylor’s expansion, we get +tη′ +ε(t) ≤ 2ηε(1) − +θ +(θ + 1)εm +� +M +f +� +|φ+ +ε + χε(φ+ +ε )| +� +|φ+ +ε + χε(φ+ +ε )|2dvolg + C|t − 1| +(3.19) +for t close to 1 with some C > 0 independent of ε. Here we have used the uniform boundedness +of η′ +ε(t) on bounded intervals. +Noticing that ηε(1) = I′ +ε(φ+ +ε )[φ+ +ε ] → 0 as ε → 0, we conclude from (3.18) and (3.19) that +there exists a small constant δ > 0 such that +η′ +ε(t) ≤ −δ for all t ∈ (1 − δ, 1 + δ) and ε small enough. +Moreover, notice that Kε(u) equals to the value of I′ +ε,u(1), it follows from Lemma 3.5 that +ηε(1 − δ) > 0 and ηε(1 + δ) < 0. Then, by the Inverse Function Theorem, tε := η−1 +ε (0) exists +and +uε := tεφ+ +ε ∈ Nε ∩ R+φ+ +ε +is well-defined for all ε small enough. Furthermore, since |η′ +ε(t)−1| is bounded by a constant, +say cδ > 0, on (1 − δ, 1 + δ), we consequently get +∥uε − φ+ +ε ∥ε = |η−1 +ε (0) − η−1 +ε (Kε(φ+ +ε ))| · ∥φ+ +ε ∥ε ≤ cδ|Kε(φ+ +ε )| · ∥φ+ +ε ∥ε. +Now the conclusion follows from Kε(φ+ +ε ) ≤ O +� +∥I′ +ε(φ+ +ε )∥ε +� +. + +19 +Corollary 3.8. For the sequence {φε} in (3.15), there exists {uε} such that uε ∈ Nε and +∥φε − uε − χε(uε)∥ε ≤ O(∥L′ +ε(φε)∥ε). Moreover, +max +t>0 Iε(tφ+ +ε ) = Iε(uε) ≤ Lε(φε) + O +� +∥L′ +ε(φε)∥2 +ε +� +. +Proof. To see this, let uε = tεφ+ +ε be as in Proposition 3.7 and set zε = φ+ +ε + χε(φ+ +ε ). Then one +obtains from Lemma 3.6 that +∥φε − uε − χε(uε)∥ε ≤ ∥φε − zε∥ε + |tε − 1| · ∥φ+ +ε ∥ε + ∥χε(φ+ +ε ) − χε(uε)∥ε +≤ O +� +∥L′ +ε(φε)∥ε +� ++ O +� +∥I′ +ε(φ+ +ε )∥ε +� +(3.20) +where we have used an easily checked inequality +∥χε(φ+ +ε ) − χε(uε)∥ε ≤ ∥χ′ +ε(τφ+ +ε )∥H+ +ε →H− +ε · ∥φ+ +ε − uε∥ε = O(|tε − 1|) +for some τ between tε and 1. Observing that I′ +ε(φ+ +ε ) = L′ +ε(zε), and using the C2 smoothness of +Lε, we have +∥I′ +ε(φ+ +ε )∥ε = ∥L′ +ε(zε)∥ε ≤ ∥L′ +ε(φε)∥ε + O(∥φε − zε∥ε) = O(∥L′ +ε(φε)∥ε). +This together with (3.20) implies +∥φε − uε − χε(uε)∥ε ≤ O(∥L′ +ε(φε)∥ε). +Now, by Talyor’s expansion, we can obtain +Lε(φε) = Lε(uε + χε(uε)) + L′ +ε(uε + χε(uε))[φε − uε − χε(uε)] + O +� +∥L′ +ε(φε)∥2 +ε +� += Iε(uε) + I′ +ε(uε)[φ+ +ε − uε] + O +� +∥L′ +ε(φε)∥2 +ε +� +. +Since uε = tεφ+ +ε ∈ Nε, we have I′ +ε(uε)[φ+ +ε − uε] ≡ 0 and this implies the last estimate. +4 +Proof of the main results +4.1 +The equation on Euclidean spaces: the bubbles +We consider solutions to the equation +˜DgRmψ + aωC ·gRm ψ = f(|ψ|)ψ +on Rm +(4.1) +belonging to the class W +1 +2,2(Rm, ˜S(Rm)), where +˜S(Rm) = +� +S(Rm) ⊕ S(Rm) +m is odd, +S(Rm) +m is even, +˜DgRm = DgRm ⊕ −DgRm if m is odd and ˜DgRm = DgRm if m is even. These solutions will cor- +respond to ”bubbles” or test spinors for our variational problem. We also assume the nonlinear +function f satisfies conditions (f1)-(f3). + +20 +First of all, let us set A = ˜DgRm + aωC·gRm. By a straightforward calculation we see that +A is a self-adjoint operator on L2 with spectrum Spec(A) = (−∞, −a] ∪ [a, +∞). Following +Amann [2] let (Eλ)λ∈R be the spectral resolution of A and define the orthogonal projections by +P = +� 0 +−∞ +dEλ, +Q = +� ∞ +0 +dEλ. +Then the decomposition of E = W +1 +2,2(Rm, ˜S(Rm)) = E+ ⊕ E− is induced by +E− = E ∩ P(L2) +and +E+ = E ∩ Q(L2). +We introduce the following operators +S = +� 0 +−∞ +|λ| +1 +2dEλ +and +T = +� ∞ +0 +|λ| +1 +2dEλ. +and the new inner product on E +⟨ψ, ϕ⟩ = Re +� +(S + T)ψ, (S + T)ϕ +� +2, +ψ, ϕ ∈ E +with ∥ · ∥ denoting the corresponding norm. We easily see that (4.1) is the Euler-Lagrange +equation of the functional +Φ(ψ) = 1 +2 +� +∥Qψ∥2 − ∥Pψ∥2� +− +� +Rm F(|ψ|)dx. +(4.2) +Lemma 4.1. If {ψn} ⊂ E is a bounded sequence such that +Φ′(ψn) → 0 +and +lim inf +n→∞ +� +Rm f(|ψn|)|ψn|2dx > 0. +then there exists ψ ̸= 0 with Φ′(ψ) = 0. +Proof. Let B0 +R denote the open ball of radius R centered at the origin. If +lim +n→∞ sup +y∈Rm +� +y+B0 +R +|ψn|2dx = 0, +∀R > 0, +then a result of Lions [31] implies ψn → 0 in Lq for all q ∈ (2, m∗) and therefore +� +Rm f(|ψn|)|ψn|2dx → +0, which is a contradiction. +Passing to a subsequence, we have +lim inf +n→∞ +� +yn+B0 +R +|ψn|2dx > 0 +for some R > 0 and {yn} ⊂ Rm. Using the invariance of the operator A under translations, we +can find R > 0 and a new sequence { ˜ψn} such that +Φ′( ˜ψn) → 0 +and +lim inf +n→∞ +� +B0 +R +| ˜ψn|2dx > 0. +Up to a subsequence if necessary, we have ˜ψn ⇀ ψ and the compact embedding E ֒→ L2 +loc +shows that ψ ̸= 0. By taking the limit in Φ′( ˜ψn) → 0, we obtain Φ′(ψ) = 0 as desired. + +21 +Corollary 4.2. There exists a nontrivial solution ψ ∈ E to Eq. (4.1). +Proof. By Lemma 4.1 and the boundedness argument in Lemma 3.3, this is a direct conse- +quence of [15, Theorem 2.1]. +Now we may define +µ0 = inf +� +Φ(ψ) : ψ ∈ E \ {0} s.t. Φ′(ψ) = 0 +� +. +(4.3) +Since the super-quadratic part in (4.2) has subcritical growth at infinity, one easily sees that +µ0 > 0 is attained. In particular, analogously to Proposition 3.4 and Lemma 3.5-Corollary 3.8, +the following reduction principle holds. +Lemma 4.3. +(1) There exists a C1 map h : E+ → E− such that Φ(u+h(u)) = max +v∈E− Φ(u+v). +(2) The critical points of the functional J(u) = Φ(u+h(u)) and those of Φ are in one-to-one +correspondence via the map u �→ u + h(u). +(3) For each u ∈ E+ \ {0}, the map t �→ J(tu) has only one maximum on (0, +∞) and +µ0 = +inf +u∈E+\{0} max +t>0 J(tu). +(4) For any bounded sequence {zn} ∈ E such that Φ(zn) → c > 0 and Φ′(zn) → 0, there +holds +max +t>0 J(tz+ +n ) ≤ Φ(zn) + O(∥Φ′(zn)∥2). +From elliptic estimates and and bootstrap arguments, we deduce that (weak) solutions of +(4.1) with bounded energy are uniformly bounded in ∩q≥2W 1,q(Rm, ˜S(Rm)). Moreover, we +have the following +Lemma 4.4. Setting B = +� +ψ ∈ E : Φ(ψ) = µ0, Φ′(ψ) = 0, |ψ(0)| = maxRm |ψ| +� +, the +following holds. +(1) B is compact in W 1,2(Rm, ˜S(Rm)). +(2) There exist C, c > 0 such that |ψ(x)| ≤ C exp(−c|x|) for all ψ ∈ B. +Proof. Clearly, B is closed in E. We show that an arbitrary sequence ψn ∈ B, n ∈ N, in B has +a convergent subsequence. +In fact, since {ψn} is bounded, we have ψn ⇀ ψ0 along a subsequence in E with clearly +ψ0 ∈ B. Hence, one has ψn → ψ0 in Lq +loc for q ∈ [2, m∗). Moreover, by the fact that +µ0 = +� +Rm +1 +2f(|ψn|)|ψn|2 − F(|ψn|)dx = +� +Rm +1 +2f(|ψ0|)|ψ0|2 − F(|ψ0|)dx, +it is easy to see that for every ǫ > 0 there is R > 0 such that +lim sup +n→∞ +� +|x|≥R +1 +2f(|ψn|)|ψn|2 − F(|ψn|)dx ≤ ǫ. +(4.4) + +22 +Setting zn = ψn − ψ0 we obtain, using Φ′(ψn) = Φ′(ψ0) = 0, +⟨Qψn, Qzn⟩ + ⟨Pψn, Pzn⟩ − Re +� +Rm f(|ψn|)(ψn, Qzn − Pzn)dx = 0 +and +⟨Qψ0, Qzn⟩ + ⟨Pψ0, Pzn⟩ − Re +� +Rm f(|ψ0|)(ψ0, Qzn − Pzn)dx = 0. +Hence, we get +∥zn∥2 = Re +� +Rm f(|ψn|)(ψn, Qzn − Pzn)dx + on(1), +(4.5) +where we have used Re +� +Rm f(|ψ0|)(ψ0, Qzn − Pzn)dx → 0 as n → ∞ which holds because +zn ⇀ 0 in Lq for q ∈ [2, m∗]. Recall that, by (f1)-(f3), for arbitrarily small δ > 0 there exists +cδ > 0 such that f(s)s ≤ δs + cδ(f(s)s2) +p−1 +p +and F(s) ≤ +1 +θ+2f(s)s2 for all s ≥ 0. Thus, (4.4) +and (4.5) imply +∥zn∥2 ≤ δ|ψn|2|Qzn − Pzn|2 + cδ +� � +|x|≥R +f(|ψn|)|ψn|2dx +� p−1 +p |Qzn − Pzn|p + on(1) +≤ δC∥zn∥ + Cδ +� +ǫ + on(1) +� p−1 +p ∥zn∥ + on(1). +Due to the arbitrariness of δ, ǫ > 0, one sees ∥zn∥ → 0 as n → ∞. +Now we prove the compactness in W 1,2(Rm, ˜S(Rm)). As a consequence of the equation +(recall A = ˜DgRm + aωC·gRm) +Aψn = f(|ψn|)ψn +and +Aψ0 = f(|ψ0|)ψ0, +we have +|A(ψn − ψn)|2 = +��f(|ψn|)ψn − f(|ψ0|)ψ0 +�� +2 +≤ +��f(|ψn|)(ψn − ψ0) +�� +2 + +��(f(|ψn|) − f(|ψ0|))ψ0 +�� +2. +From |ψn|∞ ≤ C and ψn → ψ0 in E we deduce +��f(|ψn|)(ψn − ψ0) +�� +2 ≤ f(C)|ψn − ψ0|2 = on(1), +as n → ∞ +and +� +Rm +��(f(|ψn|) − f(|ψ0|))ψ0 +��2dx = +� +|x|≤R +��(f(|ψn|) − f(|ψ0|))ψ0 +��2dx + oR(1) += on(1) + oR(1) +as n → ∞ because |ψ0(x)| → 0 as |x| → ∞. Therefore, we get |A(ψn − ψn)|2 → 0, i.e. +ψn → ψ0 in W 1,2(Rm, ˜S(Rm)). +To see the exponential decay, we rewrite (4.1) as +˜DgRmψ = −aωC ·gRm ψ + f(|ψ|)ψ. + +23 +Applying the operator ˜DgRm to both sides of the above equation and noting that ˜D2 +gRm = −∆, +we find +∆ψ = +� +a2 − f(|ψ|)2� +ψ − ∇f(|ψ|) ·gRm ψ. +Now using the fact that +∆|ψ|2 = 2 Re(∆ψ, ψ) + 2|∇ψ|2 +and that Re(∇f(|ψ|) ·gRm ψ, ψ) ≡ 0, we obtain +∆|ψ|2 = 2 +� +a2 − f(|ψ|)2� +|ψ|2 + 2|∇ψ|2 ≥ 2 +� +a2 − f(|ψ|)2� +|ψ|2. +Now for ψ ∈ B, by |ψ(x)| → 0 as |x| → ∞, we may take R > 0 large enough so that +∆|ψ|2 ≥ a2|ψ|2 +for all |x| ≥ R. +Let Γ(x) = e−a|x|. One checks easily that +∆Γ − a2Γ < 0, +for |x| > 0. +By taking C > 0 be such that |ψ(x)|2 ≤ C · Γ(x) holds on |x| = R, we may consider U = +|ψ|2 − C · Γ and get +∆U = ∆|ψ|2 − C · ∆Γ > a2U. +By elliptic estimates and the comparison principle, we can easily conclude that U(x) ≤ 0 for +all |x| ≥ R. Hence we obtain the exponential decay of |ψ(x)| at infinity. Finally, thanks to the +compactness of B, we see that the exponential decay holds uniformly for all ψ ∈ B. +4.2 +Bourguignon-Gauduchon trivialization +Our proof relies on the construction of a test spinor on M in order to show the concentration +behavior under the conditions (f1)-(f3). The test spinor comes from a spinor on Rm being cut- +off and transplanted to M so that it has support in a small neighborhood of an arbitrary point +y ∈ M. We first need to recall a construction from the paper [7] of Ammann et al. +To begin with, we fix a spinor field ψ ∈ B arbitrarily. Let r < inj(M)/2 where inj(M) > 0 +is the injectivity radius of M, and let η ∈ C∞ +c (Rm, [0, 1]) be such that |∇η| ≤ 2/r, η(x) = 1 +for |x| ≤ r and η(x) = 0 for |x| ≥ 2r. Then, we define ϕε : Rm → Sm by +ϕε(x) = η(x)ψε(x) +where +ψε(x) = ψ(x/ε). +(4.6) +In order to transplant the test spinor on M, we recall the Bourguignon-Gauduchon-trivialization. +Here we fix y ∈ M arbitrarily, and let (x1, . . . , xm) be the normal coordinates given by the ex- +ponential map +expy : Rm ∼= TyM ⊃ U → V ⊂ M, +x �→ ξ = expy(x). +For ξ ∈ V , let G(ξ) = (gij(ξ))ij denote the corresponding metric at ξ. Since G(ξ) is symmetric +and positive definite, the square root B(ξ) = (bij(ξ))ij of G(ξ)−1 is well defined, symmetric +and positive definite. It can be thought of as linear isometry +B(ξ) : (Rm ∼= Texp−1 +y (ξ)U, gRm) → (TpV, g). + +24 +We obtain an isomorphism of SO(m)-principal bundles: +PSO(U, gRm) φ +� +� +PSO(V, g) +� +TyM ⊃ U +expy � V ⊂ M +where φ{v1, . . . , vm} = {Bv1, . . . , Bvm} for an oriented frame {v1, . . . , vm} on U. Notice that +φ commutes with the right action of SO(m), hence it induces an isomorphism of spin structures: +Spin(m) × U = PSpin(U, gRm) +� +� +PSpin(V, g) ⊂ PSpin(M) +� +TyM ⊃ U +expy +� V ⊂ M +Thus we obtain an isomorphism between the spinor bundles S(U) and S(V ): +S(U) := PSpin(U, gRm) ×ρ Sm −→ S(V ) := PSpin(V, g) ×ρ Sm ⊂ S(M) +(4.7) +where (ρ, Sm) is the complex spin representation. +Let {∂1, . . . , ∂m} be the canonical frame on the Euclidean space, where ∂i = +∂ +∂xi. Setting +ei = B(∂i) = � +j bij∂j we obtain an orthonormal frame {e1, . . . , em} of (TV, g). Via (4.7), we +also have +ei ·g ¯ψ = B(∂i) ·g ¯ψ = ∂i ·gRm ψ +for ψ ∈ Sm. +Now a spinor field ϕ ∈ Γ(˜S(U)) corresponds via the isomorphim (4.7) to a spinor ¯ϕ ∈ +Γ(˜S(V )), and we will keep this notation for various spinor fields to represent such correspon- +dence. In particular, since the spinors ϕε ∈ Γ(˜S(U)) have compact support in U they correspond +to spinors ¯ϕε ∈ Γ(˜S(M)) with compact support in V . +In the sequel, in order to simplify the notation, we use ∇ and ¯∇, respectively, for the Levi- +Civita connections on (TU, gRm) and (TV, g) and for the natural lifts of these connections to the +spinor bundles ˜S(U) and ˜S(V ), respectively. By abuse of notation, we write D and ¯D instead of +the Dirac operators acting on Γ(˜S(U)) and Γ(˜S(V )), respectively. By [7, Proposition 3.2] there +holds +¯D ¯ϕε = Dϕε + W ·g ¯ϕε + X ·g ¯ϕε + +� +i,j +(bij − δij)∂i ·gRm ∇∂jϕε +(4.8) +with W ∈ Γ(Cl(TV )) and X ∈ Γ(TV ) given by +W = 1 +4 +� +i,j,k +i̸=j̸=k̸=i +� +α,β +biα(∂αbjβ)b−1 +βk ei ·g ej ·g ek, +and +X = 1 +4 +� +i,k +�¯Γi +ik − ¯Γk +ii +� +ek = 1 +2 +� +i,k +¯Γi +ikek; +here (b−1 +ij )ij denotes the inverse matrix of B, and ¯Γk +ij := g( ¯∇eiej, ek). In what follows we +identify x = (x1, . . . , xm) ∈ U ⊂ Rm with � +i xiei ∈ TyM for notational convenience. + +25 +As remarked in [17, 30], in the neighborhood of y, the metric g and its determinant have the +following expansion: +gij(expy x) = δij − 1 +3Ry(ei, x, x, ej) + O(|x|3), +(4.9) +� +det g(expy x) = 1 − 1 +6Ricy(x, x) + O(|x|3) +(4.10) +where +R(ei, ej, ek, el) = g(∇ei∇ejek, el) − g(∇ej∇eiek, el) − g(∇[ei,ej]ek, el) +is the Riemannian curvature tensor and Ric(v, w) = �m +i=1 R(ei, v, w, ei) is the Ricci curvature. +Observing that B = (G−1) +1 +2, as in [7], we have +bij = δij + 1 +6Ry(ei, x, x, ej) + O(|x|3), +W = O(|x|3) +and +X = O(|x|). +(4.11) +The main results of this subsection will be the following two lemmas. +Lemma 4.5. Let ¯ϕε be as in (4.6), then ∥L′ +ε( ¯ϕε)∥ε ≤ O(ε2) as ε → 0. +Proof. By the definition of ϕε in (4.6), together with (4.8), one checks easily that +εDϕε = ε∇η ·gRm ψε + εηDψε = ε∇η ·gRm ψε − aηωC ·Rm ψε + ηf(|ψε|)ψε +(4.12) +and +ε ¯D ¯ϕε + aωC ·g ¯ϕε − f(| ¯ϕε|) ¯ϕε = J1 + J2 + · · · + J6 ∈ H∗ +where +J1 = ε∇η ·gRm ψε, +J2 = η · +� +f(| ¯ψε|) − f(| ¯ϕε|) +� ¯ψε, +J3 = εηW ·g ¯ψε, +J4 = εηX ·g ¯ψε, +J5 = εη +� +i,j +(bij − δij)∂i ·gRm ∇∂jψε, +J6 = ε +� +i,j +(bij − δij)∂jη∂i ·gRm ψε. +In the following estimates we use that the support of η is contained in B2r(0) ⊂ Rm. Using +the exponential decay in Lemma 4.4 and (4.9)-(4.11), we have: +∥J1∥H→R ≲ +� 1 +εm +� +B2r(y) +��ε∇η ·gRm ψε +��2dvolg +� 1 +2 +≲ +� 1 +εm +� +r≤|x|≤2r +��εψε +��2dx +� 1 +2 +≲ ε +� � +r +ε ≤|x|≤ 2r +ε +��ψ +��2dx +� 1 +2 +≲ ε exp +� +− c · r +ε +� +, + +26 +∥J2∥H→R ≲ +� 1 +εm +� +B2r(y)\Br(y) +�� ¯ψε +��2dvolg +� 1 +2 ++ +� 1 +εm +� +B2r(y)\Br(y) +�� ¯ψε +��pdvolg +� p−1 +p +≲ +� � +r +ε≤|x|≤ 2r +ε +|ψ|2dx +� 1 +2 ++ +� � +r +ε≤|x|≤ 2r +ε +|ψ|pdx +� p−1 +p +≲ exp +� +− c · r +ε +� +, +∥J3∥H→R ≲ +� 1 +εm +� +B2r(y) +��εW ·g ¯ψε +��2dvolg +� 1 +2 +≲ +� 1 +εm +� +|x|≤2r +� +ε|x|3|ψε| +�2dx +� 1 +2 +≲ ε4 +� � +|x|≤ 2r +ε +|x|6|ψ|2dx +� 1 +2 +≲ ε4, +∥J4∥H→R ≲ +� 1 +εm +� +B2r(y) +��εX ·g ¯ψε +��2dvolg +� 1 +2 +≲ +� 1 +εm +� +|x|≤2r +� +ε|x||ψε| +�2dx +� 1 +2 +≲ ε2 +� � +|x|≤ 2r +ε +|x|2|ψ|2dx +� 1 +2 +≲ ε2, +∥J5∥H→R ≲ +� 1 +εm +� +|x|≤2r +� +ε|x|2|∇ψε| +�2dx +� 1 +2 +≲ ε2 +� � +|x|≤ 2r +ε +|x|4|∇ψ|2dx +� 1 +2 +≲ ε2, +∥J6∥H→R ≲ +� 1 +εm +� +|x|≤2r +� +ε|x|2|ψε| +�2dx +� 1 +2 +≲ ε3 +� � +|x|≤ 2r +ε +|x|4|ψ|2dx +� 1 +2 +≲ ε3. +Here, in the estimates of J1 and J2, the constant c > 0 comes from the exponential decay in +Lemma 4.4. From all these, we finally obtain ∥L′ +ε( ¯ϕε)∥ε ≤ O(ε2). +We define a functional Θ : M × B → R by +Θ(y, ψ) = 1 +6 +� +Rm Ricy(x, x) +�1 +2f(|ψ|)|ψ|2 − F(|ψ|) +� +dx ++ 1 +12 +� +i,j +Re +� +Rm Ry(ei, x, x, ej)(∇∂jψ, ∂i ·gRm ψ)dx +Then we have the following +Lemma 4.6. Let µ0 be the ground state energy of Eq. (4.1) defined in (4.3) and ¯ϕε be defined +in (4.6), then +Lε( ¯ϕε) = µ0 − ε2Θ(y, ψ) + o(ε2) +as ε → 0. +Proof. By (4.8) and (4.12) again, we have +1 +εm +� +M +(ε ¯D ¯ϕε, ¯ϕε)dvolg + a +εm +� +M +(ωC ·g ¯ϕε, ¯ϕε)dvolg = I1 + I2 + · · · + I7, +where +I1 = Re +εm +� +M +η · (ε∇η ·gRm ψε, ¯ψε)dvolg, + +27 +I2 = 1 +εm +� +M +f(| ¯ϕε|)| ¯ϕε|2dvolg, +I3 = 1 +εm +� +M +η2� +f(| ¯ψε|) − f(| ¯ϕε|) +� +| ¯ψε|2dvolg, +I4 = Re +εm +� +M +η2 · (εW ·g ¯ψε, ¯ψε)dvolg, +I5 = Re +εm +� +M +η2 · (εX ·g ¯ψε, ¯ψε)dvolg, +I6 = +� +i,j +Re +εm +� +M +η2(bij − δij)(ε∂i ·gRm ∇∂jψε, ¯ψε)dvolg, +I7 = +� +i,j +Re +εm +� +M +η∂jη · (bij − δij)(ε∂i ·gRm ψε, ¯ψε)dvolg. +Analogously to the arguments in Lemma 4.5, we shall use (4.9)-(4.11) and the fact (∇η ·gRm ψε, ¯ψε) ∈ +iR to obtain +I1 = 0, +I2 = 1 +εm +� +|x|≤r +f(|ψε|)|ψε|2dx − +1 +6 εm +� +|x|≤r +f(|ψε|)|ψε|2Ricy(x, x)dx + o(ε2) += +� +Rm f(|ψ|)|ψ|2dx − ε2 +6 +� +Rm f(|ψ|)|ψ|2Ricy(x, x)dx + o(ε2), +|I3| ≲ 1 +εm +� +r≤|x|≤2r +|ψε|2dx + 1 +εm +� +r≤|x|≤2r +|ψε|pdx +≲ +� +r +ε≤|x|≤ 2r +ε +|ψ|2dx + +� +r +ε≤|x|≤ 2r +ε +|ψ|pdx ≲ o(ε2), +|I4| ≲ 1 +εm +� +|x|≤2r +ε|x|3|ψε|2dx ≲ ε4 +� +|x|≤ 2r +ε +|x|3|ψ|2dx ≲ ε4, +I5 = 0, +I6 = +� +i,j +Re +6 εm +� +|x|≤r +Ry(ei, x, x, ej)(ε∂i ·gRm ∇∂jψε, ψε)dx + o(ε2) += ε2 +6 +� +i,j +Re +� +Rm Ry(ei, x, x, ej)(∂i ·gRm ∇∂jψ, ψ)dx + o(ε2) += −ε2 +6 +� +i,j +Re +� +Rm Ry(ei, x, x, ej)(∇∂jψ, ∂i ·gRm ψ)dx + o(ε2), +I7 = 0. + +28 +Combining all these estimates and the fact ψ ∈ B, we deduce that +Lε( ¯ϕε) = I1 + I2 + · · · + I7 +2 +− 1 +εm +� +M +F(| ¯ϕε|)dvolg += 1 +2 +� +Rm f(|ψ|)|ψ|2dx − +� +Rm F(|ψ|)dx − ε2Θ(y, ψ) + o(ε2) += µ0 − ε2Θ(y, ψ) + o(ε2) +as desired. +4.3 +Characterization of the concentration profile +From Proposition 3.4 and (3.14), we deduce that +µε = +inf +u∈H+ +ε \{0} +max +ψ∈Ru⊕H− +ε +Lε(ψ) = +inf +u∈H+ +ε \{0} max +t>0 Iε(tu) = inf +u∈Nε Iε(u) +is a critical value. As in Proposition 3.4, we take ¯ψε to be the corresponding critical point of Lε. +Then, as it was mentioned in (3.11), we find +1 +εm +� +M +1 +2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg = µε ≥ τ0 +(4.13) +for some τ0 > 0. In what follows, for any ξ ∈ M and r > 0, Br(ξ) ⊂ M denotes the ball of +radius r with respect to the metric g. +Lemma 4.7. There exist yε ∈ M, r0, δ0 > 0 such that +lim inf +ε→0 +1 +εm +� +Bεr0(yε) +| ¯ψε|2dvolg ≥ δ0. +Proof. Assume to the contrary that for any r > 0 +sup +ξ∈M +1 +εm +� +B2εr(ξ) +| ¯ψε|2dvolg → 0 +as ε → 0. +(4.14) +For each ξ ∈ M, we choose a smooth real cut-off function βξ,ε ≡ 1 on Bεr(ξ) and supp βξ,ε ⊂ +B2εr(ξ). Then, for s ∈ (0, 1), we consider qs = 2 + (m∗ − 2)s ∈ (2, m∗) and we have +� +B2εr(ξ) +|βξ,ε ¯ψε|qsdvolg ≤ +� � +B2εr(ξ) +|βξ,ε ¯ψε|2dvolg +�1−s� � +B2εr(ξ) +|βξ,ε ¯ψε| +2m +m−1dvolg +�s +. +Taking s = +2 +m∗ , we obtain from Lemma 3.1 that +� 1 +εm +� +B2εr(ξ) +|βξ,ε ¯ψε|m∗dvolg +�s +≤ C∥βξ,ε ¯ψε∥2 +ε. + +29 +We now cover M by balls of radius εr such that any point ξ ∈ M is contained in at most KM +balls, where KM does not depend on ε. In fact, one may take KM = 1 + dimM. Consequently +we have +1 +εm +� +M +| ¯ψε|qsdvolg ≤ C · KM +� +sup +ξ∈M +� +B2εr(ξ) +|βξ,ε ¯ψε|2dvolg +�1−s +∥ ¯ψε∥2 +ε. +Since ∥ ¯ψε∥ε is bounded, it follows from (4.14) that | ¯ψε|qs,ε → 0, and since 2 < qs < m∗, we +see easily that | ¯ψε|q,ε → 0 for all q ∈ (2, m∗) which contradicts (4.13) +Lemma 4.8. limε→0 µε = µ0. +Proof. By the estimates obtained in Corollary 3.8, Lemma 4.5 and 4.6, we only need to show +that +lim inf +ε→0 +µε ≥ µ0. +For this purpose, let us take a sequence {yε} ⊂ M and constants r0, δ0 > 0 such that Lemma +4.7 is valid. Given 0 < r < inj(M)/2 arbitrarily, let βε ∈ C∞(M, [0, 1]) be such that βε ≡ 1 +on Br(yε) and supp βε ⊂ B2r(yε). Via the Bourguignon-Gauduchon trivialization between the +spinor bundles S(Br(yε)) → S(Br(0)) and the rescaling x �→ x +ε on Rm, the spinor field βε ¯ψε +corresponds to a spinor field zε(·) on B2r/ε(0) ⊂ Rm. +Since βε ¯ψε is bounded in H with respect to the norm ∥·∥ε, one sees easily that zε is bounded +in W +1 +2,2(BR(0), ˜S(BR(0))) for any R > 0 as ε → 0. By the fact +� +|x|≤ r +ε +|zε|m∗dx ≲ 1 +εm +� +Br(yε) +| ¯ψε|m∗dvolg ≲ ∥ ¯ψε∥ε < ∞ +for all small ε, it follows that there exists z0 ∈ Lm∗(Rm, ˜S(Rm)) ∩ W +1 +2,2 +loc (Rm, ˜S(Rm)) such that +zε ⇀ z0 in W +1 +2,2 +loc (Rm, ˜S(Rm)) weakly and zε → z0 in Lq +loc(Rm, ˜S(Rm)) for 2 ≤ q < +2m +m−1. +Let ϕ ∈ W +1 +2,2(Rm, ˜S(Rm)) be such that supp ϕ is compact, i.e. supp ϕ ⊂ B0 +R for some +R > 0 large. Then, by a similar identity of (4.8) and (4.9)-(4.11), we have +� +Rm +� ˜DgRmz0 + aωC ·gRm z0 − f(|z0|)z0, ϕ +� +dx += lim +ε→0 +� +supp ϕ +� ˜DgRmzε + aωC ·gRm zε − f(|zε|)zε, ϕ +� +dvolgΘε += lim +ε→0 +1 +εm +� +BεR(ξε) +� +ε ˜Dgψε + aωC ·g ψε − f(|ψε|)ψε, ¯ϕε +� +dvolg += 0 +where ϕε(x) = ϕ( x +ε) and ¯ϕε is the spinor on M defined via the Bourguignon-Gauduchon +trivialization. Hence z0 satisfies +˜DgRmz0 + aωC ·gRm z0 = f(|z0|)z0 +on Rm. +(4.15) +As a consequence of (f1)-(f3) and by elliptic regularity, we have ˜DgRmz0 + aωC ·gRm z0 ∈ +L +m∗ +m∗−1(Rm, ˜S(Rm)). Moreover, combined with the Sobolev embedding L +p +p−1(Rm, ˜S(Rm)) ֒→ +W − 1 +2 ,2(Rm, ˜S(Rm)), we get z0 ∈ W +1 +2 ,2(Rm, ˜S(Rm)). + +30 +Now, by Lemma 4.7, one sees that z0 is a non-trivial solution to (4.15). In virtue of (4.13), +we conclude that +lim inf +ε→0 +Lε( ¯ψε) = lim inf +ε→0 +1 +εm +� +M +1 +2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg +≥ lim inf +ε→0 +� +BR(0) +1 +2f(|zε|)|zε|2 − F(|zε|)dx +≥ +� +BR(0) +1 +2f(|z0|)|z0|2 − F(|z0|)dx +where in the last inequality we have used the Fatou’s lemma. Due to the arbitrariness of R > 0, +together with (4.3), we obtain +lim inf +ε→0 +µε = lim inf +ε→0 +Lε( ¯ψε) ≥ µ0. +Now, we fix the sequence {yε} ⊂ M and constants r0, δ0 > 0 as in Lemma 4.7. Up to +a subsequence, we also assume that yε → y0 in M as ε → 0. As Lemma 4.8 alluded, there +exists a least-energy solution z0 of the limit equation (4.1) corresponding to the weak limit +of the solutions ¯ψε via Bourguignon-Gauduchon trivialization and rescaling. Without loss of +generality, one may assume yε to be the maximum point of | ¯ψε|, and then z0 ∈ B. +Choose β ∈ C∞(M, [0, 1]) be such that β ≡ 1 on Br(y0) and supp β ⊂ B2r(y0) for some +r < inj(M)/2. Set ¯φε = β¯z0,ε, where z0,ε(x) = z0(x/ε) and ¯z0,ε is the corresponding spinor +field obtained via Bourguignon-Gauduchon trivialization. We have the following asymptotic +characterization. +Lemma 4.9. ∥ ¯ψε − ¯φε∥ε → 0 as ε → 0. +Proof. Setting wε = ¯ψε − ¯φε, we have |wε|q,ε → 0 as ε → 0 for all q ∈ (2, m∗) (otherwise, we +can apply Lemma 4.7 and 4.8 for {wε} instead of { ¯ψε} to get Lε( ¯ψε) ≥ 2µ0 which is absurd). +To proceed, let us go over the proof of Lemma 4.8 again to see that +µ0 = lim +ε→0 +1 +εm +� +M +1 +2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg = +� +Rm +1 +2f(|z0|)|z0|2 − F(|z0|)dx +Similar to (4.4) (but here we need to transplant it on the manifold), it is easy to see that for every +ǫ > 0 there is R > 0 large such that +lim sup +ε→0 +� +M\BεR(yε) +1 +2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg ≤ ǫ. +(4.16) +From the fact L′ +ε( ¯ψε) = 0 and the estimate in Lemma 4.5, we have +� ¯ψ+ +ε , w+ +ε +� +ε + +� ¯ψ− +ε , w− +ε +� +ε − Re +εm +� +M +f(| ¯ψε|)( ¯ψε, w+ +ε − w− +ε )dvolg = 0 +and +�¯φ+ +ε , w+ +ε +� +ε + +�¯φ− +ε , w− +ε +� +ε − Re +εm +� +M +f(|¯φε|)(¯φε, w+ +ε − w− +ε )dvolg = oε(1). + +31 +Hence, we get +∥wε∥2 +ε = Re +εm +� +M +f(| ¯ψε|)( ¯ψε, w+ +ε − w− +ε )dvolg + oε(1), +(4.17) +where we have used that |wε|q,ε → 0 as ε → 0 for all q ∈ (2, m∗). Recall that, by (f1)- +(f3), for arbitrarily small δ > 0 there exists cδ > 0 such that f(s) ≤ δ + cδ(f(s)s2) +p−1 +p +and F(s) ≤ +1 +θ+2f(s)s2 for all s ≥ 0. Thus, it follows from (4.16), (4.17) and the Sobolev +embeddings that +∥wε∥2 +ε ≤ δC∥wε∥ε + Cδ +� +ǫ + oε(1) +� p−1 +p ∥wε∥ε + oε(1), +for some constants C, Cδ > 0. Due to the arbitrariness of δ, ǫ > 0, one sees ∥wε∥ε → 0 as +ε → ∞. +Lemma 4.10. For small ε > 0, there exist positive constants C, c > 0 such that +| ¯ψε(ξ)| ≤ C exp +� +− c +ε dist(ξ, yε) +� +, +for all ξ ∈ M. +Proof. We first show that the family { ¯ψε} decays uniformly on M in the following sense: +dist(ξ, yε) +ε +→ ∞ =⇒ | ¯ψε(ξ)| → 0, +as ε → 0. +(4.18) +Indeed, since ¯ψε solves (3.4) and Lε( ¯ψε) = µε → µ0, by the Sobolev embedding and +bootstrap arguments, we deduce that {| ¯ψε|∞} is bounded. Applying the operator ε ˜Dg on both +sides of (3.4), and using the Lichnerowicz formula, we find +ε2∆g ¯ψε − ε2Scalg +4 +¯ψε = (a2 − f(| ¯ψε|)2) ¯ψε − ε∇gf(| ¯ψε|) ·g ¯ψε +on M, +where ∆g = divg∇g is the Laplace-Beltrami operator, ∇g is the gradient and Scalg is the scalar +curvature with respect to the metric g. Using the fact +∆g| ¯ψε|2 = 2 Re(∆g ¯ψε, ¯ψε) + 2|∇g ¯ψε|2, +we have +ε2∆g| ¯ψε|2 = 2 +� +a2 + ε2Scalg +4 +− f(| ¯ψε|)2� +| ¯ψε|2 + 2|∇g ¯ψε|2 ≥ −C| ¯ψε|2 +(4.19) +for some C > 0. Then, it follows from the local boundedness of sub-solutions (see for example +[23, Theorem 4.1]) that +| ¯ψε(ξ)|2 ≤ C0 +εm +� +Bε(ξ) +| ¯ψε|2dvolg +(4.20) +with C0 > 0 independent of ξ ∈ M and ε > 0. +Assume by contradiction that there exist δ > 0 and ξε ∈ M such that dist(ξε, yε)/ε → ∞ as +ε → 0 and | ¯ψε(ξε)| ≥ δ. By (4.20) and Lemma 4.9, we deduce, as ε → 0 +δ ≤ C0 +� 1 +εm +� +Bε(ξε) +| ¯ψε|2dvolg +� 1 +2 ≤ C0| ¯ψε − ¯φε|2,ε + C0 +� 1 +εm +� +Bε(ξε) +|¯φε|2dvolg +� 1 +2 +≤ oε(1) + C0 +� � +B1 +� exp−1 +yε (ξε) +ε +� |z0|2dvolg +� 1 +2 → 0 + +32 +which is a contradiction; here ¯φε and z0 are as in Lemma 4.9. This proves (4.18). +In order to see the exponential decay, by (4.18) and (4.19), we can take R0 > 0 sufficiently +large so that +ε2∆g| ¯ψε|2 ≥ a2| ¯ψε|2 +on M \ BεR0(yε) +(4.21) +for all ε > 0 small. Our next strategy is to use the comparison principle and is very similar +to the flat case (see Lemma 4.4): we first construct a sequence of positive functions ¯Γε on +M \ BεR0(yε) which decays exponentially, then we chose C > 0 such that | ¯ψε(ξ)|2 ≤ C · ¯Γ(ξ) +for ξ ∈ ∂BεR0(yε) and finally, by setting Uε = | ¯ψε|2 − C · ¯Γ, we will show +ε2∆gUε ≥ a2Uε, +on M \ BεR0(yε). +For the sake of clarity, let us consider the (local) normal coordinates exp−1 +yε : Br(yε) → +Rm ∼= TyεM for a fixed r < inj(M)/5 and set vε(x) = | ¯ψε|2 ◦ expyε(εx). Then we find +1 +� +det g(εx) +� +j,k +∂j +� +gjk(εx) +� +det g(εx)∂kvε +� +− a2vε ≥ 0 +for R0 ≤ |x| ≤ r +ε, +where (gij)ij = G−1 is the inverse matrix of the metric g. +For each ε > 0 small, we take Γε(x) = e− a +2 Lε(|x|), where Lε : [0, ∞) → [0, ∞) is a function +defined as +Lε(ρ) = + + + + + + + + + + + + + + + +ρ +0 ≤ ρ < r +ε, +1 +2 +� +ρ + r +ε sin +�ε +rρ − 1 +�� ++ r +2ε +r +ε ≤ ρ < π + 1 +ε +r, +π + 2 +2ε r +ρ ≥ π + 1 +ε +r. +(4.22) +According to the above definition, Γε ∈ C2(Rm \{0}) is spherically symmetric, and so satisfies +∆Γε − a2 +4 Γε = −a +2e− a +2 Lε(ρ)� +L′′ +ε(ρ) − a +2(L′ +ε(ρ))2 + m − 1 +ρ +L′ +ε(ρ) + a +2 +� +for ρ = |x| > 0. +Now we see easily that +∆Γε − a2 +4 Γε = + + + + + + + +− a(m − 1) +2ρ +e− a +2 Lε(ρ) +0 ≤ ρ < r +ε, +− a2 +4 e− a +2 Lε(ρ) +ρ ≥ π + 1 +ε +r. +And for r +ε ≤ ρ < π+1 +ε r, from a direct computation, we have +L′′ +ε(ρ) − a +2(L′ +ε(ρ))2 + m − 1 +ρ +L′ +ε(ρ) + a +2 += a +2 − a +8 +� +1 + cos +�ε +rρ − 1 +��2 +− ε +2r sin +�ε +rρ − 1 +� ++ m − 1 +2ρ +� +1 + cos +�ε +rρ − 1 +�� +≥ a +2 − a +8 +� +1 + cos +�ε +rρ − 1 +��2 +− ε +2r sin +�ε +rρ − 1 +� ++ +ε +2(π + 1)r +� +1 + cos +�ε +rρ − 1 +�� += a +2 − a +8 +� +1 + cos +�ε +rρ − 1 +��2 +− ε +� +(π + 1)2 + 1 +2(π + 1)r +sin +�ε +rρ − 1 − ϑ +� ++ +ε +2(π + 1)r + +33 +where ϑ = arcsin +1 +√ +(π+1)2+1 ∈ (0, π +2). Therefore, for small ε > 0, we can derive that +∆Γε − a2 +4 Γε < 0 +on Rm \ {0}. +Moreover, about the gradient of Γε we have +|∇Γε(x)| +Γε(x) +≤ a +2 +for all |x| > 0 and ε > 0. +Hence, by taking r smaller if necessary, it follows from (4.9) that +1 +� +det g(εx) +� +j,k +∂j +� +gjk(εx) +� +det g(εx)∂kΓε +� +≤ ∆Γε + or(1) +� +|∆Γε| + |∇Γε| +� +≤ a2Γε +for R0 ≤ |x| ≤ r/ε. +Next, using (4.18), we can fix C0 > 0 such that vε(x) ≤ C0 · Γε(x) = C0e− a +2 R0 for |x| = R0 +and ε small. Via the Riemannian normal coordinates, we can transplant the functions Γε onto +M \ BεR0(yε) by introducing +¯Γε(ξ) = + + + + + +Γε +�exp−1 +yε (ξ) +ε +� +ξ ∈ B5r(yε) \ BεR0(yε), +e− a(π+2) +4ε +r +M \ B5r(yε). +(4.23) +Then, from (4.22), we see that ¯Γε > 0 is a well-defined C2-smooth function on M \ BεR0(yε), +and in particular, +ε2∆g¯Γε − a2¯Γε ≤ 0 +on M \ BεR0(yε). +Setting Uε = | ¯ψε|2 − C0 · ¯Γε(x), we can see from (4.21) that +ε2∆gUε − a2Uε ≥ 0 +on M \ BεR0(yε). +And it is standard to show from the comparison principle that Uε ≤ 0 on M \ BεR0(yε). +Turning back to the definition (4.23), we see that +¯Γε(ξ) ≤ exp +� +− c +ε dist(ξ, yε) +� +for ξ ∈ Br(yε). +and +¯Γε(ξ) ≤ exp +� +− c · r +ε +� +for ξ ∈ M \ Br(yε). +Setting dM = sup{dist(ξ1, ξ2) : ξ1, ξ2 ∈ M}, we see easily that dM ≥ inj(M) and so +¯Γε(ξ) ≤ exp +� +− c · r +ε · dM +dist(ξ, yε) +� +for all ξ ∈ M. +This implies the exponential decay for | ¯ψε| by simply taking the square root. + +34 +Recall that in the proof of Lemma 4.8, we have taken βε ∈ C∞(M, [0, 1]) be such that βε ≡ 1 +on Br(yε) and supp βε ⊂ B2r(yε) for some r < inj(M). Via the Bourguignon-Gauduchon +trivialization between the spinor bundles S(Br(yε)) → S(Br(0)) and the rescaling x �→ x +ε on +Rm, the spinor field βε ¯ψε corresponds to a spinor field zε on B2r/ε(0) ⊂ Rm. And, by Lemma +4.9 and bootstrap arguments, zε converges in W 1,q(Rm, ˜S(Rm)) to some z0 ∈ B as ε → 0, for +q ≥ 2. Thanks to the fast decay rate of ¯ψε, in addition to Lemma 4.8, we have the following +refined lower bound estimate for the critical level µε. +Lemma 4.11. Let yε be a maximum point of | ¯ψε|. Up to a subsequence if necessary, assume +yε → y0 ∈ M as ε → 0 with respect to the Riemannian metric g. Then +µε ≥ µ0 − ε2Θ(y0, z0) + o(ε2). +Proof. Notice that L′ +ε( ¯ψε) = 0 and +Lε( ¯ψε) − Lε(βε ¯ψε) = 1 +εm +� +M +1 +2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg +− 1 +εm +� +M +β2 +ε +2 f(| ¯ψε|)| ¯ψε|2 − F(|βε ¯ψε|)dvolg. +By (f1)-(f2), for each fixed s ≥ 0, we deduce that the function t �→ +t2 +2 f(s)s2 − F(ts) is +non-decreasing for t ∈ [0, 1]. Hence, by βε(ξ) ∈ [0, 1] for all ξ ∈ M, one sees easily +Lε(βε ¯ψε) ≤ Lε( ¯ψε) = µε. +On the other hand, since βε ¯ψε corresponds to a spinor field zε on B2r/ε(0) ⊂ Rm through the +Bourguignon-Gauduchon trivialization and rescaling, by developing the relationship in (4.8) for +βε ¯ψε and zε, we obtain the following correspondence of spinors +ε ¯D(βε ¯ψε) ←→ Dzε + ε3W ·gRm zε + εX ·gRm zε + ε2 � +i,j +(bij − δij)∂i ·gRm ∇∂jzε. +Now, we can argue similarly to the proof of Lemma 4.5 and 4.6 to obtain Φ′(zε) = o(ε) and +Lε(βε ¯ψε) = 1 +εm +� +M +1 +2(ε ¯D(βε ¯ψε), βε ¯ψε) + a +2(ωC ·g βε ¯ψε, βε ¯ψε) − F(|βε ¯ψε|)dvolg += +� +Rm +1 +2(Dzε, zε) + a +2(ωC ·gRm zε, zε) − F(|zε|)dx +− ε2 +12 +� +Rm Ricyε(x, x)(Dzε, zε)dx − a ε2 +12 +� +Rm Ricyε(x, x)(ωC ·gRm zε, zε)dx ++ ε2 +6 +� +Rm Ricyε(x, x)F(|zε|)dx +− ε2 +12 +� +i,j +Re +� +Rm Ryε(ei, x, x, ej)(∇∂jzε, ∂i ·gRm zε)dx + o(ε2) + +35 +Since yε → y0 and zε → z0 in W 1,q(Rm, ˜S(Rm)) as ε → 0, for q ≥ 2, and since |zε| decays +exponentially, we have +� +Rm Ricyε(x, x) +� +(Dzε, zε) + (aωC ·gRm zε, zε) − 2F(|zε|) +� +dx += +� +Rm Ricy0(x, x) +� +(Dz0, z0) + (aωC ·gRm z0, z0) − 2F(|z0|) +� +dx + oε(1) += +� +Rm Ricy0(x, x) +� +f(|z0|)|z0|2 − 2F(|z0|) +� +dx + oε(1) +and +� +i,j +Re +� +Rm Ryε(ei, x, x, ej)(∇∂jzε, ∂i ·gRm zε)dx += +� +i,j +Re +� +Rm Ry0(ei, x, x, ej)(∇∂jz0, ∂i ·gRm z0)dx + oε(1) +Note that, by Lemma 4.3 (4), we also have +µ0 = +inf +u∈E+\{0} max +t>0 J(tu) ≤ Φ(zε) + O(∥Φ′(zε)∥2). +Hence, it follows directly that +µε ≥ Lε(βε ¯ψε) ≥ µ0 − ε2Θ(y0, z0) + o(ε2). +Proof of the main theorem. We first see from Corollary 3.8, Lemma 4.5 and 4.6 that +µε = Lε( ¯ψε) ≤ µ0 − ε2 +max +(y,ψ)∈M×B Θ(y, ψ) + o(ε2) +for small ε > 0. Hence, by Lemma 4.11 and taking the limit ε → 0, we have +Θ(y0, z0) ≥ +max +(y,ψ)∈M×B Θ(y, ψ). +Therefore, we conclude that (yε, zε) → (y0, z0) in M × B such that +lim +ε→0 Θ(yε, zε) = +max +(y,ψ)∈M×B Θ(y, ψ), +and +Lε( ¯ψε) = µ0 − ε2 +max +(y,ψ)∈M×B Θ(y, ψ) + o(ε2). +In the 2-dimensional case, the Ricci tensor determines the whole curvature. Specifically, we +have +Ric(x, x) = R(e1, x, x, e1) + R(e2, x, x, e2) = R(e1, e2, e2, e1)|x|2 +and +Scalg = +2 +� +j=1 +2 +� +i=1 +R(ej, ei, ei, ej) = 2R(e1, e2, e2, e1). + +36 +Hence, by noting that the scalar curvature is twice the Gaussian curvature for surfaces, we have +Θ(y, ψ) = Kg(y) +6 +� +R2 +�1 +2f(|ψ|)|ψ|2 − F(|ψ|) +� +|x|2dx ++ Kg(y) +12 +Re +� +R2 +� +(x2∇∂1 − x1∇∂2)ψ, (x2∂1 − x1∂2) ·gR2 ψ +� +dx, +where Kg denotes the Gaussian curvature of (M, g). +Finally, by substituting f(|ψ|)|ψ|2 = |ψ|n∗ and F(|ψ|) = +1 +n∗|ψ|n∗ into the above formulas, +one completes the proof. +Acknowledgements. The authors wish to express their gratitude to the anonymous reviewer +for his/her positive comments and constructive suggestions. +References +[1] R. Adams, Sobolev Spaces, Academic Press, New York 1975. +[2] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 +(1979), no. 2, 127-166. +[3] B. 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Phys. 35, 1117 (1966). +THOMAS BARTSCH +MATHEMATISCHES INSTITUT, UNIVERSIT ¨AT GIESSEN +35392 GIESSEN, GERMANY +Thomas.Bartsch@math.uni-giessen.de +TIAN XU +CENTER FOR APPLIED MATHEMATICS, TIANJIN UNIVERSITY +TIANJIN, 300072, CHINA +xutian@amss.ac.cn + diff --git a/R9E4T4oBgHgl3EQfLAxE/content/tmp_files/load_file.txt b/R9E4T4oBgHgl3EQfLAxE/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a94914e08d731005b922b23d2d54421911188fff --- /dev/null +++ b/R9E4T4oBgHgl3EQfLAxE/content/tmp_files/load_file.txt @@ -0,0 +1,1226 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf,len=1225 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='04934v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='DG] 12 Jan 2023 Curvature effect in the spinorial Yamabe problem on product manifolds Thomas Bartsch, Tian Xu* Abstract Let (M1, g(1)), (M2, g(2)) be closed Riemannian spin manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We study the existence of solutions of the Spinorial Yamabe problem on the product M1 × M2 equipped with a family of metrics ε−2g(1) ⊕ g(2), ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Via variational methods and blow-up techniques, we prove the existence of solutions which depend only on the factor M1, and which exhibit a spike layer as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, we locate the asymptotic position of the peak points of the solutions in terms of the curvature tensor on (M1, g(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' MSC 2010: Primary: 53C27;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Secondary: 35B40, 35Q40, 35R01, 58E30, 58J60 Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Spinorial Yamabe equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' strongly indefinite functional;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' blow-up solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' spike layer solutions 1 Introduction Let N be an n-dimensional closed spin manifold, n ≥ 2, with Riemannian metric g and a fixed spin structure σ : PSpin(N) → PSO(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Denoted by ρ : Spin(n) → End(Sn) the spin representation, we write S(N) = PSpin(N) ×ρ Sn for the spinor bundle over N and DN g : C∞(N, S(N)) → C∞(N, S(N)) for the (Atiyah-Singer) Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' A spinorial analogue of the Yamabe equation can be written as DN g ψ = |ψ|n∗−2 g ψ, on (N, g, σ) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) where n∗ := 2n n−1 — in fact, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) is conformally invariant and is the Euler-Lagrange equa- tion of a variational problem similar to Yamabe’s problem (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In case ψ ∈ C1(N, S(N)) is a non-trivial solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1), the (generalized) conformal metric ˜g = |ψ| 4 n−1 g g induces a spinor field ϕ on (N, ˜g, σ) such that DN ˜g ϕ = ϕ, |ϕ|˜g ≡ 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2) on N \\ ψ−1({0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As an important geometric application, a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2) on a two- dimensional manifold N corresponds to the existence of an isometric immersion ( �N, g) → R3 Supported by the National Science Foundation of China (NSFC 11601370) and the Alexander von Humboldt Foundation of Germany 1 2 of the universal covering � N into the Euclidean 3-space with constant mean curvature (see [3,21] and references therein for more geometric backgrounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' A series of works of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Ammann and his group [3–9] have provided a brief picture of how variational method is employed to the study of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' From the view point of analysis, as it was pointed out in [3], standard variational methods do not directly imply the existence of a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This is due to the criticality of the nonlinearity in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Indeed, the exponent n∗ = 2n n−1 is critical in the sense that the Sobolev embedding involved is precisely the one for which the compactness is lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Similar to the idea in solving the Yamabe problem, it is possible to find a criterion which recovers compactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here the crucial observation is that a spinorial analogue of Aubin’s inequality holds (see [7]): λ+ min(N, [g], σ) := inf ˜g∈[g] λ+ 1 (˜g)Vol(N, ˜g) 1 n ≤ λ+ min(Sn, [gSn], σSn) = n 2ω 1 nn (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) where [g] = {f 2g : f ∈ C1(N), f ≥ 0, supp f = N} is the (generalized) conformal class of g and, for each ˜g ∈ [g], λ+ 1 (˜g) > 0 denotes the smallest positive eigenvalue of the associated Dirac operator DN ˜g , (Sn, gSn, σSn) is the n-dimensional sphere equipped with its canonical metric gSn and spin structure σSn, and ωn is the volume of (Sn, gSn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The quantity λ+ min(N, [g], σ) is known as the B¨ar-Hijazi-Lott invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' One of the main results obtained in [3] shows that if inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) is strict then the spinorial Yamabe problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) has a nontrivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' However, the strict inequality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) is only verified for some special cases and a general result is still lacking (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' [6,9,22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The purpose of this paper is to establish existence results for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) on products of compact spin manifolds without knowing whether the strict inequality in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, we are also interested in the effect of the curvature tensors in our existence results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, given closed spin manifolds (M1, g(1), σ1) and (M2, g(2), σ2), with fixed spin structures, let us consider a family of metrics gε on the product N = M1 × M2 defined by gε = ε−2g(1) ⊕ g(2), ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let m1 and m2 be the dimensions of M1 and M2 respectively, we will be interested in solutions of the spinorial Yamabe equation on the product manifold (N, gε): DN gεφ = |φ|n∗−2 gε φ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) where n∗ = 2n n−1 and n := dim N = m1+m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In the sense of separation of variables, we restrict our study to spinor fields of the form φ = ψ ⊗ϕ ∈ C1(N, S(N)) such that ψ ∈ C1(M1, S(M1)) and ϕ ∈ C1(M2, S(M2)) are spinor fields on M1 and M2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In order to describe our main results, it is useful to recall some notation and definitions in differential geometry, see for instance [17, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For a closed Riemannian m-manifold (M, g), let exp : TM → M be the exponential map defined on the tangent bundle TM of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since M is closed, there exists r > 0 such that expξ : Br(0) ⊃ Rm ∼= TξM → Br(ξ) ⊂ M is a diffeomorphism for any ξ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Throughout the paper, Br(0) will denote the ball in Rm centered at 0 with radius r and, for ξ ∈ M, Br(ξ) will denote the ball in M centered at ξ with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For vector fields X, Y, W, Z on M, the Riemannian curvature tensor R is given by R(X, Y, W, Z) = g(∇X∇Y W, Z) − g(∇Y ∇XW, Z) − g(∇[X,Y ]W, Z) 3 where ∇ is the Riemannian connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For an orthonormal basis {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , em} of TξM, ξ ∈ M, the Ricci tensor Ric : TξM × TξM → R is given by the trace of R, that is Ric(X, Y ) = m � j=1 R(ej, X, Y, ej) and the scalar curvature and Gaussian curvature will be denoted by Scalg(ξ) and Kg(ξ) re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' On spin manifolds, since the tangent bundle is embedded in the bundle of Clifford algebra, vector fields have two different actions on spinors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' the Clifford multiplications and the covariant derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here, to distinguish these two actions on a spinor ψ, we denote respec- tively ∂j ·g ψ the Clifford multiplication of ∂j and ∇∂jψ the covariant derivative, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , m, with respect to the background metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We also adopt the notation fε ≲ gε for two ε-dependent functions fε and gε, when there exists a constant C > 0 independent of ε such that fε ≤ Cgε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1 An illustrative Example We begin by describing an example when n = 3, where N = Σ × S1 for a closed Riemann surface Σ and S1 the standard circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let g denote the background metric on Σ and dτ denote the standard metric on S1 with total length 2π, then we are concerned with the product metrics ε−2g ⊕ dτ, ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We also equip Σ and S1 with spin structures σΣ and σS1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In this setting, we have n∗ = 3 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) reads as DN gεφ = |φ|gεφ, φ : N → S(N) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5) where the spinor bundle S(N) is identified as the tensor product S(N) = S(Σ) ⊗ S(S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since the associated Dirac operator on S1 is simply i d dτ , and since we are looking for a solution of the form φ = ψ ⊗ ϕ, we can take ϕ = e−iλτ to be an eigen-spinor on S1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' i d dτ ϕ = λϕ) for some eigenvalue λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then φ = ψ ⊗ e−iλτ is a solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5) if and only if ψ : Σ → S(Σ) is a solution to the following reduced equation (see Section 3 for a detailed explanation) εDΣ g ψ + λωΣ C ·g ψ = |ψ|gψ, on Σ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6) where ωΣ C is the chirality operator in the Clifford bundle Cl(TΣ) and “ ·g ” denotes the Clifford multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We also introduce the following equation which corresponds to a limiting equation to prob- lem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6) as ε goes to zero: DgR2ψ + λωC ·gR2 ψ = |ψ|ψ, ψ : R2 → S(R2) ∼= C2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) where gR2 denotes the standard Euclidean metric and ωC = i∂1 ·gR2 ∂2 is the corresponding chirality operator with “ ·gR2 ” being the Clifford multiplication and ∂1 = ∂ ∂x1, ∂2 = ∂ ∂x2 are the canonical base in the tangent bundle TR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The blow-up profiles (the so-called concentration phenomenon) appearing in solution se- quences of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6) (as ε → 0) are described by rescaled solutions of the above limit equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As we will see in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) has a variational structure, of strongly indefinite type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Ground state solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' solutions with minimal energy) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) can be obtained via a standard linking arguments, moreover, these solutions decay exponentially at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 4 Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) is invariant by translation, we denote B the set of ground state solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) having maximum modulus at the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' |ψ(0)| = maxx∈R2 |ψ(x)| for ψ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As explained in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4, B is compact in W 1,q(R2, S(R2)), q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now we are ready to state the results for N = Σ × S1: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' There exists ε0 > 0 such that, for any ε ∈ (0, ε0), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6) has a solution ψε ∈ C1(Σ, S(Σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Furthermore, there is a maximum point yε ∈ Σ such that (1) for a constant c > 0, |ψε(ξ)|g ≲ exp � − c ε dist(ξ, yε) � , for all ξ ∈ Σ where dist is the distance induced by the metric g;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (2) as ε → 0, up to a subsequence, the transformed spinor zε(x) ≡ ψε ◦ expyε(εx) converges uniformly to a ground state solution z0 ∈ B of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) and yε → y0 in (Σ, g) such that Θ(y0, z0) = max (y,ψ)∈Σ×B Θ(y, ψ), where Θ : Σ × B → R is a functional defined by Θ(y, ψ) = Kg(y) 36 � R2 |ψ|3|x|2dx + Kg(y) 12 Re � R2 � (x2∇∂1 − x1∇∂2)ψ, (x2∂1 − x1∂2) ·gR2 ψ � dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) The result presented above implies that the concentrating behavior of a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5) is affected by the Gaussian curvature Kg on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since Σ × B is compact, a maximizer of Θ does exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The spin structure of Euclidean spaces is quite explicit and equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) can be rewritten in matrix notation as � γ1 ∂ ∂x1 + γ2 ∂ ∂x2 � ψ + λγ3ψ = |ψ|ψ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9) where γk, k = 1, 2, 3, are the 2 × 2 Pauli matrices γ1 = � 0 i i 0 � , γ2 = � 0 1 −1 0 � , γ3 = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Passing to polar coordinates in R2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (x1, x2) �→ (r, ϑ), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9) reads as \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 e−iϑ� i ∂ ∂r + 1 r ∂ ∂ϑ � ψ2 = � |ψ1|2 + |ψ2|2 ψ1 − λψ1 eiϑ� i ∂ ∂r − 1 r ∂ ∂ϑ � ψ1 = � |ψ1|2 + |ψ2|2 ψ2 + λψ2 5 where ψ = �ψ1 ψ2 � ∈ C2, and this suggests the following special ansatz (see [18]) ψ(r, ϑ) = � v(r)eiSϑ iu(r)ei(S+1)ϑ � , r > 0, ϑ ∈ [0, 2π) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10) with u, v real-valued and S ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Plugging such ansatz into the functional Θ, we find that the second term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' we get a simpler expression of Θ as Θ(y, ψ) = πKg(y) 18 � ∞ 0 � u2 + v2� 3 2r3dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This would give a simplified view of the concentration phenomenon in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' y0 must be a global maximum point of the Gaussian curvature Kg on (Σ, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Unfortunately, we find no evidence that ground state solutions to the limit equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) should be in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This may lead to conjecture that, up to translations and certain group actions (for instance, the multi- plication by eiω for ω ∈ [0, 2π]) , the ground state solution ψ to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) is uniquely determined and takes the form of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10) (or may be other symmetric ansatz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We add that solutions of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10) to the 2D nonlinear Dirac equation with Kerr-type critical nonlinearity |ψ|2ψ have been studied in [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Furthermore, ground state solutions to the spinorial Yamabe equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) on Rn has been recently classified in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2 Full statement of the results In order to develop an existence and concentration theory for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) on general product spaces N = M1 × M2, we first introduce explicitly the spinor bundle S(N) over N in terms of the factors M1 and M2, that is S(N) = � (S(M1) ⊕ S(M1)) ⊗ S(M2) both m1 and m2 are odd, S(M1) ⊗ S(M2) else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In this setting, there is no difficulty to understand that a spinor φ ∈ S(N) has the form φ = ψ⊗ϕ where ϕ ∈ S(M2) and ψ = ψ1 ⊕ ψ2 ∈ S(M1) ⊕ S(M1) if both m1 and m2 are odd, and ψ ∈ S(M1) if m1 or m2 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Motivated by the example mentioned previously, we impose the following hypothesis on the second factor (M2, g(2), σ2): (H) there is a solution ϕλ with constant length |ϕλ|g(2) = 1 of the Dirac equation DM2 g(2)ϕ = λϕ, for some λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Under this hypothesis, φ = ψ ⊗ ϕλ is a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) if and only if ψ is a solution of the following reduced equation ε ˜DM1 g(1)ψ + λωM1 C g(1) ψ = |ψ|n∗−2 g(1) ψ, on M1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) where ˜DM1 g(1) = �DM1 g(1) ⊕ −DM1 g(1) both m1 and m2 are odd, DM1 g(1) if m1 is even, 6 and ωM1 C ·g(1) denotes the action of the chirality operator with respect to the metric g(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In case m1 is odd and m2 is even, one may interchange M1 and M2 to get the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The corresponding limit equation associated to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) is the following one: ˜DgRm1 ψ + λωC ·gRm1 ψ = |ψ|n∗−2ψ on Rm1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) where gRm1 is the standard Euclidean metric, ˜DgRm1 = � DgRm1 ⊕ −DgRm1 if both m1 and m2 are odd, DgRm1 if m1 is even, and ωC = i[ m1+1 2 ]∂1 ·gRm1 · · · ·gRm1 ∂m1 is the corresponding chirality operator in the Clifford algebra Cl(TRm1) with “ ·gRm1 ” being the Clifford multiplication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∂1 = ∂ ∂x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , ∂m1 = ∂ ∂xm1 are the canonical base in the tangent bundle TRm1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) can be regarded as the Euler-Lagrange equation for the functional L(ψ) = 1 2 � Rm1 � ˜DgRm1 ψ + λωC ·gRm1 ψ, ψ � dx − 1 n∗ � Rm1 |ψ|n∗dx defined for ψ ∈ H1/2(Rm1, S(Rm1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since the spectrum of the linear differential operator ˜Dλ = ˜DgRm1 + λωC is given by Spec( ˜Dλ) = � −∞, −|λ| � ∪ � |λ|, +∞ � , the above functional is strongly indefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Several techniques have been introduced to handle such situations (see for instance [10, 11, 15, 27, 35] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Notice that we have set n = m1 + m2 and n∗ = 2n n−1, we see the Sobolev embedding H1/2(Rm1, S(Rm1)) ֒→ Ln∗(Rm1, S(Rm1)) is locally compact (due to n∗ < m∗ 1 = 2m1 m1−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This means that Palais-Smale sequences for the functional L possess local strong convergence in Ln∗ and thus in H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In the framework of Concentration-Compactness theory, one obtains the existence of ground state solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) via standard variational arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For ease of notation, we still denote B the set of all ground state solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) satis- fying |ψ(0)| = maxx∈Rm1 |ψ(x)| for ψ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We know from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4 that B is compact in W 1,q(Rm1, S(Rm1)), q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then our main result reads as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Assume dim M1 = m1 ≥ 2 and (M2, g(2), σ2) satisfies hypothesis (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let m1 be even when n = m1 + m2 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then there exists ε0 > 0 such that, for any ε ∈ (0, ε0), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) has a solution of the form ψε ∈ C1(M1, S(M1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Furthermore, there is a maximum point yε ∈ M1 such that (1) for a constant c > 0, |ψε(ξ)|g(1) ≲ exp � − c ε dist(1)(ξ, yε) � , for all ξ ∈ M1 where dist(1) is the distance induced by the metric g(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 7 (2) as ε → 0, up to a subsequence, the transformed spinor zε(x) ≡ ψℓ ◦ expyε(εx) converges uniformly to a ground state solution z0 ∈ B of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) and yε → y0 in (M1, g(1)) such that Θ(y0, z0) = max (y,ψ)∈M1×B Θ(y, ψ), where Θ : M1 × B → R is a functional defined by Θ(y, ψ) = 1 12n � Rm1 Ricy(x, x)|ψ| 2n n−1dx + 1 12 � j,k Re � Rm1 Ry(ej, x, x, ek)(∇∂kψ, ∂j ·gRm1 ψ)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The case m1 = m2 = 1, which corresponds to N = S1 × S1, is rather simple and does not reflect the effect of curvature since the problem is reduced to an ordinary differential equation on the first circle (see [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' a) The hypothesis (H) is rather harmless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This is satisfied by a large class of manifolds including the circle S1, the m-spheres and many conformally flat manifolds (see for instance [6,9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' b) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3 does not treat directly the case of m1 odd and m2 even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As the statements show, the concentration phenomenon will be obtained on the even dimensional factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This is mainly due to the specific spin-representation (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) below) when n = m1+m2 is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' c) Analogously to Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2, if we have additionally a symmetric characterization of ground state solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) then the functional Θ can be intensively simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In higher dimensions, it is still not very clear how to give a general symmetric characteriza- tion of the solutions (while the Laplacian commutes with rotations, it is not the case for Dirac operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' However in 3D Dirac equations, it is known that there is one candidate symmetric ansatz (see in [20,36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Finally we would like to compare problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) with its counterpart of elliptic type: − ε2∆gu + u = up−1, u > 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) on a smooth closed Riemannian manifold (M, g), with dim M = m ≥ 3 and p ∈ (2, 2m m−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' An interesting observation is that our results about (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) depend on both curvature tensors and the ground state solutions of the limit equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In fact, the point y0 ∈ M1 in our result locates the blow-up (or concentration) while the ground state solution z0 ∈ B gives the profile of the blowing-up bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Unlike the well-known results about (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) in [16,19,32] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' where only scalar curvature enters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The functional Θ in our theorems appear to be complicated and mysterious, in particular, the second term in Θ is not very clear to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This is because there is very little information available for the ground state solutions of the strongly indefinite problems (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since the limit equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) on the Euclidean space Rm is explicitly understood, for which there exists a unique positive solution (up to translations) and is radially symmetric, the results for positive solutions of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) only depends on geometric quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' It would be very interesting to characterize those ground state solutions for Dirac equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12), and then one may have a better understanding for the functional Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3 Outline of the paper The proof of our results will be carried out in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' First, in Section 2, we recall some preliminaries and also fix our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, we will formally provide the spinor bun- dles and Dirac operators on product spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In Section 3, we explicitly introduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) as the reduced equation of Spinorial Yamabe equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4), and set up the associated variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The existence result is standard since the nonlinearity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) has subcriti- cal growth (in the sense of Sobolev embedding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In this case the concentration phenomenon manifests itself in the difficulty of locating the behavior of the solutions when the parameter ε is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here, the key point lies in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8 which provides a refined upper bound esti- mate for the critical levels in our variational framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In fact, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8 makes it possible to compute a asymptotic expansion of the critical levels in terms of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Section 4 is devoted to give the complete proof of our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In this section, we first collect basic proper- ties of the ground state solutions of the limit equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) since they perform as bubbles in the concentration phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The analysis of the concentration phenomenon is quite delicate and it requires a careful asymptotic expansion of the critical levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' With the help of a well adopted spinor bundle trivialization (the so-called Bourguignon-Gauduchon trivialization) and our Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8, we establish the critical level expansion in terms of ε in which the effect of curvature tensors enters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 2 Spinor bundles and Dirac operators on product spaces In this section, we collect some basic notations from spin geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Instructional material can be found in [29, Chapter I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 5 and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1 Algebraic preliminaries Let us denote by {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , em} the canonical basis of an oriented Euclidean space V and by Cℓ(V ) the complex Clifford algebra of V with its multiplication being denoted by “·”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In case the dimension m of V is even, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' m = 2k, the Clifford algebra is isomorphic to the alge- bra M(2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' C) of all complex 2k × 2k matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence Cℓ(V ) has precisely one irreducible module, the spinor module S2k with dimC S2k = 2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' When restricting this representation to the even subalgebra Cℓ0(V ), the module S2k splits into two irreducible unitary representations S2k = S+ 2k ⊕ S− 2k, given by the eigensubspaces of the endomorphism ωV C := ike1 · · · em to the eigenvalues ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In the context, we will call ωV C the “chirality operator” or the “complex volume element”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In case m is odd, that is m = 2k+1, the Clifford algebra Cℓ(V ) is isomorphic to M(2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' C)⊕ M(2k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And thus, we obtain two 2k-dimensional irreducible spinor modules S0 2k+1 and S1 2k+1 if we project the Clifford multiplication onto the first and second component respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Simi- lar to the splitting in even dimensions, the two modules S0 2k+1 and S1 2k+1 can be distinguished by the chirality operator ωV C := ik+1e1 · · ·em in the sense that on Sj 2k+1 it acts as (−1)j, j = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' It will cause no confusion if we simply identify S0 2k+1 and S1 2k+1 as the same vector space, that is S2k+1 = S0 2k+1 = S1 2k+1, and equip them with Clifford multiplications of opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let V and W be two oriented Euclidean spaces with dim V = m1 and dim W = m2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We denote Cℓ(V ) and Cℓ(W) the associated Clifford algebras of V and W respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By an 9 abuse of notation, we use the same symbol “·” for the Clifford multiplication in Cℓ(V ), Cℓ(W) and in their representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As it is well known, the Clifford algebra of the sum of two vector spaces is the Z2-graded tensor product of the Clifford algebras of the two summands, that is Cℓ(V ⊕ W) = Cℓ(V )�⊗Cℓ(W) (see [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Therefore, we can construct the spinor module of V ⊕ W from those of V and W as Sm1+m2 = � (Sm1 ⊕ Sm1) ⊗ Sm2 if both m1 and m2 are odd, Sm1 ⊗ Sm2 if m1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) Here, as mentioned before, we simply excluded the case where m1 is odd and m2 is even because V and W can be interchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As for the representation of Clifford multiplications on Sm1+m2, let ξ ∈ V , ζ ∈ W, ϕ ∈ Sm2 and ψ = ψ1 ⊕ ψ2 ∈ Sm1 ⊕ Sm1 if both m1 and m2 are odd, and ψ ∈ Sm1 if m1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We set (ξ ⊕ ζ) · (ψ ⊗ ϕ) = (ξ · ψ) ⊗ ϕ + (ωV C · ψ) ⊗ (ζ · ϕ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2) where, in case both m1 and m2 odd, ξ · ψ = (ξ · ψ1) ⊕ (−ξ · ψ2) and ωV C · ψ = i(ψ2 ⊕ −ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' With this notation, one easily checks that (ξ ⊕ ζ) · (ξ ⊕ ζ) · (ψ ⊗ ϕ) = −|ξ ⊕ ζ|2(ψ ⊗ ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thus Sm1+m2 is a nontrivial Cℓ(V ⊕ W)-module of (complex) dimension 2[ m1+m2 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, in case m1 + m2 is even, the splitting of Sm1+m2 into half-spinor modules is given by S+ m1+m2 = � (ψ ⊕ ψ) ⊗ ϕ : ψ ∈ Sm1, ϕ ∈ Sm2 � , S− m1+m2 = � (ψ ⊕ −ψ) ⊗ ϕ : ψ ∈ Sm1, ϕ ∈ Sm2 � for both m1 and m2 odd and S+ m1+m2 = (S+ m1 ⊗ S+ m2) ⊕ (S− m1 ⊗ S− m2), S− m1+m2 = (S+ m1 ⊗ S− m2) ⊕ (S− m1 ⊗ S+ m2) for both m1 and m2 even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Next, let us turn to the manifold setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let (M1, g(1)) and (M2, g(2)) be two oriented Rie- mannian manifolds of dimensions m1 and m2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We henceforth suppose that both manifolds are equipped with a fixed spin structure (for details about spin structures, we refer to [21, 29] or to the well written self-contained introduction [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This induces a unique spin structure on the Riemannian product (N = M1 × M2, g = g(1) ⊕ g(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Indeed, let πM1 and πM2 denote the projections on M1 and M2, the tangent bundle of N can be decomposed as TN = π∗ M1TM1 ⊕ π∗ M2TM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For simplicity, we omit the projections and write TN = TM1 ⊕ TM2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And such splitting is orthogonal with respect to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence the frame bundle of N can be reduced to a SO(m1) × SO(m2)-principal bundle, and this is isomorphic to the product of the frame bundles over M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2 The Dirac operator Fix the spin structures σM1 and σM2, let us consider the Clifford bundles (with Clifford multipli- cations) (Cl(TM1), ·g(1)), (Cl(TM2), ·g(2)) and spinor bundles S(M1), S(M2) over M1 and M2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' From the previous considerations in the algebraic settings, we know for the spinor bundles that S(N) = � (S(M1) ⊕ S(M1)) ⊗ S(M2) if both m1 and m2 are odd, S(M1) ⊗ S(M2) if m1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For X ∈ TM1, Y ∈ TM2, ϕ ∈ Γ(S(M2)) and ψ = ψ1 ⊕ ψ2 ∈ Γ(S(M1) ⊕ S(M1)) for both m1 and m2 odd and ψ ∈ Γ(S(M1)) for m1 even, we have (X ⊕ Y ) ·g (ψ ⊗ ϕ) = (X ·g(1) ψ) ⊗ ϕ + (ωM1 C g(1) ψ) ⊗ (Y ·g(2) ϕ) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) where in case m1 and m2 odd we set X ·g(1) ψ = (X ·g(1) ψ1) ⊕ (−X ·g(1) ψ2) and ωM1 C g(1) ψ = i(ψ2 ⊕ −ψ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let ∇S(M1) and ∇S(M2) be the (lifted) Levi-Civita connections on S(M1) and S(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By ∇S(M1)⊗S(M2) = ∇S(M1) ⊗ IdS(M2) + IdS(M1) ⊗ ∇S(M2) we mean the tensor product connection on S(M1)⊗S(M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' If we take {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , Xm1} a locally positively oriented orthonormal frame of (M1, g(1)), then the Dirac operator on M1 is (locally) defined by DM1 g(1) = �m1 j=1 Xj ·g(1) ∇S(M1) Xj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Similarly, if we take {Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , Ym2} a locally positively oriented orthonormal frame of (M2, g(2)), we have DM2 g(2) = �m2 j=1 Yj ·g(2) ∇S(M2) Yj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Evidently, in the product setting, {X1 ⊕ 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , Xm1 ⊕ 0, 0 ⊕ Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , 0 ⊕ Ym2} is a local section of the frame bundle of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) yields DN g := m1 � j=1 (Xj ⊕ 0) ·g(1) ∇S(M1)⊗S(M2) Xj⊕0 + m2 � j=1 (0 ⊕ Yj) ·g(2) ∇S(M1)⊗S(M2) 0⊕Yj = ˜DM1 g(1) ⊗ IdS(M2) + (ωM1 C g(1) IdS(M1)) ⊗ DM2 g(2) which defines the Dirac operator on N = M1 × M2, where ˜DM1 g(1) = DM1 g(1) ⊕ −DM1 g(1) if both m1 and m2 are odd and ˜DM1 g(1) = DM1 g(1) if m1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For the case m1 + m2 even, we have the decomposition S(N) = S(N)+ ⊕ S(N)− and, moreover, when restrict DN g on those half-spinor spaces we get DN g : Γ(S(N)±) → Γ(S(N)∓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 3 Variational setting In what follows, we always consider the case N = M1 × M2, m1 = dim M1 ≥ 2 and m2 = dim M2 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In order to give unified expressions in odd and even cases, we will write simply S(N) = ˜S(M1) ⊗ S(M2) with ˜S(M1) = � S(M1) ⊕ S(M1) if m1 is odd, S(M1) if m1 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 11 and denote ψ ⊗ ϕ for a spinor field in S(N) when no confusion can arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let us consider the warped metric gε := ε−2g(1) ⊕ g(2), where ε > 0 is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Accord- ing to the discussions in the previous section, we know for the Dirac operators that DN gε = ˜DM1 ε−2g(1) ⊗ IdS(M2) + (ωM1 C ε−2g(1) Id˜S(M1)) ⊗ DM2 g(2) where ωM1 C denotes the chirality operator and ”·ε−2g(1)” denotes the Clifford multiplication on M1 associated to the conformal metric ε−2g(1) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Turning to the nonlinear problems, let us denote | · |ε−2g(1) and | · |g(2) the natural hermitian metrics on S(M1) and S(M2) respectively and | · |gε the induced metric on S(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Recall the notation n = m1 + m2 and n∗ = 2n n−1, we can expand the spinorial Yamabe equation DN gεφ = |φ|n∗−2 gε φ, φ = ¯ψ ⊗ ϕ ∈ S(N) into ( ˜DM1 ε−2g(1) ¯ψ) ⊗ ϕ + (ωM1 C ε−2g(1) ¯ψ) ⊗ (DM2 g(2)ϕ) = � | ¯ψ|ε−2g(1)|ϕ|g(2) �n∗−2 ¯ψ ⊗ ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) We will now show how the assumption (H) on (M2, g(2), σM2) enters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In fact, if M2 pos- sesses a nontrivial eigenspinor ϕM2 of constant length for some λ ̸= 0, then by substituting ¯ψ ⊗ ϕM2 into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) we get the equivalent problem ˜DM1 ε−2g(1) ¯ψ + λωM1 C ε−2g(1) ¯ψ = � | ¯ψ|ε−2g(1) �n∗−2 ¯ψ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2) which is sitting on M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here, we adopt the convention that λ > 0 since (up to a change of orientation on M1) the proof for λ < 0 is exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Notice that the Dirac operator behaves very nicely under conformal changes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' [24, 26]): Let g0 and g = e2ug0 be two conformal metrics on a Riemannian spin m-manifold M, then there exists an isomorphism of vector bundles ι : S(M, g0) → S(M, g) which is a fiberwise isometry such that DM g � ι(ψ) � = ι � e− m+1 2 uDM g0 � e m−1 2 uψ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thus Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2) is conformally equivalent to ε ˜DM1 g(1)ψ + λωM1 C g(1) ψ = |ψ|n∗−2 g(1) ψ on M1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) where ωM1 C ·g(1) denotes the action of the chirality operator with respect to the metric g(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1 The configuration space and the Lagrangian Plainly, our goal is reduced to find solutions of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) for varying ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Notice that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) is well-defined on M1, and n∗ = 2n n−1 < 2m1 m1−1 = m∗ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' It is not necessary to carry the super- and sub-scripts in (M1, g(1)), ˜DM1 g(1) and ωM1 C during the proofs, hence in order to simplify the notation, we consider the following generalized problem ε ˜Dgψ + aωC ·g ψ = f(|ψ|g)ψ, ψ : M → ˜S(M) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) 12 on a closed spin m-manifold (M, g, σ), where a > 0 is a constant and f : [0, ∞) → [0, ∞) is the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Clearly, f(s) = sn∗−2 is our primary concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Unless otherwise stated, we will occasionally drop the subscript of | · |g on ˜S(M) for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In the sequel, for q > 1, let us denote Lq := Lq(M, ˜S(M)) with the norm | · |q q := � M | · |qdvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, for q = 2, we have L2 is a Hilbert space with inner product (·, ·)2 = Re � M(·, ·)dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For each fixed ε > 0, let Aε := ε ˜Dg + aωC·g denote the self-adjoint operator on L2 with domain D(Aε) = H1 ≡ H1(M, ˜S(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' It is already known that, on a closed spin manifold (M, g), the spectrum Spec(Aε) ⊂ (−∞, −a] ∪ [a, +∞) is symmetric about the origin and consists of an unbounded discrete sequence of eigenvalues (with finite multiplicity for each eigenvalue), see [34] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thus, from the classical spectral theory of elliptic self-adjoint operators, we may choose a complete orthonormal basis ψε ±1, ψε ±2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' of L2 consisting of the eigenspinors of Aε, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Aεψε ±k = λ±k(ε)ψε ±k and the spectrum Spec(Aε) will be denoted as · · ≤ λ−2(ε) ≤ λ−1(ε) < 0 < λ1(ε) ≤ λ2(ε) ≤ · · · , where each eigenvalue appears with its multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, we have λk(ε) = −λ−k(ε) and |λ±k(ε)| → +∞ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Define the unbounded operator |Aε|s : L2 → L2, s ≥ 0, by |Aε|sψ = ∞ � k=−∞ |λk(ε)|sαkψε k where ψ = �∞ k=−∞ αkψε k ∈ L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In this way, we can introduce the domain of |Aε|s in L2 as H s := � ψ = ∞ � k=1 αkψε k ∈ L2 : ∞ � k=−∞ |λk(ε)|2s|αk|2 < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' It is worth pointing out that H 1 2 coincides with the Sobolev space of order 1 2, that is W 1 2 ,2(M, ˜S(M)) (see for instance [1,3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, we can equip H := H 1 2 with the inner product ⟨ψ, ϕ⟩ε := 1 εm Re � M � |Aε|1/2ψ, |Aε|1/2ϕ � dvolg (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5) and the induced norm ∥ · ∥ε such that (H, ⟨·, ·⟩ε) becomes a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Remark that, in the above notations, we have emphasized the dependence on the parameter ε because it appears in the differential operator and its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The dual space of H will be denoted by H∗ = W − 1 2 ,2(M, ˜S(M)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Identifying H with H∗, we will use the same notation ⟨·, ·⟩ε to denote the norm on H∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Recall that we have an (·, ·)2-orthogonal decomposition L2 = L+ ε ⊕ L− ε , ψ = ψ+ + ψ− with L+ ε := +∞ � k=1 ker(Aε − λk(ε)) and L− ε := ∞ � k=1 ker(Aε − λ−k(ε)) 13 so that Aε is positive definite on L+ ε and negative definite on L− ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This leads to the orthogonal decomposition of H with respect to the inner product ⟨·, ·⟩ε as H = H+ ε ⊕ H− ε , H± ε = H ∩ L± ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' On the Banach space Lq, q > 1, we introduce the new norm |ψ|q,ε = � 1 εm � M |ψ|qdvolg � 1 q for each ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, recall m∗ = 2m m−1, we have (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' [34]) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' If ε > 0 is small, then for any q ∈ [2, m∗] the embedding IdH : (H, ∥ · ∥ε) ֒→ (Lq, | · |q,ε) is bounded independent of ε, that is, there exists cq > 0 such that |ψ|q,ε ≤ cq∥ψ∥ε for all ψ ∈ H and all ε > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, the embedding is compact for q ∈ [2, m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The proof of this lemma is a combination of the Lichnerowicz formula ( ˜Dg)2 = ∇∗∇ + 1 4Scalg, where Scalg is the scalar curvature of (M, g), and an application of the Calder´on-Lions inter- polation theorem (see [33]) between H 1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In fact, for ψ ∈ C∞(M, ˜S(M)), we have ��|Aε|ψ ��2 2 = � M ε2|∇ψ|2 + � a2 + ε2 4 Scalg � |ψ|2dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6) It follows that, for large positive ε, the influence of the scalar curvature Scalg enters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In this situation, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6) may no longer be a norm when Scalg possesses certain negative parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And this fact will probably affect the embedding constant of H = H 1 2 ֒→ Lm∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' With Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1, we introduce the following conditions for the nonlinearity f in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) which contains the power function f(s) = sp−2 as a special case: (f1) f(0) = 0, f ∈ C1(0, ∞) and f ′(s) > 0 for s > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (f2) there exists p ∈ (2, m∗), c > 0 such that f ′(s)s ≤ c(1 + sp−2) for s ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (f3) there exists θ > 0 such that f(s) ≤ 1 θf ′(s)s for s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let F(s) be the primitive function of f(s)s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' F(s) = � s 0 f(t)t dt, it is standard to see that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) is the Euler-Lagrange equation of the functional Lε(ψ) = 1 εm � M �1 2(Aεψ, ψ) − F(|ψ|) � dvolg = 1 2 � ∥ψ+∥2 ε − ∥ψ−∥2 ε � − 1 εm � M F(|ψ|)dvolg (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) defined on H = H+ ε ⊕ H− ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1, we have Lε ∈ C2(H, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We emphasize that the relation between F and f is F ′(s) = f(s)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And by an abuse of nota- tion, we simply write ψ = ψ+ +ψ− for the orthogonal decomposition of H without mentioning its dependence on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' However, one should always keep in mind that, for different values of ε, this decomposition of a spinor ψ is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2 The reduced action Let us begin with the compactness of the functional Lε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For each ε > 0 small, Lε satisfies the (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' )c-condition for c ≥ 0, that is, Lε(ψn) → c L′ ε(ψn) → 0 � ⇒ {ψn} possesses a convergent subsequence in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, ψn → 0 if and only if c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since L′ ε(ψn) → 0 in H∗, we have c + o(∥ψn∥ε) = Lε(ψn) − 1 2L′ ε(ψn)[ψn] = 1 εm � M 1 2f(|ψn|)|ψn|2 − F(|ψn|)dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) We also have o(∥ψn∥ε) = L′ ε(ψn)[ψ+ n − ψ− n ] = ∥ψn∥2 ε − 1 εm Re � M f(|ψn|)(ψn, ψ+ n − ψ− n )dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By (f1)-(f3), one checks easily that for arbitrarily small δ > 0 there exists cδ > 0 such that f(s)s ≤ δs + cδ(f(s)s2) p−1 p and F(s) ≤ 1 θ+2f(s)s2 for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' From this, together with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1, we obtain ∥ψn∥2 ε ≤ 1 εm � M f(|ψn|)|ψn| · |ψ+ n − ψ− n |dvolg + o(∥ψn∥ε) ≤ δ|ψn|2,ε|ψ+ n − ψ− n |2,ε + cδ � 1 εm � M f(|ψn|)|ψn|2dvolg � p−1 p |ψ+ n − ψ− n |p,ε + o(∥ψn∥ε) ≤ δc2∥ψn∥2 ε + Cp,δ � c + o(∥ψn∥ε) � p−1 p ∥ψn∥ε + o(∥ψn∥ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9) By suitable choosing δ > 0 small, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9) that {ψn} is bounded in H with respect to the norm ∥ · ∥ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And due to the compact embedding of H ֒→ Lp, one easily checks that {ψn} is compact in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Finally, we mention that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9) together imply: ψn → 0 if and only if c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' One may see from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) that the quadratic part in the functional Lε is of strongly indefinite type, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' positive- and negative-definite on infinite dimensional subspaces of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, in order to obtain a critical point of Lε, it is now crucial to find a suitable min-max scheme for Lε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Fortunately, due to our requests on the nonlinearity f (see conditions (f1)-(f3)), we have a very good geometric behavior of Lε in the following sense: (i) Since the function F is non-negative, for each fixed u ∈ H+ ε , the functional Lε(u + ·) : H− ε → R, w �→ Lε(u + w) is anti-coercive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lε(u + w) → −∞ as ∥w∥ε → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 15 (ii) Since F ′′(s) = f ′(s)s+f(s) > 0 for s > 0, the quadratic form L′′ ε(u+w)[·, ·] is negative definite on H− ε , in other words, the above functional Lε(u + ·) is strictly concave on H− ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (iii) Since (f3) implies that f(s) ≥ csθ for some constant c > 0 and all s ≥ 1, the function F has super-quadratic growth at infinity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' F(s) ≥ c 2+θs2+θ as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And hence, for each fixed u ∈ H+ ε \\ {0}, Lε(tu + w) → −∞ as |t| + ∥w∥ε → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10) Combining all the above three properties, for each u ∈ H+ ε \\ {0}, we are able to maximize the functional Lε on R+u ⊕ H− ε , where R+ = (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We remark that u is varying in the space H+ ε \\ {0}, so we can restrict ourselves to the choice u ∈ S+ ε := {u ∈ H+ ε : ∥u∥ε = 1} without changing the maxima of Lε on R+u ⊕ H− ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By (f1)-(f2), one deduces that F(s) ≤ δ 2s2 + Cδsp for arbitrarily small δ > 0 and hence (by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) max R+u⊕H− ε Lε ≥ max t>0 Lε(tu) ≥ max t>0 �1 − δ 2 t2 − Cδtp� = τ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) Therefore, we have found a candidate min-max scheme for Lε: we first maximize the functional Lε on R+u ⊕ H− ε and then minimize with respect to u ∈ H+ ε \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By summarizing the above observations, we have the following basic conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For each ε > 0 small the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1) There exists χε ∈ C1(H+ ε , H− ε ) such that for u ∈ H+ ε w ∈ H− ε , w ̸= χε(u) ⇒ Lε(u + w) < Lε(u + χε(u));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' that is χε(u) is the unique maximizer of the functional w �→ Lε(u+w) on H− ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, for u ∈ H+ ε ∥χε(u)∥2 ε ≤ 2 εm � M F(|u|)dvolg and L′ ε(u + χε(u))[w] ≡ 0 for all w ∈ H− ε ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (2) If {un} is a (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' )-sequence for the reduced functional Iε : H+ ε → R, Iε(u) = Lε(u + χε(u)), then {un + χε(un)} is a (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' )-sequence for Lε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3) There exists uε ∈ H+ ε such that I′ ε(uε) = 0 and Iε(uε) = µε := inf γ∈Γε max t∈[0,1] Iε(γ(t)) > 0, where Γε = � γ ∈ C([0, 1], H+ ε ) : γ(0) = 0, Iε(γ(1)) < 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, ¯ψε = uε + χε(uε) is a non-trivial critical point of Lε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Similar assertions can be found in [15, Section 2] where a certain abstract theory has been set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We only mention here that the existence and uniqueness of the maximizer χε(u) in assertion (1) follows directly from the anti-coerciveness and strict concavity of the functional Lε(u+·) on H− ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The C1-smoothness of χε is a consequence of the Implicit Function Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We next provide an upper bound estimate of the critical level µε obtained in the above proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For every u ∈ H+ ε \\ {0}, the map Iε,u : R → R, Iε,u(t) = Iε(tu), is of class C2 and satisfies Iε,u(0) = I′ ε,u(0) = 0 and I′′ ε,u(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, there holds I′ ε,u(t) = 0, t > 0 =⇒ I′′ ε,u(t) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' First of all, we can use the fact that I′ ε,u(t) = L′ ε(tu + χε(tu))[u] to see that Iε,u is of class C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By (f1) − (f3) there holds Iε(tu) ≥ Lε(tu) ≥ (1 − δ)t2 2 ∥u∥2 ε − Cp,δ tp∥u∥p ε ∀u ∈ H+ ε \\ {0} and t > 0, for any fixed δ > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence Iε(0) = 0 is a strict local minimum in a neighborhood of 0 ∈ H+ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And in particular, we can see that Iε,u(0) = I′ ε,u(0) = 0 and I′′ ε,u(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, to complete the proof, it is sufficient to show that I′ ε(u)[u] = 0, u ̸= 0 =⇒ I′′ ε (u)[u, u] < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) For simplicity, let us denote Ψε : H → R by Ψε(ψ) = 1 εm � M F(|ψ|)dvolg and set ψ = u+χε(u) and w = χ′ ε(u)[u] − χε(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By using L′ ε(u + χε(u))|H− ε ≡ 0, we see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) is a direct consequence of the following computation: I′′ ε (u)[u, u] = L′′ ε(ψ)[u + χ′ ε(u)[u], u] = L′′ ε(ψ)[ψ + w, ψ + w] = L′′ ε(ψ)[ψ, ψ] + 2L′′ ε(ψ)[ψ, w] + L′′ ε(ψ)[w, w] = I′ ε(u)[u] + � Ψ′ ε(ψ)[ψ] − Ψ′′ ε(ψ)[ψ, ψ] � + 2 � Ψ′ ε(ψ)[w] − Ψ′′ ε(ψ)[ψ, w] � −Ψ′′ ε(ψ)[w, w] − ∥w∥2 ε ≤ I′ ε(u)[u] − 1 εm � M f(|ψ|)f ′(|ψ|)|ψ|3 f(|ψ|) + f ′(|ψ|)|ψ|dvolg − ∥w∥2 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) Since f(s)f ′(s)s ≤ (f(s) + f ′(s)s)2 for all s ≥ 0, and the equality holds iff s = 0, we have 0 < f(|ψ|)f ′(|ψ|)|ψ|3 f(|ψ|) + f ′(|ψ|)|ψ| ≤ f(|ψ|)|ψ|2 + f ′(|ψ|)|ψ|3 for ψ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Therefore, the integration in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) is well-defined due to (f1)-(f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The above lemma indicates that, for each u ∈ H+ ε \\ {0}, the function Iε,u(·) has at most one critical point tε,u ∈ (0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10), one checks that such tε,u exists for every u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And then we can define a natural constraint for the reduced functional Iε as Nε := � u ∈ H+ ε \\ {0} : I′ ε(u)[u] = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 17 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5 also implies Nε is a smooth submanifold of codimension 1 in H+ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And conse- quently, the critical point found in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4 (3) can be characterized by µε := Lε( ¯ψε) = inf u∈H+ ε \\{0} max ψ∈Ru⊕H− ε Lε(ψ) = inf u∈H+ ε \\{0} max t>0 Iε(tu) = inf u∈Nε Iε(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='14) For later purposes, it is worth to remind that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) implies the existence of some τ0 > 0 independent of ε such that µε ≥ τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In what follows, we intend to pass to the limit ε → 0 and consider the asymptotic behavior of the min-max level µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The idea is to provide an upper bound estimate around an arbitrary (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' )-sequence, so that one may substitute certain test spinors in the functional Lε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Without loss of generality, we assume that {φε} ⊂ H is an arbitrary sequence such that c1 ≤ Lε(φε) ≤ c2 and ∥L′ ε(φε)∥ε → 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='15) as ε → 0 for some constants c1, c2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here, we will identify the dual space H∗ with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Under (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='15), we have: (1) ∥φε∥ε is uniformly bounded in ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (2) ∥φ− ε − χε(φ+ ε )∥ε ≤ O � ∥L′ ε(φε)∥ε � as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3) I′ ε(φ+ ε ) → 0 as ε → 0 in the dual space of H+ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For the boundedness, we recall that Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1 implies that the embedding constant for H ֒→ Lp∗ is independent of ε, and hence the arguments in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3 can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For (2), let us first set zε = φ+ ε + χε(φ+ ε ) and vε = φ− ε − χε(φ+ ε ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then we have vε ∈ H− ε and, by the definition of χε, 0 = L′ ε(zε)[vε] = − � χε(φ+ ε ), vε � ε − 1 εm Re � M f(|zε|)(zε, vε)dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since ∥L′ ε(φε)∥ε → 0 as ε → 0, it follows that o(∥vε∥ε) = L′ ε(φε)[vε] = − � φ− ε , vε � − 1 εm Re � M f(|φε|)(φε, vε)dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And hence, we get o(∥vε∥ε) = ∥vε∥2 ε + 1 εm Re � M f(|φε|)(φε, vε)dvolg − 1 εm Re � M f(|zε|)(zε, vε)dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='16) Since the map ψ → F(|ψ|) is convex by (f1), we have 1 εm Re � M f(|φε|)(φε, vε)dvolg − 1 εm Re � M f(|zε|)(zε, vε)dvolg ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thus, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='16), we can infer that ∥vε∥ε ≤ O � ∥L′ ε(φε)∥ε � as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In order to check (3) we compute I′ ε(φ+ ε ) = L′ ε � φ+ ε +χε(φ+ ε ) � , which implies that ∥I′ ε(φ+ ε )∥ε → 0 as ε → 0 is a direct consequence of (2) and the C2 smoothness of Lε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 18 Next, let us introduce the functional Kε : H+ ε → R by Kε(u) = I′ ε(u)[u].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, it is clear that Kε is C1 and its derivative is given by the formula K′ ε(u)[w] = I′ ε(u)[w] + I′′ ε (u)[u, w] for u, w ∈ H+ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We also have Nε = K−1 ε (0) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13), there holds K′ ε(u)[u] ≤ 2Kε(u) − 1 εm � M f(|ψ|)f ′(|ψ|)|ψ|3 f(|ψ|) + f ′(|ψ|)|ψ|dvolg, for u ∈ H+ ε where ψ = u + χε(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By virtue of (f3), one checks easily that f(s)f′(s)s f(s)+f′(s)s ≥ θ θ+1f(s) for s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thus the above estimate implies K′ ε(u)[u] ≤ 2Kε(u) − θ (θ + 1)εm � M f(|u + χε(u)|)|u + χε(u)|2dvolg, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='17) for u ∈ H+ ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For the sequence {φε} in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='15), there exists {tε} ⊂ R such that tεφ+ ε ∈ Nε and |tε − 1| ≤ O � ∥I′ ε(φ+ ε )∥ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We begin with the observation that F(s) ≤ 1 θ+2f(s)s2 for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Due to the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='15) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6 (3), it follows directly that lim inf ε→0 1 εm � M f � |φ+ ε + χε(φ+ ε )| � |φ+ ε + χε(φ+ ε )|2dvolg ≥ c0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='18) for some constant c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let us set ηε : (0, ∞) → R by ηε(t) = Kε(tφ+ ε ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' One easily checks that tη′ ε(t) = K′ ε(tφ+ ε )[tφ+ ε ] for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='17) and Taylor’s expansion, we get tη′ ε(t) ≤ 2ηε(1) − θ (θ + 1)εm � M f � |φ+ ε + χε(φ+ ε )| � |φ+ ε + χε(φ+ ε )|2dvolg + C|t − 1| (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='19) for t close to 1 with some C > 0 independent of ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here we have used the uniform boundedness of η′ ε(t) on bounded intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Noticing that ηε(1) = I′ ε(φ+ ε )[φ+ ε ] → 0 as ε → 0, we conclude from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='19) that there exists a small constant δ > 0 such that η′ ε(t) ≤ −δ for all t ∈ (1 − δ, 1 + δ) and ε small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, notice that Kε(u) equals to the value of I′ ε,u(1), it follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5 that ηε(1 − δ) > 0 and ηε(1 + δ) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, by the Inverse Function Theorem, tε := η−1 ε (0) exists and uε := tεφ+ ε ∈ Nε ∩ R+φ+ ε is well-defined for all ε small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Furthermore, since |η′ ε(t)−1| is bounded by a constant, say cδ > 0, on (1 − δ, 1 + δ), we consequently get ∥uε − φ+ ε ∥ε = |η−1 ε (0) − η−1 ε (Kε(φ+ ε ))| · ∥φ+ ε ∥ε ≤ cδ|Kε(φ+ ε )| · ∥φ+ ε ∥ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now the conclusion follows from Kε(φ+ ε ) ≤ O � ∥I′ ε(φ+ ε )∥ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 19 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For the sequence {φε} in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='15), there exists {uε} such that uε ∈ Nε and ∥φε − uε − χε(uε)∥ε ≤ O(∥L′ ε(φε)∥ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, max t>0 Iε(tφ+ ε ) = Iε(uε) ≤ Lε(φε) + O � ∥L′ ε(φε)∥2 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' To see this, let uε = tεφ+ ε be as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7 and set zε = φ+ ε + χε(φ+ ε ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then one obtains from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6 that ∥φε − uε − χε(uε)∥ε ≤ ∥φε − zε∥ε + |tε − 1| · ∥φ+ ε ∥ε + ∥χε(φ+ ε ) − χε(uε)∥ε ≤ O � ∥L′ ε(φε)∥ε � + O � ∥I′ ε(φ+ ε )∥ε � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='20) where we have used an easily checked inequality ∥χε(φ+ ε ) − χε(uε)∥ε ≤ ∥χ′ ε(τφ+ ε )∥H+ ε →H− ε · ∥φ+ ε − uε∥ε = O(|tε − 1|) for some τ between tε and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Observing that I′ ε(φ+ ε ) = L′ ε(zε), and using the C2 smoothness of Lε, we have ∥I′ ε(φ+ ε )∥ε = ∥L′ ε(zε)∥ε ≤ ∥L′ ε(φε)∥ε + O(∥φε − zε∥ε) = O(∥L′ ε(φε)∥ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='20) implies ∥φε − uε − χε(uε)∥ε ≤ O(∥L′ ε(φε)∥ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now, by Talyor’s expansion, we can obtain Lε(φε) = Lε(uε + χε(uε)) + L′ ε(uε + χε(uε))[φε − uε − χε(uε)] + O � ∥L′ ε(φε)∥2 ε � = Iε(uε) + I′ ε(uε)[φ+ ε − uε] + O � ∥L′ ε(φε)∥2 ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since uε = tεφ+ ε ∈ Nε, we have I′ ε(uε)[φ+ ε − uε] ≡ 0 and this implies the last estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 4 Proof of the main results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1 The equation on Euclidean spaces: the bubbles We consider solutions to the equation ˜DgRmψ + aωC ·gRm ψ = f(|ψ|)ψ on Rm (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) belonging to the class W 1 2,2(Rm, ˜S(Rm)), where ˜S(Rm) = � S(Rm) ⊕ S(Rm) m is odd, S(Rm) m is even, ˜DgRm = DgRm ⊕ −DgRm if m is odd and ˜DgRm = DgRm if m is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' These solutions will cor- respond to ”bubbles” or test spinors for our variational problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We also assume the nonlinear function f satisfies conditions (f1)-(f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 20 First of all, let us set A = ˜DgRm + aωC·gRm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By a straightforward calculation we see that A is a self-adjoint operator on L2 with spectrum Spec(A) = (−∞, −a] ∪ [a, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Following Amann [2] let (Eλ)λ∈R be the spectral resolution of A and define the orthogonal projections by P = � 0 −∞ dEλ, Q = � ∞ 0 dEλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then the decomposition of E = W 1 2,2(Rm, ˜S(Rm)) = E+ ⊕ E− is induced by E− = E ∩ P(L2) and E+ = E ∩ Q(L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We introduce the following operators S = � 0 −∞ |λ| 1 2dEλ and T = � ∞ 0 |λ| 1 2dEλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' and the new inner product on E ⟨ψ, ϕ⟩ = Re � (S + T)ψ, (S + T)ϕ � 2, ψ, ϕ ∈ E with ∥ · ∥ denoting the corresponding norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We easily see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) is the Euler-Lagrange equation of the functional Φ(ψ) = 1 2 � ∥Qψ∥2 − ∥Pψ∥2� − � Rm F(|ψ|)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' If {ψn} ⊂ E is a bounded sequence such that Φ′(ψn) → 0 and lim inf n→∞ � Rm f(|ψn|)|ψn|2dx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' then there exists ψ ̸= 0 with Φ′(ψ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let B0 R denote the open ball of radius R centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' If lim n→∞ sup y∈Rm � y+B0 R |ψn|2dx = 0, ∀R > 0, then a result of Lions [31] implies ψn → 0 in Lq for all q ∈ (2, m∗) and therefore � Rm f(|ψn|)|ψn|2dx → 0, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Passing to a subsequence, we have lim inf n→∞ � yn+B0 R |ψn|2dx > 0 for some R > 0 and {yn} ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Using the invariance of the operator A under translations, we can find R > 0 and a new sequence { ˜ψn} such that Φ′( ˜ψn) → 0 and lim inf n→∞ � B0 R | ˜ψn|2dx > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Up to a subsequence if necessary, we have ˜ψn ⇀ ψ and the compact embedding E ֒→ L2 loc shows that ψ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By taking the limit in Φ′( ˜ψn) → 0, we obtain Φ′(ψ) = 0 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 21 Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' There exists a nontrivial solution ψ ∈ E to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1 and the boundedness argument in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3, this is a direct conse- quence of [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now we may define µ0 = inf � Φ(ψ) : ψ ∈ E \\ {0} s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Φ′(ψ) = 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) Since the super-quadratic part in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2) has subcritical growth at infinity, one easily sees that µ0 > 0 is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, analogously to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5-Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8, the following reduction principle holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1) There exists a C1 map h : E+ → E− such that Φ(u+h(u)) = max v∈E− Φ(u+v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (2) The critical points of the functional J(u) = Φ(u+h(u)) and those of Φ are in one-to-one correspondence via the map u �→ u + h(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (3) For each u ∈ E+ \\ {0}, the map t �→ J(tu) has only one maximum on (0, +∞) and µ0 = inf u∈E+\\{0} max t>0 J(tu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4) For any bounded sequence {zn} ∈ E such that Φ(zn) → c > 0 and Φ′(zn) → 0, there holds max t>0 J(tz+ n ) ≤ Φ(zn) + O(∥Φ′(zn)∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' From elliptic estimates and and bootstrap arguments, we deduce that (weak) solutions of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) with bounded energy are uniformly bounded in ∩q≥2W 1,q(Rm, ˜S(Rm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, we have the following Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Setting B = � ψ ∈ E : Φ(ψ) = µ0, Φ′(ψ) = 0, |ψ(0)| = maxRm |ψ| � , the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (1) B is compact in W 1,2(Rm, ˜S(Rm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (2) There exist C, c > 0 such that |ψ(x)| ≤ C exp(−c|x|) for all ψ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Clearly, B is closed in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We show that an arbitrary sequence ψn ∈ B, n ∈ N, in B has a convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In fact, since {ψn} is bounded, we have ψn ⇀ ψ0 along a subsequence in E with clearly ψ0 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, one has ψn → ψ0 in Lq loc for q ∈ [2, m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, by the fact that µ0 = � Rm 1 2f(|ψn|)|ψn|2 − F(|ψn|)dx = � Rm 1 2f(|ψ0|)|ψ0|2 − F(|ψ0|)dx, it is easy to see that for every ǫ > 0 there is R > 0 such that lim sup n→∞ � |x|≥R 1 2f(|ψn|)|ψn|2 − F(|ψn|)dx ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) 22 Setting zn = ψn − ψ0 we obtain, using Φ′(ψn) = Φ′(ψ0) = 0, ⟨Qψn, Qzn⟩ + ⟨Pψn, Pzn⟩ − Re � Rm f(|ψn|)(ψn, Qzn − Pzn)dx = 0 and ⟨Qψ0, Qzn⟩ + ⟨Pψ0, Pzn⟩ − Re � Rm f(|ψ0|)(ψ0, Qzn − Pzn)dx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, we get ∥zn∥2 = Re � Rm f(|ψn|)(ψn, Qzn − Pzn)dx + on(1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5) where we have used Re � Rm f(|ψ0|)(ψ0, Qzn − Pzn)dx → 0 as n → ∞ which holds because zn ⇀ 0 in Lq for q ∈ [2, m∗].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Recall that, by (f1)-(f3), for arbitrarily small δ > 0 there exists cδ > 0 such that f(s)s ≤ δs + cδ(f(s)s2) p−1 p and F(s) ≤ 1 θ+2f(s)s2 for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thus, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5) imply ∥zn∥2 ≤ δ|ψn|2|Qzn − Pzn|2 + cδ � � |x|≥R f(|ψn|)|ψn|2dx � p−1 p |Qzn − Pzn|p + on(1) ≤ δC∥zn∥ + Cδ � ǫ + on(1) � p−1 p ∥zn∥ + on(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Due to the arbitrariness of δ, ǫ > 0, one sees ∥zn∥ → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now we prove the compactness in W 1,2(Rm, ˜S(Rm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As a consequence of the equation (recall A = ˜DgRm + aωC·gRm) Aψn = f(|ψn|)ψn and Aψ0 = f(|ψ0|)ψ0, we have |A(ψn − ψn)|2 = ��f(|ψn|)ψn − f(|ψ0|)ψ0 �� 2 ≤ ��f(|ψn|)(ψn − ψ0) �� 2 + ��(f(|ψn|) − f(|ψ0|))ψ0 �� 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' From |ψn|∞ ≤ C and ψn → ψ0 in E we deduce ��f(|ψn|)(ψn − ψ0) �� 2 ≤ f(C)|ψn − ψ0|2 = on(1), as n → ∞ and � Rm ��(f(|ψn|) − f(|ψ0|))ψ0 ��2dx = � |x|≤R ��(f(|ψn|) − f(|ψ0|))ψ0 ��2dx + oR(1) = on(1) + oR(1) as n → ∞ because |ψ0(x)| → 0 as |x| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Therefore, we get |A(ψn − ψn)|2 → 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ψn → ψ0 in W 1,2(Rm, ˜S(Rm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' To see the exponential decay, we rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) as ˜DgRmψ = −aωC ·gRm ψ + f(|ψ|)ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 23 Applying the operator ˜DgRm to both sides of the above equation and noting that ˜D2 gRm = −∆, we find ∆ψ = � a2 − f(|ψ|)2� ψ − ∇f(|ψ|) ·gRm ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now using the fact that ∆|ψ|2 = 2 Re(∆ψ, ψ) + 2|∇ψ|2 and that Re(∇f(|ψ|) ·gRm ψ, ψ) ≡ 0, we obtain ∆|ψ|2 = 2 � a2 − f(|ψ|)2� |ψ|2 + 2|∇ψ|2 ≥ 2 � a2 − f(|ψ|)2� |ψ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now for ψ ∈ B, by |ψ(x)| → 0 as |x| → ∞, we may take R > 0 large enough so that ∆|ψ|2 ≥ a2|ψ|2 for all |x| ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let Γ(x) = e−a|x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' One checks easily that ∆Γ − a2Γ < 0, for |x| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By taking C > 0 be such that |ψ(x)|2 ≤ C · Γ(x) holds on |x| = R, we may consider U = |ψ|2 − C · Γ and get ∆U = ∆|ψ|2 − C · ∆Γ > a2U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By elliptic estimates and the comparison principle, we can easily conclude that U(x) ≤ 0 for all |x| ≥ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence we obtain the exponential decay of |ψ(x)| at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Finally, thanks to the compactness of B, we see that the exponential decay holds uniformly for all ψ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2 Bourguignon-Gauduchon trivialization Our proof relies on the construction of a test spinor on M in order to show the concentration behavior under the conditions (f1)-(f3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The test spinor comes from a spinor on Rm being cut- off and transplanted to M so that it has support in a small neighborhood of an arbitrary point y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We first need to recall a construction from the paper [7] of Ammann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' To begin with, we fix a spinor field ψ ∈ B arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let r < inj(M)/2 where inj(M) > 0 is the injectivity radius of M, and let η ∈ C∞ c (Rm, [0, 1]) be such that |∇η| ≤ 2/r, η(x) = 1 for |x| ≤ r and η(x) = 0 for |x| ≥ 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, we define ϕε : Rm → Sm by ϕε(x) = η(x)ψε(x) where ψε(x) = ψ(x/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6) In order to transplant the test spinor on M, we recall the Bourguignon-Gauduchon-trivialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here we fix y ∈ M arbitrarily, and let (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , xm) be the normal coordinates given by the ex- ponential map expy : Rm ∼= TyM ⊃ U → V ⊂ M, x �→ ξ = expy(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For ξ ∈ V , let G(ξ) = (gij(ξ))ij denote the corresponding metric at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since G(ξ) is symmetric and positive definite, the square root B(ξ) = (bij(ξ))ij of G(ξ)−1 is well defined, symmetric and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' It can be thought of as linear isometry B(ξ) : (Rm ∼= Texp−1 y (ξ)U, gRm) → (TpV, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 24 We obtain an isomorphism of SO(m)-principal bundles: PSO(U, gRm) φ � � PSO(V, g) � TyM ⊃ U expy � V ⊂ M where φ{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , vm} = {Bv1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , Bvm} for an oriented frame {v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , vm} on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Notice that φ commutes with the right action of SO(m), hence it induces an isomorphism of spin structures: Spin(m) × U = PSpin(U, gRm) � � PSpin(V, g) ⊂ PSpin(M) � TyM ⊃ U expy � V ⊂ M Thus we obtain an isomorphism between the spinor bundles S(U) and S(V ): S(U) := PSpin(U, gRm) ×ρ Sm −→ S(V ) := PSpin(V, g) ×ρ Sm ⊂ S(M) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) where (ρ, Sm) is the complex spin representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let {∂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , ∂m} be the canonical frame on the Euclidean space, where ∂i = ∂ ∂xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Setting ei = B(∂i) = � j bij∂j we obtain an orthonormal frame {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , em} of (TV, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Via (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7), we also have ei ·g ¯ψ = B(∂i) ·g ¯ψ = ∂i ·gRm ψ for ψ ∈ Sm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now a spinor field ϕ ∈ Γ(˜S(U)) corresponds via the isomorphim (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7) to a spinor ¯ϕ ∈ Γ(˜S(V )), and we will keep this notation for various spinor fields to represent such correspon- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In particular, since the spinors ϕε ∈ Γ(˜S(U)) have compact support in U they correspond to spinors ¯ϕε ∈ Γ(˜S(M)) with compact support in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In the sequel, in order to simplify the notation, we use ∇ and ¯∇, respectively, for the Levi- Civita connections on (TU, gRm) and (TV, g) and for the natural lifts of these connections to the spinor bundles ˜S(U) and ˜S(V ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By abuse of notation, we write D and ¯D instead of the Dirac operators acting on Γ(˜S(U)) and Γ(˜S(V )), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By [7, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='2] there holds ¯D ¯ϕε = Dϕε + W ·g ¯ϕε + X ·g ¯ϕε + � i,j (bij − δij)∂i ·gRm ∇∂jϕε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) with W ∈ Γ(Cl(TV )) and X ∈ Γ(TV ) given by W = 1 4 � i,j,k i̸=j̸=k̸=i � α,β biα(∂αbjβ)b−1 βk ei ·g ej ·g ek, and X = 1 4 � i,k �¯Γi ik − ¯Γk ii � ek = 1 2 � i,k ¯Γi ikek;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' here (b−1 ij )ij denotes the inverse matrix of B, and ¯Γk ij := g( ¯∇eiej, ek).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In what follows we identify x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' , xm) ∈ U ⊂ Rm with � i xiei ∈ TyM for notational convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 25 As remarked in [17, 30], in the neighborhood of y, the metric g and its determinant have the following expansion: gij(expy x) = δij − 1 3Ry(ei, x, x, ej) + O(|x|3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9) � det g(expy x) = 1 − 1 6Ricy(x, x) + O(|x|3) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10) where R(ei, ej, ek, el) = g(∇ei∇ejek, el) − g(∇ej∇eiek, el) − g(∇[ei,ej]ek, el) is the Riemannian curvature tensor and Ric(v, w) = �m i=1 R(ei, v, w, ei) is the Ricci curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Observing that B = (G−1) 1 2, as in [7], we have bij = δij + 1 6Ry(ei, x, x, ej) + O(|x|3), W = O(|x|3) and X = O(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) The main results of this subsection will be the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let ¯ϕε be as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6), then ∥L′ ε( ¯ϕε)∥ε ≤ O(ε2) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By the definition of ϕε in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6), together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8), one checks easily that εDϕε = ε∇η ·gRm ψε + εηDψε = ε∇η ·gRm ψε − aηωC ·Rm ψε + ηf(|ψε|)ψε (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) and ε ¯D ¯ϕε + aωC ·g ¯ϕε − f(| ¯ϕε|) ¯ϕε = J1 + J2 + · · · + J6 ∈ H∗ where J1 = ε∇η ·gRm ψε, J2 = η · � f(| ¯ψε|) − f(| ¯ϕε|) � ¯ψε, J3 = εηW ·g ¯ψε, J4 = εηX ·g ¯ψε, J5 = εη � i,j (bij − δij)∂i ·gRm ∇∂jψε, J6 = ε � i,j (bij − δij)∂jη∂i ·gRm ψε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In the following estimates we use that the support of η is contained in B2r(0) ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Using the exponential decay in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' we have: ∥J1∥H→R ≲ � 1 εm � B2r(y) ��ε∇η ·gRm ψε ��2dvolg � 1 2 ≲ � 1 εm � r≤|x|≤2r ��εψε ��2dx � 1 2 ≲ ε � � r ε ≤|x|≤ 2r ε ��ψ ��2dx � 1 2 ≲ ε exp � − c · r ε � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 26 ∥J2∥H→R ≲ � 1 εm � B2r(y)\\Br(y) �� ¯ψε ��2dvolg � 1 2 + � 1 εm � B2r(y)\\Br(y) �� ¯ψε ��pdvolg � p−1 p ≲ � � r ε≤|x|≤ 2r ε |ψ|2dx � 1 2 + � � r ε≤|x|≤ 2r ε |ψ|pdx � p−1 p ≲ exp � − c · r ε � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∥J3∥H→R ≲ � 1 εm � B2r(y) ��εW ·g ¯ψε ��2dvolg � 1 2 ≲ � 1 εm � |x|≤2r � ε|x|3|ψε| �2dx � 1 2 ≲ ε4 � � |x|≤ 2r ε |x|6|ψ|2dx � 1 2 ≲ ε4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∥J4∥H→R ≲ � 1 εm � B2r(y) ��εX ·g ¯ψε ��2dvolg � 1 2 ≲ � 1 εm � |x|≤2r � ε|x||ψε| �2dx � 1 2 ≲ ε2 � � |x|≤ 2r ε |x|2|ψ|2dx � 1 2 ≲ ε2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∥J5∥H→R ≲ � 1 εm � |x|≤2r � ε|x|2|∇ψε| �2dx � 1 2 ≲ ε2 � � |x|≤ 2r ε |x|4|∇ψ|2dx � 1 2 ≲ ε2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∥J6∥H→R ≲ � 1 εm � |x|≤2r � ε|x|2|ψε| �2dx � 1 2 ≲ ε3 � � |x|≤ 2r ε |x|4|ψ|2dx � 1 2 ≲ ε3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Here, in the estimates of J1 and J2, the constant c > 0 comes from the exponential decay in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' From all these, we finally obtain ∥L′ ε( ¯ϕε)∥ε ≤ O(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We define a functional Θ : M × B → R by Θ(y, ψ) = 1 6 � Rm Ricy(x, x) �1 2f(|ψ|)|ψ|2 − F(|ψ|) � dx + 1 12 � i,j Re � Rm Ry(ei, x, x, ej)(∇∂jψ, ∂i ·gRm ψ)dx Then we have the following Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let µ0 be the ground state energy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3) and ¯ϕε be defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6), then Lε( ¯ϕε) = µ0 − ε2Θ(y, ψ) + o(ε2) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='12) again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' we have 1 εm � M (ε ¯D ¯ϕε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ϕε)dvolg + a εm � M (ωC ·g ¯ϕε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ϕε)dvolg = I1 + I2 + · · · + I7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' where I1 = Re εm � M η · (ε∇η ·gRm ψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ψε)dvolg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 27 I2 = 1 εm � M f(| ¯ϕε|)| ¯ϕε|2dvolg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I3 = 1 εm � M η2� f(| ¯ψε|) − f(| ¯ϕε|) � | ¯ψε|2dvolg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I4 = Re εm � M η2 · (εW ·g ¯ψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ψε)dvolg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I5 = Re εm � M η2 · (εX ·g ¯ψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ψε)dvolg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I6 = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re εm � M η2(bij − δij)(ε∂i ·gRm ∇∂jψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ψε)dvolg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I7 = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re εm � M η∂jη · (bij − δij)(ε∂i ·gRm ψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ψε)dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Analogously to the arguments in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5, we shall use (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11) and the fact (∇η ·gRm ψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ¯ψε) ∈ iR to obtain I1 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I2 = 1 εm � |x|≤r f(|ψε|)|ψε|2dx − 1 6 εm � |x|≤r f(|ψε|)|ψε|2Ricy(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x)dx + o(ε2) = � Rm f(|ψ|)|ψ|2dx − ε2 6 � Rm f(|ψ|)|ψ|2Ricy(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x)dx + o(ε2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' |I3| ≲ 1 εm � r≤|x|≤2r |ψε|2dx + 1 εm � r≤|x|≤2r |ψε|pdx ≲ � r ε≤|x|≤ 2r ε |ψ|2dx + � r ε≤|x|≤ 2r ε |ψ|pdx ≲ o(ε2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' |I4| ≲ 1 εm � |x|≤2r ε|x|3|ψε|2dx ≲ ε4 � |x|≤ 2r ε |x|3|ψ|2dx ≲ ε4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I5 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I6 = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re 6 εm � |x|≤r Ry(ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ej)(ε∂i ·gRm ∇∂jψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ψε)dx + o(ε2) = ε2 6 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re � Rm Ry(ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ej)(∂i ·gRm ∇∂jψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ψ)dx + o(ε2) = −ε2 6 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re � Rm Ry(ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ej)(∇∂jψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∂i ·gRm ψ)dx + o(ε2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' I7 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 28 Combining all these estimates and the fact ψ ∈ B, we deduce that Lε( ¯ϕε) = I1 + I2 + · · · + I7 2 − 1 εm � M F(| ¯ϕε|)dvolg = 1 2 � Rm f(|ψ|)|ψ|2dx − � Rm F(|ψ|)dx − ε2Θ(y, ψ) + o(ε2) = µ0 − ε2Θ(y, ψ) + o(ε2) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3 Characterization of the concentration profile From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='14), we deduce that µε = inf u∈H+ ε \\{0} max ψ∈Ru⊕H− ε Lε(ψ) = inf u∈H+ ε \\{0} max t>0 Iε(tu) = inf u∈Nε Iε(u) is a critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4, we take ¯ψε to be the corresponding critical point of Lε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, as it was mentioned in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11), we find 1 εm � M 1 2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg = µε ≥ τ0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) for some τ0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In what follows, for any ξ ∈ M and r > 0, Br(ξ) ⊂ M denotes the ball of radius r with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' There exist yε ∈ M, r0, δ0 > 0 such that lim inf ε→0 1 εm � Bεr0(yε) | ¯ψε|2dvolg ≥ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Assume to the contrary that for any r > 0 sup ξ∈M 1 εm � B2εr(ξ) | ¯ψε|2dvolg → 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='14) For each ξ ∈ M, we choose a smooth real cut-off function βξ,ε ≡ 1 on Bεr(ξ) and supp βξ,ε ⊂ B2εr(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, for s ∈ (0, 1), we consider qs = 2 + (m∗ − 2)s ∈ (2, m∗) and we have � B2εr(ξ) |βξ,ε ¯ψε|qsdvolg ≤ � � B2εr(ξ) |βξ,ε ¯ψε|2dvolg �1−s� � B2εr(ξ) |βξ,ε ¯ψε| 2m m−1dvolg �s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Taking s = 2 m∗ , we obtain from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1 that � 1 εm � B2εr(ξ) |βξ,ε ¯ψε|m∗dvolg �s ≤ C∥βξ,ε ¯ψε∥2 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 29 We now cover M by balls of radius εr such that any point ξ ∈ M is contained in at most KM balls, where KM does not depend on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In fact, one may take KM = 1 + dimM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Consequently we have 1 εm � M | ¯ψε|qsdvolg ≤ C · KM � sup ξ∈M � B2εr(ξ) |βξ,ε ¯ψε|2dvolg �1−s ∥ ¯ψε∥2 ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since ∥ ¯ψε∥ε is bounded, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='14) that | ¯ψε|qs,ε → 0, and since 2 < qs < m∗, we see easily that | ¯ψε|q,ε → 0 for all q ∈ (2, m∗) which contradicts (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' limε→0 µε = µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By the estimates obtained in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6, we only need to show that lim inf ε→0 µε ≥ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For this purpose, let us take a sequence {yε} ⊂ M and constants r0, δ0 > 0 such that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7 is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Given 0 < r < inj(M)/2 arbitrarily, let βε ∈ C∞(M, [0, 1]) be such that βε ≡ 1 on Br(yε) and supp βε ⊂ B2r(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Via the Bourguignon-Gauduchon trivialization between the spinor bundles S(Br(yε)) → S(Br(0)) and the rescaling x �→ x ε on Rm, the spinor field βε ¯ψε corresponds to a spinor field zε(·) on B2r/ε(0) ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Since βε ¯ψε is bounded in H with respect to the norm ∥·∥ε, one sees easily that zε is bounded in W 1 2,2(BR(0), ˜S(BR(0))) for any R > 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By the fact � |x|≤ r ε |zε|m∗dx ≲ 1 εm � Br(yε) | ¯ψε|m∗dvolg ≲ ∥ ¯ψε∥ε < ∞ for all small ε, it follows that there exists z0 ∈ Lm∗(Rm, ˜S(Rm)) ∩ W 1 2,2 loc (Rm, ˜S(Rm)) such that zε ⇀ z0 in W 1 2,2 loc (Rm, ˜S(Rm)) weakly and zε → z0 in Lq loc(Rm, ˜S(Rm)) for 2 ≤ q < 2m m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let ϕ ∈ W 1 2,2(Rm, ˜S(Rm)) be such that supp ϕ is compact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' supp ϕ ⊂ B0 R for some R > 0 large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, by a similar identity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11), we have � Rm � ˜DgRmz0 + aωC ·gRm z0 − f(|z0|)z0, ϕ � dx = lim ε→0 � supp ϕ � ˜DgRmzε + aωC ·gRm zε − f(|zε|)zε, ϕ � dvolgΘε = lim ε→0 1 εm � BεR(ξε) � ε ˜Dgψε + aωC ·g ψε − f(|ψε|)ψε, ¯ϕε � dvolg = 0 where ϕε(x) = ϕ( x ε) and ¯ϕε is the spinor on M defined via the Bourguignon-Gauduchon trivialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence z0 satisfies ˜DgRmz0 + aωC ·gRm z0 = f(|z0|)z0 on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='15) As a consequence of (f1)-(f3) and by elliptic regularity, we have ˜DgRmz0 + aωC ·gRm z0 ∈ L m∗ m∗−1(Rm, ˜S(Rm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, combined with the Sobolev embedding L p p−1(Rm, ˜S(Rm)) ֒→ W − 1 2 ,2(Rm, ˜S(Rm)), we get z0 ∈ W 1 2 ,2(Rm, ˜S(Rm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 30 Now, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7, one sees that z0 is a non-trivial solution to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In virtue of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='13), we conclude that lim inf ε→0 Lε( ¯ψε) = lim inf ε→0 1 εm � M 1 2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg ≥ lim inf ε→0 � BR(0) 1 2f(|zε|)|zε|2 − F(|zε|)dx ≥ � BR(0) 1 2f(|z0|)|z0|2 − F(|z0|)dx where in the last inequality we have used the Fatou’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Due to the arbitrariness of R > 0, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3), we obtain lim inf ε→0 µε = lim inf ε→0 Lε( ¯ψε) ≥ µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now, we fix the sequence {yε} ⊂ M and constants r0, δ0 > 0 as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Up to a subsequence, we also assume that yε → y0 in M as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' As Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8 alluded, there exists a least-energy solution z0 of the limit equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1) corresponding to the weak limit of the solutions ¯ψε via Bourguignon-Gauduchon trivialization and rescaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Without loss of generality, one may assume yε to be the maximum point of | ¯ψε|, and then z0 ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Choose β ∈ C∞(M, [0, 1]) be such that β ≡ 1 on Br(y0) and supp β ⊂ B2r(y0) for some r < inj(M)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Set ¯φε = β¯z0,ε, where z0,ε(x) = z0(x/ε) and ¯z0,ε is the corresponding spinor field obtained via Bourguignon-Gauduchon trivialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We have the following asymptotic characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∥ ¯ψε − ¯φε∥ε → 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Setting wε = ¯ψε − ¯φε, we have |wε|q,ε → 0 as ε → 0 for all q ∈ (2, m∗) (otherwise, we can apply Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='7 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8 for {wε} instead of { ¯ψε} to get Lε( ¯ψε) ≥ 2µ0 which is absurd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' To proceed, let us go over the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8 again to see that µ0 = lim ε→0 1 εm � M 1 2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg = � Rm 1 2f(|z0|)|z0|2 − F(|z0|)dx Similar to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) (but here we need to transplant it on the manifold), it is easy to see that for every ǫ > 0 there is R > 0 large such that lim sup ε→0 � M\\BεR(yε) 1 2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='16) From the fact L′ ε( ¯ψε) = 0 and the estimate in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5, we have � ¯ψ+ ε , w+ ε � ε + � ¯ψ− ε , w− ε � ε − Re εm � M f(| ¯ψε|)( ¯ψε, w+ ε − w− ε )dvolg = 0 and �¯φ+ ε , w+ ε � ε + �¯φ− ε , w− ε � ε − Re εm � M f(|¯φε|)(¯φε, w+ ε − w− ε )dvolg = oε(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 31 Hence, we get ∥wε∥2 ε = Re εm � M f(| ¯ψε|)( ¯ψε, w+ ε − w− ε )dvolg + oε(1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='17) where we have used that |wε|q,ε → 0 as ε → 0 for all q ∈ (2, m∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Recall that, by (f1)- (f3), for arbitrarily small δ > 0 there exists cδ > 0 such that f(s) ≤ δ + cδ(f(s)s2) p−1 p and F(s) ≤ 1 θ+2f(s)s2 for all s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thus, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='17) and the Sobolev embeddings that ∥wε∥2 ε ≤ δC∥wε∥ε + Cδ � ǫ + oε(1) � p−1 p ∥wε∥ε + oε(1), for some constants C, Cδ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Due to the arbitrariness of δ, ǫ > 0, one sees ∥wε∥ε → 0 as ε → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For small ε > 0, there exist positive constants C, c > 0 such that | ¯ψε(ξ)| ≤ C exp � − c ε dist(ξ, yε) � , for all ξ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We first show that the family { ¯ψε} decays uniformly on M in the following sense: dist(ξ, yε) ε → ∞ =⇒ | ¯ψε(ξ)| → 0, as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='18) Indeed, since ¯ψε solves (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4) and Lε( ¯ψε) = µε → µ0, by the Sobolev embedding and bootstrap arguments, we deduce that {| ¯ψε|∞} is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Applying the operator ε ˜Dg on both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4), and using the Lichnerowicz formula, we find ε2∆g ¯ψε − ε2Scalg 4 ¯ψε = (a2 − f(| ¯ψε|)2) ¯ψε − ε∇gf(| ¯ψε|) ·g ¯ψε on M, where ∆g = divg∇g is the Laplace-Beltrami operator, ∇g is the gradient and Scalg is the scalar curvature with respect to the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Using the fact ∆g| ¯ψε|2 = 2 Re(∆g ¯ψε, ¯ψε) + 2|∇g ¯ψε|2, we have ε2∆g| ¯ψε|2 = 2 � a2 + ε2Scalg 4 − f(| ¯ψε|)2� | ¯ψε|2 + 2|∇g ¯ψε|2 ≥ −C| ¯ψε|2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='19) for some C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then, it follows from the local boundedness of sub-solutions (see for example [23, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='1]) that | ¯ψε(ξ)|2 ≤ C0 εm � Bε(ξ) | ¯ψε|2dvolg (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='20) with C0 > 0 independent of ξ ∈ M and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Assume by contradiction that there exist δ > 0 and ξε ∈ M such that dist(ξε, yε)/ε → ∞ as ε → 0 and | ¯ψε(ξε)| ≥ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='20) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9, we deduce, as ε → 0 δ ≤ C0 � 1 εm � Bε(ξε) | ¯ψε|2dvolg � 1 2 ≤ C0| ¯ψε − ¯φε|2,ε + C0 � 1 εm � Bε(ξε) |¯φε|2dvolg � 1 2 ≤ oε(1) + C0 � � B1 � exp−1 yε (ξε) ε � |z0|2dvolg � 1 2 → 0 32 which is a contradiction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' here ¯φε and z0 are as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In order to see the exponential decay, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='18) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='19), we can take R0 > 0 sufficiently large so that ε2∆g| ¯ψε|2 ≥ a2| ¯ψε|2 on M \\ BεR0(yε) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='21) for all ε > 0 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Our next strategy is to use the comparison principle and is very similar to the flat case (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='4): we first construct a sequence of positive functions ¯Γε on M \\ BεR0(yε) which decays exponentially, then we chose C > 0 such that | ¯ψε(ξ)|2 ≤ C · ¯Γ(ξ) for ξ ∈ ∂BεR0(yε) and finally, by setting Uε = | ¯ψε|2 − C · ¯Γ, we will show ε2∆gUε ≥ a2Uε, on M \\ BεR0(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For the sake of clarity, let us consider the (local) normal coordinates exp−1 yε : Br(yε) → Rm ∼= TyεM for a fixed r < inj(M)/5 and set vε(x) = | ¯ψε|2 ◦ expyε(εx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then we find 1 � det g(εx) � j,k ∂j � gjk(εx) � det g(εx)∂kvε � − a2vε ≥ 0 for R0 ≤ |x| ≤ r ε, where (gij)ij = G−1 is the inverse matrix of the metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' For each ε > 0 small, we take Γε(x) = e− a 2 Lε(|x|), where Lε : [0, ∞) → [0, ∞) is a function defined as Lε(ρ) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ρ 0 ≤ ρ < r ε, 1 2 � ρ + r ε sin �ε rρ − 1 �� + r 2ε r ε ≤ ρ < π + 1 ε r, π + 2 2ε r ρ ≥ π + 1 ε r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='22) According to the above definition, Γε ∈ C2(Rm \\{0}) is spherically symmetric, and so satisfies ∆Γε − a2 4 Γε = −a 2e− a 2 Lε(ρ)� L′′ ε(ρ) − a 2(L′ ε(ρ))2 + m − 1 ρ L′ ε(ρ) + a 2 � for ρ = |x| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now we see easily that ∆Γε − a2 4 Γε = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 − a(m − 1) 2ρ e− a 2 Lε(ρ) 0 ≤ ρ < r ε, − a2 4 e− a 2 Lε(ρ) ρ ≥ π + 1 ε r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And for r ε ≤ ρ < π+1 ε r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' from a direct computation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' we have L′′ ε(ρ) − a 2(L′ ε(ρ))2 + m − 1 ρ L′ ε(ρ) + a 2 = a 2 − a 8 � 1 + cos �ε rρ − 1 ��2 − ε 2r sin �ε rρ − 1 � + m − 1 2ρ � 1 + cos �ε rρ − 1 �� ≥ a 2 − a 8 � 1 + cos �ε rρ − 1 ��2 − ε 2r sin �ε rρ − 1 � + ε 2(π + 1)r � 1 + cos �ε rρ − 1 �� = a 2 − a 8 � 1 + cos �ε rρ − 1 ��2 − ε � (π + 1)2 + 1 2(π + 1)r sin �ε rρ − 1 − ϑ � + ε 2(π + 1)r 33 where ϑ = arcsin 1 √ (π+1)2+1 ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' π 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Therefore, for small ε > 0, we can derive that ∆Γε − a2 4 Γε < 0 on Rm \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Moreover, about the gradient of Γε we have |∇Γε(x)| Γε(x) ≤ a 2 for all |x| > 0 and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, by taking r smaller if necessary, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9) that 1 � det g(εx) � j,k ∂j � gjk(εx) � det g(εx)∂kΓε � ≤ ∆Γε + or(1) � |∆Γε| + |∇Γε| � ≤ a2Γε for R0 ≤ |x| ≤ r/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Next, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='18), we can fix C0 > 0 such that vε(x) ≤ C0 · Γε(x) = C0e− a 2 R0 for |x| = R0 and ε small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Via the Riemannian normal coordinates, we can transplant the functions Γε onto M \\ BεR0(yε) by introducing ¯Γε(ξ) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Γε �exp−1 yε (ξ) ε � ξ ∈ B5r(yε) \\ BεR0(yε), e− a(π+2) 4ε r M \\ B5r(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='23) Then, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='22), we see that ¯Γε > 0 is a well-defined C2-smooth function on M \\ BεR0(yε), and in particular, ε2∆g¯Γε − a2¯Γε ≤ 0 on M \\ BεR0(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Setting Uε = | ¯ψε|2 − C0 · ¯Γε(x), we can see from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='21) that ε2∆gUε − a2Uε ≥ 0 on M \\ BεR0(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And it is standard to show from the comparison principle that Uε ≤ 0 on M \\ BεR0(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Turning back to the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='23), we see that ¯Γε(ξ) ≤ exp � − c ε dist(ξ, yε) � for ξ ∈ Br(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' and ¯Γε(ξ) ≤ exp � − c · r ε � for ξ ∈ M \\ Br(yε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Setting dM = sup{dist(ξ1, ξ2) : ξ1, ξ2 ∈ M}, we see easily that dM ≥ inj(M) and so ¯Γε(ξ) ≤ exp � − c · r ε · dM dist(ξ, yε) � for all ξ ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' This implies the exponential decay for | ¯ψε| by simply taking the square root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 34 Recall that in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8, we have taken βε ∈ C∞(M, [0, 1]) be such that βε ≡ 1 on Br(yε) and supp βε ⊂ B2r(yε) for some r < inj(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Via the Bourguignon-Gauduchon trivialization between the spinor bundles S(Br(yε)) → S(Br(0)) and the rescaling x �→ x ε on Rm, the spinor field βε ¯ψε corresponds to a spinor field zε on B2r/ε(0) ⊂ Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' And, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='9 and bootstrap arguments, zε converges in W 1,q(Rm, ˜S(Rm)) to some z0 ∈ B as ε → 0, for q ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Thanks to the fast decay rate of ¯ψε, in addition to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8, we have the following refined lower bound estimate for the critical level µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Let yε be a maximum point of | ¯ψε|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Up to a subsequence if necessary, assume yε → y0 ∈ M as ε → 0 with respect to the Riemannian metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Then µε ≥ µ0 − ε2Θ(y0, z0) + o(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Notice that L′ ε( ¯ψε) = 0 and Lε( ¯ψε) − Lε(βε ¯ψε) = 1 εm � M 1 2f(| ¯ψε|)| ¯ψε|2 − F(| ¯ψε|)dvolg − 1 εm � M β2 ε 2 f(| ¯ψε|)| ¯ψε|2 − F(|βε ¯ψε|)dvolg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' By (f1)-(f2), for each fixed s ≥ 0, we deduce that the function t �→ t2 2 f(s)s2 − F(ts) is non-decreasing for t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, by βε(ξ) ∈ [0, 1] for all ξ ∈ M, one sees easily Lε(βε ¯ψε) ≤ Lε( ¯ψε) = µε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' On the other hand, since βε ¯ψε corresponds to a spinor field zε on B2r/ε(0) ⊂ Rm through the Bourguignon-Gauduchon trivialization and rescaling, by developing the relationship in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8) for βε ¯ψε and zε, we obtain the following correspondence of spinors ε ¯D(βε ¯ψε) ←→ Dzε + ε3W ·gRm zε + εX ·gRm zε + ε2 � i,j (bij − δij)∂i ·gRm ∇∂jzε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Now, we can argue similarly to the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6 to obtain Φ′(zε) = o(ε) and Lε(βε ¯ψε) = 1 εm � M 1 2(ε ¯D(βε ¯ψε),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' βε ¯ψε) + a 2(ωC ·g βε ¯ψε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' βε ¯ψε) − F(|βε ¯ψε|)dvolg = � Rm 1 2(Dzε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' zε) + a 2(ωC ·gRm zε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' zε) − F(|zε|)dx − ε2 12 � Rm Ricyε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x)(Dzε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' zε)dx − a ε2 12 � Rm Ricyε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x)(ωC ·gRm zε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' zε)dx + ε2 6 � Rm Ricyε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x)F(|zε|)dx − ε2 12 � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re � Rm Ryε(ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ej)(∇∂jzε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∂i ·gRm zε)dx + o(ε2) 35 Since yε → y0 and zε → z0 in W 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='q(Rm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ˜S(Rm)) as ε → 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' for q ≥ 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' and since |zε| decays exponentially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' we have � Rm Ricyε(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x) � (Dzε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' zε) + (aωC ·gRm zε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' zε) − 2F(|zε|) � dx = � Rm Ricy0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x) � (Dz0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' z0) + (aωC ·gRm z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' z0) − 2F(|z0|) � dx + oε(1) = � Rm Ricy0(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x) � f(|z0|)|z0|2 − 2F(|z0|) � dx + oε(1) and � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re � Rm Ryε(ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ej)(∇∂jzε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∂i ·gRm zε)dx = � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='j Re � Rm Ry0(ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ej)(∇∂jz0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' ∂i ·gRm z0)dx + oε(1) Note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='3 (4), we also have µ0 = inf u∈E+\\{0} max t>0 J(tu) ≤ Φ(zε) + O(∥Φ′(zε)∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, it follows directly that µε ≥ Lε(βε ¯ψε) ≥ µ0 − ε2Θ(y0, z0) + o(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Proof of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' We first see from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='8, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='6 that µε = Lε( ¯ψε) ≤ µ0 − ε2 max (y,ψ)∈M×B Θ(y, ψ) + o(ε2) for small ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Hence, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='11 and taking the limit ε → 0, we have Θ(y0, z0) ≥ max (y,ψ)∈M×B Θ(y, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Therefore, we conclude that (yε, zε) → (y0, z0) in M × B such that lim ε→0 Θ(yε, zε) = max (y,ψ)∈M×B Θ(y, ψ), and Lε( ¯ψε) = µ0 − ε2 max (y,ψ)∈M×B Θ(y, ψ) + o(ε2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' In the 2-dimensional case, the Ricci tensor determines the whole curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Specifically, we have Ric(x, x) = R(e1, x, x, e1) + R(e2, x, x, e2) = R(e1, e2, e2, e1)|x|2 and Scalg = 2 � j=1 2 � i=1 R(ej, ei, ei, ej) = 2R(e1, e2, e2, e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' 36 Hence, by noting that the scalar curvature is twice the Gaussian curvature for surfaces, we have Θ(y, ψ) = Kg(y) 6 � R2 �1 2f(|ψ|)|ψ|2 − F(|ψ|) � |x|2dx + Kg(y) 12 Re � R2 � (x2∇∂1 − x1∇∂2)ψ, (x2∂1 − x1∂2) ·gR2 ψ � dx, where Kg denotes the Gaussian curvature of (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Finally, by substituting f(|ψ|)|ψ|2 = |ψ|n∗ and F(|ψ|) = 1 n∗|ψ|n∗ into the above formulas, one completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content=' The authors wish to express their 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Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='Bartsch@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='uni-giessen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='de TIAN XU CENTER FOR APPLIED MATHEMATICS, TIANJIN UNIVERSITY TIANJIN, 300072, CHINA xutian@amss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} +page_content='cn' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9E4T4oBgHgl3EQfLAxE/content/2301.04934v1.pdf'} diff --git a/R9FJT4oBgHgl3EQfLCxT/content/2301.11467v1.pdf 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R. Offringa1, 2, B. Adebahr3, A. Kutkin1, E. A. K. Adams1, 2, T. A. Oosterloo1, 2, J. M. van der Hulst2, H. Dénes1, +C. G. Bassa1, D. L. Lucero4, 1, W. J. G. Blok1, 5, 2, K. M. Hess6, 1, 2, J. van Leeuwen1, G. M. Loose1, Y. Maan7, 1, +L. C. Oostrum1, 8, 9, E. Orrú1, D. Vohl8, 1, and J. Ziemke1, 10 +1 ASTRON, the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4,7991 PD Dwingeloo, The Netherlands +e-mail: offringa@astron.nl +2 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands +3 Astronomisches Institut der Ruhr-Universität Bochum (AIRUB), Universitätsstrasse 150, 44780 Bochum, Germany +4 Department of Physics, Virginia Polytechnic Institute and State University, 50 West Campus Drive, Blacksburg, VA 24061, USA +5 Dept. of Astronomy, Univ. of Cape Town, Private Bag X3, Rondebosch 7701, South Africa +6 Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía s/n, 18008 Granada, Spain +7 National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Pune 411007, Maharashtra, India +8 Anton Pannekoek Institute, University of Amsterdam, Postbus 94249, 1090 GE Amsterdam, The Netherlands +9 Netherlands eScience Center, Science Park 402, 1098 XH Amsterdam, The Netherlands +10 University of Oslo Center for Information Technology, P.O. Box 1059, 0316 Oslo, Norway +Received September 20, 2022; accepted January 3, 2023 +ABSTRACT +Context. Apertif is a multi-beam receiver system for the Westerbork Synthesis Radio Telescope that operates at 1.1-1.5 GHz, which +overlaps with various radio services, resulting in contamination of astronomical signals with radio-frequency interference (RFI). +Aims. We analyze approaches to mitigate Apertif interference and design an automated detection procedure for its imaging mode. +Using this approach, we present long-term RFI detection results of over 300 Apertif observations. +Methods. Our approach is based on the AOFlagger detection approach. We introduce several new features, including ways to deal with +ranges of invalid data (e.g. caused by shadowing) in both the SumThreshold and scale-invariant rank operator steps; pre-calibration +bandpass calibration; auto-correlation flagging; and HI flagging avoidance. These methods are implemented in a new framework that +uses the Lua language for scripting, which is new in AOFlagger version 3. +Results. Our approach removes RFI fully automatically, and is robust and effective enough for further calibration and (continuum) +imaging of these data. Analysis of 304 observations show an average of 11.1% of lost data due to RFI with a large spread. We +observe 14.6% RFI in auto-correlations. Computationally, AOFlagger achieves a throughput of 370 MB/s on a single computing +node. Compared to published machine learning results, the method is one to two orders of magnitude faster. +Key words. +Instrumentation: interferometers; Methods: observational; Techniques: interferometric; Surveys; Radio continuum: +general +1. Introduction +Technical advancement of mankind is driving an increase of +man-made radio-frequency transmitters, both terrestrial and in +space. This raises the bar for radio astronomical studies that +try to detect sky signals that are many orders of magnitude +fainter than man-made transmissions. Now that radio-astronomy +is evolving into a science where it is the norm to measure data +volumes in petabytes, mitigation of radio-frequency interference +(RFI) needs to be computationally efficient and fully automated. +Apertif is a receiver system upgrade for the Westerbork Syn- +thesis Radio Telescope (WSRT) that makes use of phased-array +feeds to allow for 40 simultaneous adjacent beams on the sky +(Van Cappellen et al. 2022). Observations are performed at a +central frequency of 1280 or 1370 MHz with an instantaneous +bandwidth of 300 MHz. +The data volume produced by Apertif is considerable. Volt- +ages from the 12 dishes with Apertif receivers are correlated +for all beams, typically integrated for 30 seconds and recorded +with four polarizations. The bandwidth of 300 MHz is split into +384 sub-bands, each with 64 channels of 12.2 kHz. Because of +the large bandwidth, it overlaps with various services, includ- +ing GPS and air-traffic communications. Although the WSRT +resides in a radio protected zone, it is not shielded from satel- +lites and air-traffic. Moreover, starting 2020, 5G transmissions +make use of the 1452 – 1492 MHz bandwidth. For these reasons, +Apertif requires an efficient approach to deal with RFI. Due to +the large amount of data, such an approach has to work fully +automatically. +The most common method to deal with RFI, is to detect data +samples that have a significant contribution of RFI and ignore +affected data in the processing (e.g. Winkel et al. 2006; Middel- +berg 2006; Offringa et al. 2010a; Prasad & Chengalur 2012; Peck +& Fenech 2013; Yang et al. 2020; Sun et al. 2022). This process +is referred to as data flagging, and is also our method of choice +for dealing with RFI in Apertif data in this work. Our detec- +tion methodology builds upon the RFI detection pipelines for the +Low-Frequency Array (LOFAR; Van Haarlem et al. 2013; Of- +fringa et al. 2010b) and the Murchison Widefield Array (MWA; +Article number, page 1 of 15 +arXiv:2301.01562v1 [astro-ph.IM] 4 Jan 2023 + +A&A proofs: manuscript no. apertif-rfi +Tingay et al. 2013; Offringa et al. 2015). Those pipelines inte- +grate an aoflagger flagging strategy, which combines filtering, +sumthresholding, morphological operations and heuristics. De- +tails of the aoflagger approach will be discussed in §2.1. +Apertif supports a transient (beam-formed) mode and an +imaging mode. The RFI detection approach for these two modes +are fundamentally different. In this work we aim at RFI detection +in imaging mode, i.e., after having correlated and integrated the +voltages from all the antennas. See Sclocco et al. (2019) for an +approach to mitigate RFI in beam-forming mode. Our approach +is part of a fully automated Apertif imaging pipeline called aper- +cal (Adebahr et al. 2022). +A multi-beam receiver makes it possible to perform spatial +filtering techniques to suppress interference (Kocz et al. 2010, +2012; Hellbourg et al. 2014). This requires fast dedicated com- +puting hardware that processes the raw signals from all the +beams, which for Apertif is not available. Spatial filtering is also +mainly used to filter out a limited number of known transmit- +ters, which for Apertif is likely not sufficient by itself, although +it might save some part of the bandwidth. +Another approach to detect interference is by using the spec- +tral kurtosis statistic (Gary et al. 2010; Taylor et al. 2019; Purver +et al. 2021). This has shown results that are competitive with +amplitude-based detection. However, this requires a specialized +correlator and a doubling of the data volume to be able to calcu- +late the kurtosis. +Recently, machine learning has been used to address the is- +sue of RFI detection (Harrison & Mishra 2019; Yang et al. 2020; +Xiao et al. 2022; Sun et al. 2022). Yang et al. (2020) argue that +convolutional neural networks can achieve an accuracy that is +higher than that of their sumthreshold implementation. For this +comparison, the authors use their own customized implementa- +tion of the sumthreshold method, whereas in platforms such as +aoflagger the method is typically applied iteratively and com- +bined with filters (Offringa et al. 2010a,b) and morphological +operators (Offringa et al. 2012; Van de Gronde et al. 2016) to en- +hance the accuracy. With these additions, it has been shown that +pipelines such as aoflagger typically detect all interference that +astronomers would manually flag. In this work, we will show- +case what can be achieved with traditional methods — including +their computational requirements — thereby providing an up- +dated base-line to compare against. +In this paper, we introduce a flagging strategy for Apertif +data using the AOFlagger framework, and demonstrate our de- +signed strategy on Apertif data. In §2, we will start by introduc- +ing the AOFlagger steps used to construct the Apertif approach, +and introduce several new operations that are integrated into the +Apertif flagging strategy. In §3, results of applying this strat- +egy are presented, including long-term statistics and the compu- +tational requirements. Finally, in §4 we discuss the results and +draw conclusions. +2. Method +For this work, we have designed an interference detection ap- +proach for Apertif based on the existing aoflagger approach and +integrated this into the apercal pipeline. apercal is an automated +processing pipeline for Apertif imaging observations (Adebahr +et al. 2022), consisting of common steps such as data formatting, +interference detection, calibration and imaging. Interference de- +tection is one of the first steps during data reduction and is funda- +mental for achieving a good and persistent calibration and image +quality and later steps of the processing. +To improve the detection quality, several modifications to +aoflagger are required. This consists of extensions of existing +algorithms and optimizing parameters for apertif, which we will +discuss in this section. We will start with an overview of the de- +tection approach. +2.1. Overview +Fig. 1 shows an overview of the steps that the default aoflagger +strategy performs. The aoflagger approach to RFI detection in +a subset can be summarized as i) estimation and subtraction of +the sky signal by applying a Gaussian high-pass filter in time- +frequency space (see §2.5); and ii) detection of excessive values, +with increased sensitivity towards spectral-lines and broadband +features. The detection is performed with the sumthreshold al- +gorithm (Offringa et al. 2010a). Steps i) and ii) are typically iter- +ated three times with increased sensitivity to make sure that the +final sky signal estimate is minimally biased by interference. As +a final step, the flags from different polarizations are combined +and are extended in time and frequency, using the scale-invariant +rank (SIR) operator (Offringa et al. 2012; Van de Gronde et al. +2016). This latter step improves detection of interference that ta- +pers off below the noise floor and fills gaps in the flag mask when +a persistent transmitter is not fully detected. +With aoflagger, detection of interference is performed inde- +pendently on subsets of the data, and the pipeline of Fig. 1 runs +independently for each subset. For Apertif, such a subset was +chosen to contain the data from all four linearly polarized cor- +relations (XX, XY, YX, YY), the full bandwidth (300 MHz), an +interval of typically half an hour for a single beam and a single +correlated baseline. Hence, the detection of interference for dif- +ferent beams, baselines and time intervals is independently per- +formed, even though these are part of the same observation. The +motivation for flagging these subsets independently is two fold: +– It improves performance: it allows parallel and distributed +detection of subsets. The independent flagging of beams and +time intervals matches with the format of the data. Despite +this, data access is still not ideal, because the data for one +baseline is stored dispersed over the time direction. +– Combined detection does not significantly improve detec- +tion: the added value of detection on combined subsets of +data is small, i.e., one subset contains little information about +the RFI in another subset. This is because the impact of RFI +can vary greatly between different beams and different base- +lines. Furthermore, it rarely occurs that RFI which affects +image quality is not detectable in half an hour of data, but is +detectable when multiple half hour intervals are combined. +Performing detection on integrated baselines has, in some +cases, been shown to make faint RFI detectable (Offringa et al. +2015; Wilensky et al. 2019). Early tests with Apertif data, how- +ever, indicated that there is no gain in combining baselines. We +have also performed tests that flag after integrating over multiple +beams, but again found no improvement in doing so. These tests +were not exhaustive and it could be that combined detection on +baselines or beams could still improve the accuracy somewhat. +aoflagger aims to take out RFI that requires raw, high- +resolution data flagging. Because of the high resolution of pro- +cessed data, the computational performance of detection is crit- +ical. It is important to perform high-resolution flagging early, +because it results in the highest accuracy and the impact of flag- +ging is reduced compared to low-resolution flagging (Offringa +et al. 2013). On the other hand, some phenomena cause the loss +Article number, page 2 of 15 + +A. R. Offringa et al.: An interference detection strategy for Apertif based on AOFlagger 3 +Fig. 1. The default aoflagger strategy for RFI-detection (before modifications for Apertif). These steps are independently performed on smaller +subsets of the data. The input data of one independent run through these steps typically consists of approximately an hour of correlations from a +single pair of antennas and a single beam, with the full bandwidth and all four linearly polarized cross-correlations present. +of large time intervals or frequency ranges. Common instrumen- +tal causes are correlator failures, temporary local RFI or strong +broadband transmitters. Detection of such issues does not require +the high-resolution data, and it is therefore less critical to detect +such issues in the first aoflagger detection run. Such issues can +be found in post-processing of lower-resolution data for which +the performance is less critical. +2.2. Invalid data +There are several instrumental issues that may result in data with +invalid values that interrupt the data in time or frequency. A few +examples of such issues are correlator malfunctions, dish shad- +owing, incorrectly set sub-band gains, network failures (between +stations and the correlator) or data corruption. Such instrumen- +tal issues result in visibilities that may have non-physical values +for certain times, frequencies, feeds or antennas, or could lead +to visibilities with a not-a-number (NaN) value. We will refer to +such data as invalid data. +In most cases, invalid data can be detected and flagged early +in the processing. For example, shadowing can be determined +from the target direction and the layout of the array, and miss- +ing sub-band data caused by network congestion can be detected +by the correlator. In this paper, we consider the detection of such +issues outside the context of interference detection. It does, how- +ever, make it necessary for the detector to continue to work in the +presence of (pre-detected) invalid data, which may affect only +specific times, frequencies or some other selection of data. +Making the aoflagger algorithm aware of invalid data is one +of the changes that was required for Apertif. The aoflagger al- +gorithm was originally designed to work on raw high-resolution +single-subband LOFAR data. It rarely happens that such a span +of data is partially invalid, and initially aoflagger algorithms +therefore do not take invalid data into account. In the case of +Apertif, the full bandwidth is offered to aoflagger, and the loss +or corruption of a single subband causes therefore gaps in the +bandwidth. Being a different instrument, Apertif is also affected +by different issues that may not affect LOFAR, such as shadow- +ing. For these reasons, we have extended the aoflagger algo- +rithm to take invalid data into account. This requires changes to +the sumthreshold and sir-operator steps of the algorithm, which +we will discuss in the next two sections. +2.3. Extension of the sumthreshold algorithm +The sumthreshold algorithm is a combinatorial thresholding +method that detects line-like structures in the time-frequency +data (Offringa et al. 2010a). This method is effective for the de- +tection of RFI, because most RFI raises the amplitude of consec- +utive time or frequency samples. The method iteratively thresh- +olds the average over an increasing number of neighbouring +samples with a decreasing threshold. With i the zero-indexed it- +eration number, Mi the number of samples, χi the threshold and +ρ a constant normally chosen to be 1.5, +Mi += +2i +(1) +χi += +χ0 ρ− log2 Mi. +(2) +χ0 is a user parameter that controls the total sensitivity of the +method. The various default aoflagger algorithms use values of +χ0 = 6...8.5 σ. The mode of the noise σ is determined from the +data that is (at that point in the detection) determined to be RFI +free, and is estimated by calculating the truncated mode of the +RFI free data, skipping 20% of the outlier values (the 10% min- +imum and maximum values), thereby assuming that the inner +80% follow a Rayleigh distribution. Assuming that the contribu- +tion of the noise is Gaussian distributed in the real and imaginary +components of the visibilities, this results in a stable estimate of +its standard deviation (Fridman 2008). +A single iteration consists of thresholding all sequences of +size Mi in both the time and the frequency direction (unless +Mi = 1), possibly with different thresholds for the two dimen- +sions, to separately control the sensitivity towards spectral line +RFI and transient broadband RFI. Typically, a total of 9 of these +iterations are performed, giving a maximum size of M8 = 256. +A sample that is flagged in an earlier iteration or direction, is +(temporarily) replaced by the mean of the non-flagged samples +in the sequence. The following description demonstrates the first +three iterations, using χ0 = 6 and ρ = 1.5: +1. Flag samples with an absolute value ≥ 6σ. +2. (a) Flag every sequence of 2 consecutive samples in time +with an absolute average ≥ 4σ (because χ2 = 6σ × +1.5− log2(22) = 4σ). +(b) Flag every sequence of 2 consecutive samples in fre- +quency with an absolute average ≥ 4σ. +3. Repeat 2.(a) and (b) with 4 samples and a threshold of χ4 = +6σ × 1.5− log2(23) = 2 2 +3σ. +Article number, page 3 of 15 + +A&A proofs: manuscript no. apertif-rfi +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Frequency index +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Time index +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Frequency index +40 +45 +50 +55 +Time index +Include bad data +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Frequency index +40 +45 +50 +55 +Time index +Set bad data to zero +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Frequency index +40 +45 +50 +55 +Time index +Exclude bad data +Fig. 2. Three methods of handling invalid data in the sumthreshold step. The top image shows the simulated input data, which consists of Gaussian +complex noise, spectral line RFI every 10 channels that increases in strength in frequency direction, and a block of invalid data (time indices 50– +100), simulating e.g. a temporary correlator failure. The bottom images show a zoom in on the left edge of the invalid data. Flagged data is marked +in yellow. Bottom-left: normal sumthreshold without using knowledge of the invalid data; bottom-centre: invalid samples are set to zero before +sumthreshold; bottom-right: invalid samples are removed before sumthreshold. +Subsequent iterations will threshold sequences of 8, 16, 32, . . . +samples with an average above χ8 ≈ 1.8σ, χ16 ≈ 1.2σ, etc. +In the form described by Offringa et al. (2010a), pre-existing +classification of invalid data is not taken into account in the +sumthreshold method. An example of such a case is shown +in Fig. 2, which considers a simulated observation with 200 +timesteps and 100 channels. The observation contains spectral- +line interference that affects one channel out of every ten chan- +nels and increases power at higher frequencies. Timesteps 50— +100 are known to be invalid data, and are set to high values by +raising them with 10 times the standard deviation. +The second row of Fig. 2 zooms in on time indices 30-60. +The first image of the second row shows the result of a basic +application of sumthreshold. For this result, the knowledge that +some data was invalid is not used. As a result, the invalid data is +considered to be RFI, and samples before and after the block of +invalid data are flagged with an increased sensitivity. As a result, +the false-positive rate is clearly increased. +A simple approach to mitigate this is to consider invalid val- +ues to be zero when applying the sumthreshold method. This re- +sults in the plot shown in the middle of the second row of Fig. 2. +This result does not show increased false positives because of +the invalid data. With this approach, information about flagged +samples on either side (before/after) of the missing data does not +(significantly) aid detection, because the invalid data is consid- +ered to be zero, and this lowers the average absolute sum in the +iterations of the sumthreshold method that consider longer con- +secutive ranges. This results in a higher false-negative rate than +would theoretically be possible if the information on both sides +of the invalid data would have been used together. In particu- +lar, the faintest interfering line at channel index 5 is no longer +detected. +While the loss in accuracy is minimal, there is a simple +method to aid the detection of interference on one side of the +block of invalid data with information from the other block: by +completely skipping data in the summed direction (time or fre- +quency). In other words, samples that are directly before and +after a block of invalid data are treated as if they are consec- +utive. The result of this is shown in the third column of Fig. 2, +which indeed shows a lower false-negative rate. In particular, the +faintest spectral line at channel 5 is now fully detected. +When comparing these two approaches to deal with invalid +data, the approach to exclude the invalid data leads to a small +increase in false-positive detections when the RFI is not con- +sistently present in time or frequency, i.e. when it is present on +one side of the invalid data block and not present on the other +side. This should be weighted against the increased sensitivity +when the RFI is consistently present. The optimal choice there- +fore depends on the behaviour of the RFI. Because persistent +RFI is common, and because it is more important to avoid false +negatives in persistent RFI (which might negatively affect later +processing steps) over avoiding false negatives in transient RFI +(which would lead to a small loss of data), we use the method of +excluding invalid data in our Apertif strategy. +We have implemented this in two ways: i) stack all valid +data into a temporary storage, run the normal sumthreshold +algorithm on these data and reverse the stacking operation on +the resulting mask; and ii) skip over the invalid data inside the +sumthreshold method. We have timed these two implementa- +tions on simulated complex Gaussian data with 10,000 timesteps +× 256 channels. Each run is repeated 100 times. The first im- +plementation runs about 2.5× faster (0.18 s per data set) com- +pared to the second implementation (0.45 s per data set). The +first method is still 6× slower compared to the regular algorithm +Article number, page 4 of 15 + +A. R. Offringa et al.: An interference detection strategy for Apertif based on AOFlagger 3 +(which takes 0.03 s per data set). This can be explained by the +extra copying of data that is required in each iteration (both for +the time and for the frequency direction). +2.4. Extension of the scale-invariant rank operator +The SIR-operator is a morphological operation that is used in +aoflagger to extend the detected RFI mask in the time and fre- +quency direction. It is an effective step to follow threshold-based +methods to detect faint RFI based on the morphology of detected +flags (Offringa et al. 2012; Van de Gronde et al. 2016). It is scale +invariant, which implies that the fractional increase in flags in +one dimension is constant, i.e., independent of the scale of that +feature in that dimension. +The SIR-operator is essentially a one-dimensional operator +that can be applied to a sequence of flag values. To apply it to +radio interferometric data, Offringa et al. (2012) apply the op- +erator in both the time and frequency dimensions: in time it is +separately applied to all the channels, and in frequency it is ap- +plied separately to all timesteps. The union of these to steps is +taken as the result. +Assume that X is a single sequence of flag values, such that +X[i] holds a Boolean value that represents the state of the flag. +The output ρ(X) of the SIR-operator applied to X, is defined as +the union of all subsequences of the input X, for which +#i: j +F ≥ (1 − η) ( j − i) . +(3) +Here, #i: j +F is brief for #F (X[i : j]), which is the count-operator +that returns the number of flagged samples in a sequence. X[i : j] +is the subsequence of samples consisting of all elements X[k] for +i ≤ k < j and η ∈ [0 . . . 1] is a tunable parameter that sets the +aggressiveness of the operator. +Eq. (3) implies that a sequence of flags caused by invalid data +is extended on both sides by a ratio of η. An example of this is +given in the centre-left panel of Fig. 3. This behaviour is undesir- +able because, unlike most RFI signals, invalid data typically has +a sharp boundary and should be flagged like that. The extension +of invalid data causes a high number of false positives. +A simple solution is to count invalid data as unflagged data in +the SIR operator. This implies that Eq. (3) is modified so that the +count operator only counts the number of flags corresponding to +valid data: +#i: j +F V ≥ (1 − η) ( j − i) , +(4) +where #F V is the number of valid samples that are flagged in the +interval X[i : j] (as opposed to #F , which counts flagged val- +ues that can both be valid or invalid). Because the right side is +unchanged and the left side remains equal or is decreased com- +pared to Eq. (3), this modification always flags an equal or fewer +amount of samples. An application of this approach is demon- +strated in the centre-right panel of Fig. 3. This approach reme- +dies the extending of flags around invalid data. +The downside of the approach of Eq. (5) is that a continu- +ous transmitter is assumed not to be present in the invalid data +range, causing flags on either side to have a decreased proba- +bility of getting flagged. For example, in case a correlator fails +for a minute during which a transmitter remains present in one +channel with decreasing power, the transmitter is less likely to +be flagged after the correlator failure. To address this, we further +modify Eq. (3) to: +#i: j +F ≥ (1 − η) #i: j +V, +(5) +where #i: j +V is the number of valid (flagged or unflagged) samples +in interval X[i : j]. This approach is effectively the same as re- +moving the invalid samples from the sequence before applying +Eq. (3). Therefore, a transmitter that gets interrupted by invalid +data receives a higher probability to get flagged. An example of +this approach is given in the bottom-left panel of Fig. 3. Invalid +samples are skipped in this approach, and flagged samples on +one side of a sequence of invalid samples may increase the prob- +ability of samples on the other side of the sequence, irregardless +of the size of the invalid sample sequence. +The approach of Eq. (5) can overstep its goal of using in- +formation from before and after a sequence of invalid data, in +particular in the case of very long sequences of invalid samples. +For example, when considering a transmitter that is active for +one minute before the receiving antenna is shadowed for 6 hours +(causing invalid data), it is undesirable that samples after shad- +owing receive higher detection probability because of what hap- +pened 6 hours ago. A final modification to the SIR operator we +consider is therefore to introduce a penalty parameter ρ that can +balance between Eqs. (4) and (5): +#i: j +F ≥ (1 − η) +� +( j − i)ρ + #i: j +V(1 − ρ) +� +. +(6) +With ρ = 0, invalid samples are skipped, making the method +equal to Eq. (5) and with ρ = 1, invalid samples are counted +as unflagged samples, making the method equal to Eq. (4). A +value of ρ = 0.2 implies that five invalid samples count as one +unflagged sample, thereby lowering the probability of flagging +through a block of invalid data, but still transferring some of the +flag information from before to after the invalid data and vice +versa. This method is demonstrated with a setting of ρ = 0.1 in +the bottom-right panel of Fig. 3. +Considering the results of all approaches in Fig. 3, it is +clearly undesirable to generally extend invalid data using the tra- +ditional SIR-operator defined in Eq. 3. Any of the three different +variations of the algorithm (Eqs. 4, 5 and 6), which can be de- +scribed by choosing different ρ-values in Eq. (6), solve this is- +sue. The different values of ρ do not cause significant changes. +We have tested values of ρ on a few observations, some with ar- +tificially added invalid data, and visually compared the flagging +results. Based on these results and the arguments given earlier +about finding a balance between Eqs. (4) and (5), we use ρ = 0.1. +Introducing the parameter for invalid-data weighting ρ has +no significant effect on the speed of the algorithm. The original +algorithm can be implemented with a computational complex- +ity of O(N) (Offringa et al. 2012), and the same holds for the +algorithm that includes the invalid-data penalty parameter. +2.5. High-pass filtering +The high-pass filter that is applied to remove astronomical +source contribution before thresholding is, for computational +reasons, implemented as a Gaussian low-pass filter followed by +subtracting the difference between the input and the low-pass fil- +tered result. The high frequency resolution of Apertif makes it +necessary to use a large filtering kernel in the frequency direc- +tion. Effectively, a kernel with a Gaussian sigma of 875 channels +and 2.5 timesteps is used. Before filtering, the data is averaged +in the frequency direction by a factor of 175, and after low-pass +filtering, the result is upscaled to the original resolution using +nearest neighbour resampling. This allows a convolution with a +much smaller kernel, improving the speed of this operation. The +result is an approximate of a Gaussian high-pass filter, but for the +purpose of removing the sky signal, this is sufficiently accurate. +Article number, page 5 of 15 + +A&A proofs: manuscript no. apertif-rfi +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Channel +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Time +Input +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Channel +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Time +Flag invalid data before SIR operator +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Channel +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Time +Unflag invalid data before SIR operator +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +4 +4.5 +5 +5.5 +6 +0 +10 +20 +30 +40 +50 +60 +70 +80 +90 +Channel +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Time +Exclude invalid data before SIR operator +Fig. 3. Different ways of handling invalid data in the sir-operator step on a simulated data set with a Gaussian burst of interference in a few +channels. Purple marks invalid data, yellow is detected as interference. The SIR-operator operates on the flag mask, hence the visibility values are +not used. Top: input data. Centre-left: Invalid data is counted as flagged data. Centre-right: Invalid data is counted as unflagged data. Bottom-left: +Invalid data is removed before applying the SIR-operator. Bottom-right: Invalid data is penalized with ρ = 0.1. +2.6. Bandpass correction +In the Apercal Apertif processing pipeline, the entire bandwidth +of Apertif is used at once during RFI detection. This is different +from the original LOFAR strategy, that flagged small (200 KHz) +subbands independently. Using the entire bandwidth has the ben- +efit that broadband RFI that covers several sub-bands can be de- +tected. This is relevant for Apertif observations, which are af- +fected by broadband transmitting satellites and radar. +Because the bandwidth of Apertif is subdivided into sub- +bands using a poly-phase filter bank, the band shape of the poly- +phase filter is imprinted on the data. An example of this is shown +in the top-left panel of Fig. 4. This is corrected for during cali- +bration, but during flagging (which needs to be done before cal- +ibration) the shape is still present. +Performing detection using the entire bandwidth but without +correcting for the poly-phase filter bank causes sub-band edge +channels to be flagged, because the edges cause sharp transi- +tions that trigger the detector. Moreover, the deviations in the +data caused by the band-edges decrease the sensitivity of the de- +tection toward actual RFI. The top-right panel of Fig. 4 shows an +example of flagging without bandpass correction. +To remedy this, we implement a sub-band band-pass correc- +tion step in the detector. This step corrects the poly-phase filter +shape using a static, observation-independent correction. We de- +termine the shape by performing gain-calibration on a clean re- +gion of the band, and average the solutions over the subbands. +The bottom-left panel of Fig. 4 shows the resulting corrected +data set, and the bottom-right panel of Fig. 4 shows the result +of flagging the bandpass. As can be seen, the band-pass cor- +Article number, page 6 of 15 + +A. R. Offringa et al.: An interference detection strategy for Apertif based on AOFlagger 3 +Fig. 4. Static sub-band band-pass correction before flagging with the Apertif flagging strategy. Top-left: input before correction; top-right: flagged +without correction; bottom-left: input after correction; bottom-right: flagged with correction. +rection has decreased the number of false detections consider- +ably. Some edge channels are still flagged, even after correction. +This is caused by aliasing in the sub-band edge channels, which +change the statistics of those edge channels slightly. This can +lead to artefacts which are very similar to RFI, hence they are +occasionally flagged. This flagging is normally of limited con- +cern, because those sub-band edge channels that are flagged are +of lower quality. Because of this, they are often discarded during +imaging. +2.7. Flagging of auto-correlations +Given the output voltage of the two feeds of the same antenna, +e = +� +ex, ey +� +, auto-correlated visibilities are formed by taking the +product eHe (i.e., the outer product e ⊗ e) and integrating, re- +sulting in XX, XY, YX and YY visibilities. While auto-correlations +are not often used for scientific data products, they are useful for +system monitoring and quantifying the system noise. For such +analyses, it is desirable to flag RFI. +Compared to cross-correlated visibilities, auto-correlated +visibilities have different properties: in the XX and YY corre- +lations, system noise and RFI will not decorrelate, and auto- +correlated visibilities are sensitive to the global sky signal in- +stead of fluctuations in the sky signal. +An example of auto-correlated dynamic spectrum from +Apertif is shown in the top image of Fig. 5 (after sub-band +band-pass correction as described in §2.6). Compared to cross- +correlations such as shown in Fig. 4, the dynamic spectrum of +auto-correlated visibilities appears much smoother, is system- +atically offset from zero and contains stronger structure in the +frequency direction. +The flagging strategy that was optimized for the cross- +correlations detects RFI by comparing high-passed filtered am- +plitudes of visibilities to the variance of these amplitudes. Be- +cause the amplitude variance is much lower compared to cross- +correlations, this results in flagging auto-correlations with in- +creased sensitivity. At the same time, the auto-correlations con- +tain stronger instrumental frequency-structure. These two effects +combined causes the cross-correlation flagging strategy to flag +all of the visibilities of the auto-correlations of Fig. 5. +To solve this, we use a different flagging configuration for +the auto-correlations. The difference with the cross-correlation +strategy is as follows: +- The time-direction sumthreshold step (sensitive to consis- +tently high values in the time direction, e.g. band-pass struc- +ture) is reduced in sensitivity by a factor of 6. +- The frequency-direction sumthreshold step (sensitive to +consistently high values in the frequency direction, e.g. +broad-band RFI) is reduced in sensitivity by a factor of 2. +Article number, page 7 of 15 + +A&A proofs: manuscript no. apertif-rfi +1280 +1300 +1320 +1340 +1360 +1380 +1400 +1420 +Frequency (MHz) +14:00 +16:00 +18:00 +20:00 +22:00 +0:00 +Time (UTC, hh:mm) +1280 +1300 +1320 +1340 +1360 +1380 +1400 +1420 +Frequency (MHz) +14:00 +16:00 +18:00 +20:00 +22:00 +0:00 +Time (UTC, hh:mm) +1280 +1300 +1320 +1340 +1360 +1380 +1400 +1420 +Frequency (MHz) +14:00 +16:00 +18:00 +20:00 +22:00 +0:00 +Time (UTC, hh:mm) +Fig. 5. Flagging of auto-correlations. Top image: input after sub-band +band-pass correction; centre image: same after iterative high-pass fil- +tering and with 10x more sensitive colour scale; bottom image: af- +ter flagging with the auto-correlation specific strategy. Because auto- +correlations have different properties compared to cross-correlations, +they require a specialized flagging strategy. +- The size of the high-pass filter kernel is reduced by 3.5 in +the frequency direction, to filter out more of the spectral gain +fluctuations of the instrument. +- The number of iterations is increased from 3 to 5. This in- +creases the required computations but improves robustness +in the presence of a large dynamic range, as is the case for +auto-correlations. +- Only the XX, XY and YY correlations are used for detection, to +reduce unnecessary computations. YX correlations are equal +to the conjugated XY correlations, and using these for flag- +ging does not provide additional information. +A result of this auto-correlations strategy is shown in the bot- +tom image of Fig. 5. Visual inspection shows that all visible RFI +is indeed detected, and the number of false detections appears +low. Because we do not have a ground truth, we do not try to +quantify these results. Similar to the cross-correlation strategy, +the auto-correlation strategy flags parts of the sub-band edges. +The centre image of Fig. 5 shows the high-pass filtered data of +the final iteration. +2.8. Avoiding HI removal +In observations that cover bright nearby galaxies or the Galactic +plane, the 1420 MHz HI line may be detectable in the visibil- +ities from a single cross-correlated baseline. For example, the +top-left image of Fig. 6 shows one baseline from a M31 ob- +servation, which clearly shows a contribution from HI-emission +around 1420 MHz. This poses a challenge for RFI detection, be- +cause such a fine, spectrally-consistent signal is quite similar to +RFI. As shown in the top-right image of Fig. 6, when standard +flagging is performed on these data, the HI emission is detected +as RFI. +We analyze different ways to mitigate this. In the Nether- +lands, frequencies between 1400-1427 MHz are reserved for ra- +dio astronomy and other forms of passive research1, and trans- +mitting inside this band is not allowed. As a result, these frequen- +cies are almost free of man-made emission. A simple mitigation +strategy is therefore to disable RFI detection inside this band. +Unfortunately, the recorded visibilities do occasionally contain +strong, non-astronomical values inside this band. The three ver- +tical lines in the images of Fig. 6 are an example of such an +observation. Most frequently, these are caused by saturation of a +receiver, causing a broadband-like signal in the recorded visibili- +ties, although they might occasionally be caused by RFI emitted +at these frequencies (e.g. from a sparking device or lightning). +Leaving these broadband contaminants in the data causes degra- +dation of the images. In particular, they cause visible stripes in +continuum, full bandwidth images. +Another approach is to flag only based on Stokes Q, U and +V. Man-made RFI is often polarized, whereas the sky emission +in these polarizations is generally much fainter. The result of +this approach is shown in the bottom-left image of Fig. 6. While +a part of the HI emission has been left intact, it is still bright +enough in these polarizations to get detected. This is even the +case when flagging on only one of these polarizations: the HI +emission is present in all of the polarizations. Moreover, we oc- +casionally observe RFI that is only visible in Stokes I, and re- +moving any of the polarizations decreases the effectiveness of +RFI detection. In Fig. 6, the transmitter around 1425 MHz / 0:00 +UTC is for example not as well detected in this approach com- +pared to standard flagging. +Because none of these approaches give good results, we con- +sider another approach, and run the flagger twice: in run A) we +flag the data with the normal detection strategy, and in run B) +we run the detection with a strategy that is insensitive to spectral +lines. For frequencies outside the HI range we use the flags from +run A), and inside the HI range (1418–1424 MHz) we use B). +The result of this approach is shown in the bottom-right image +of Fig. 6. With this approach, broadband structures have been +detected as RFI and HI emission is left in the data. +To avoid flagging spectral lines in run B), we adjust the fol- +lowing flagging settings during this run): +1 The Dutch spectrum allocations can be found at https://www. +agentschaptelecom.nl/ +Article number, page 8 of 15 + +A. R. Offringa et al.: An interference detection strategy for Apertif based on AOFlagger 3 +0 +200 +400 +600 +800 +1000 +1200 +Visibility (Jy) +1405 +1410 +1415 +1420 +1425 +Frequency (MHz) +20:00 +22:00 +0:00 +2:00 +4:00 +Time (UTC, hh:mm) +0 +200 +400 +600 +800 +1000 +1200 +Visibility (Jy) +1405 +1410 +1415 +1420 +1425 +Frequency (MHz) +20:00 +22:00 +0:00 +2:00 +4:00 +Time (UTC, hh:mm) +0 +200 +400 +600 +800 +1000 +1200 +Visibility (Jy) +1405 +1410 +1415 +1420 +1425 +Frequency (MHz) +20:00 +22:00 +0:00 +2:00 +4:00 +Time (UTC, hh:mm) +0 +200 +400 +600 +800 +1000 +1200 +Visibility (Jy) +1405 +1410 +1415 +1420 +1425 +Frequency (MHz) +20:00 +22:00 +0:00 +2:00 +4:00 +Time (UTC, hh:mm) +Fig. 6. Band-pass corrected M31 data from WSRT RT9 × RTA with a strong HI signal. Top-left image: input data. The bright emission around +1420 MHz is from HI and should not be flagged. The vertical lines are instrument or RFI artefacts that should be flagged. Top-right image: +after RFI detection without HI modifications, showing in pink what is flagged. Bottom-left image: after RFI detection using Stokes Q, U and V. +Bottom-right image: after RFI detection using a specialized strategy for 1418-1424 MHz. +- The high-pass filter in frequency direction is set to have a +kernel size of one channel, to filter out fluctuations in fre- +quency. +- The sensitivity of the time-direction sumthreshold step is de- +creased by a factor of 4, to reduce flagging of line-like struc- +tures. +- The sensitivity of the frequency-direction sumthreshold step +is decreased by a factor of 2. This reduces flagging of tem- +poral fringes in HI emission. +- The number of iterations is increased to remain robust in the +presence of strong HI emission. +On overall, the resulting strategy is almost entirely insensitive to +spectral-line-like structures. The sensitivity to broadband struc- +tures will also be reduced because of these changes, but given +that this strategy remains sensitive to faint broadband structures +such as shown in Fig. 6, we consider this tolerable. +Because run B) requires only a small part of the full band- +width, the second flagging run is relatively fast, hence the in- +crease in computations caused by this is modest (about 20%). +2.9. Reading overhead and memory considerations +During the AOFlagger stage of the apercal pipeline, observa- +tions are stored in the Casacore Measurement Set format. In this +format, the data of an observation is lexicographically sorted in +time, and then in baseline and frequency. While this ordering +is suitable for calibration, flagging requires the data baseline by +baseline. Unfortunately, the data for a single baseline is spread +throughout the file. Therefore, reading a baseline requires read- +ing the file from beginning to end. Because of the block size and +caching of storage media, it is inefficient to read the baselines +one by one with this approach. +AOFlagger supports three methods for accessing the data: +- Direct reading. In this mode, the data is directly read from the +measurement set just before they are needed. Because multi- +ple baselines are processed in parallel using multi-threading, +a few baselines are read from the measurement set at once. +This mode results in scanning through the input data multiple +times, which is computationally costly. +- Reorder before processing. In this mode, the whole measure- +ment set is reordered by baseline, frequency and then time +and rewritten to disk in a binary, internal format before pro- +cessing is started. This results in reading the data only twice +and is generally faster than the direct reading mode, but re- +quires disk space to store the copy of the data. +- In-memory data. In this mode, the whole measurement set is +read into memory before starting processing. This results in +reading the data only once and is generally the fastest mode, +but requires a considerable amount of memory. +Apertif data sets are large and expensive to read: reading the data +more than once is undesirable. As a result, the only acceptable +reading mode is the in-memory mode. In the particular comput- +ing mode where Apercal runs, the amount of memory required +by this mode is a considerable constraint, and requires a dedi- +cated node for each flagging operation performed. +Other observatories have solved this issue by integrating +aoflagger into a multi-step preprocessing pipeline that stream +through the data, split the data in time for flagging and hand +these data over part by part to AOFlagger via its application pro- +gramming interface. Examples of such pipelines are cotter (Of- +fringa et al. 2015) and DP3 (Van Diepen et al. 2018), which are +preprocessing pipelines for the Murchison Widefield Array and +the Low-Frequency Array, respectively. In this approach, sev- +eral tasks (e.g. conversion, phase rotation, flagging, averaging, +compression) can be applied with a single read through the data, +thereby reducing the read overhead. In the case of Apertif, such +Article number, page 9 of 15 + +A&A proofs: manuscript no. apertif-rfi +a streaming pipeline does not exist. Instead, aoflagger runs as a +stand-alone tool inside Apercal. +To solve the memory and reading issue for Apertif, we imple- +mented a time-chunking approach into aoflagger. In this mode, +aoflagger reads small chunks in time and flags these indepen- +dently. This makes it possible to use the memory reading mode, +because the data for individual chunks is small enough to fit in +memory. It does imply that the algorithm has less information +available to do its RFI detection. Therefore, it is important to +let time chunks still have a significant size, because AOFlagger +would otherwise not be able to find faint RFI, that is persistent +in time, but not detectable in a small chunk. For Apertif, we use +a chunk size corresponding to about half an hour of data. +2.10. Use of Lua +Before AOFlagger version 3, AOFlagger strategies were written +in the extensible markup language (XML). An XML file speci- +fies a sequence of steps and is interpreted by AOFlagger, and this +sequence is executed separately for the data from every baseline. +The sequences run multi-threaded, and reading and writing of +data is done outside of the strategy. Examples of XML steps are +to calculate visibility amplitudes; running sumthreshold or sir +operations on the data; or to combine the flags of all polariza- +tions. +Over the years, the use of AOFlagger extended to more and +more use-cases: different telescopes, flagging after calibration, +high-resolution flagging, etc. It became desirable to make the +strategies more flexible. In particularly, it became desirable to +support standard scripting structures such as loops, condition- +als, variables and to provide standardized documentation of the +steps. The idea was therefore formed to embed a standard in- +terpreter into AOFlagger and provide a function interface for +each step. The data-intensive computations are still performed +by high-performance precompiled C++ code, while these are +glued together using an interpreted script, thereby combining +flexibility with high performance. +Our first approach was to embed it into Python, because of +its popularity in astronomical data science. After having im- +plemented a prototype that embeds the Python interpreter into +AOFlagger, it turns out some of the features of the Python inter- +preter conflict with how AOFlagger runs these scripts. Particular +challenges were to deal with the global interpret lock; memory +management; and fast restarts of the interpreter. While there are +various ways to work around these issues, the design goals of +the Python language and interpreters do not focus specifically to +make the language embeddable. +Lua2 is a scripting language that is widely used for em- +bedding scripts in applications, notably in computer games to +implement scripted game sequences. This scenario is close to +the AOFlagger use-case: the interpreter is integrated into such +games, called many times and supports multi-threaded script ex- +ecution. Algorithmic code that requires high performance can +be implemented in compiled languages (C++ in the AOFlagger +case). With this idea in mind, we decided to integrate the Lua +interpreter into AOFlagger and implement all steps as Lua func- +tions. +The use of a full scripting language has increased the pos- +sibilities inside the flagging strategies considerably. For exam- +ple, it is now possible to adapt the strategy based on properties +such as the baseline length, frequency, auto- or cross-correlation, +etc. A consequence of the new interface is that existing strate- +2 https://www.lua.org/ +gies need to be rewritten, which can not be done automatically. +All default strategies have been rewritten to use Lua, which cur- +rently includes specialized scripts for 11 observatories (Aartfaac, +APERTIF, Arecibo, ATCA, Bighorns, JVLA, MWA, WSRT, +LOFAR, NenuFAR). These have all been verified to produce the +same result as the old XML-based strategies. Because the new +function interface gives better control over what steps need to +be run, the speed of the new strategies is slightly higher (several +percent). We do not notice any significant overhead from using +Lua: the computational time is dominated by the computations +inside the function calls. +3. Results +Apertif observations are processed by the automated Apercal +pipeline. This pipeline includes the flagging strategy as de- +scribed in Sec. 2. In this section, we present results of the full +flagging step on Apertif observations. The data that we look at +has been recorded between 2019 and 2022. Science products +from the first year of observing have been described in the first +Apertif data release (Adams et al. in press; Kutkin et al. in press). +3.1. RFI detection examples +The detection strategy described in Sec. 2 runs fully automated, +and does not require further flagging before calibration and con- +tinuum imaging. In general, manual inspection of data after RFI +detection shows no residual RFI and few false positives. Fig. 7 +shows the 1280–1430 MHz range of a typical observation. The +top plot shows the data before RFI detection, and the bottom plot +shows in white what has been detected as RFI. Fig. 8 shows a +challenging case with wider bandwidth, with a moderate amount +of RFI, missing data (1200–1220 MHz) and strong fringes. Top +and bottom plots show again before and after detection. This +also demonstrates the challenging situation for radio astronomi- +cal science between 1150 and 1300 MHz. +For continuum imaging, it is often useful (or at least prag- +matic) to take out any visibility that appears to have a contribu- +tion from RFI. For spectral imaging, a flagging result such as +shown in Fig. 8 is problematic, because many channels are fully +removed. In those cases, it is possible to reduce the sensitivity of +the RFI detection. The sensitivity is specified as a variable in the +script. For the detection result shown in Fig. 9, the sensitivity was +decreased by a factor of 3. Compared with the result in Fig. 8, +this reduced the flagging from 49% to 33%. This takes out the +strongest RFI, but leaves weak (but visible) RFI in the data. De- +creasing the sensitivity further continues to trade the availability +of visibilities with a lower quality of those visibility. +3.2. RFI characteristics and long-term statistics +During the flagging step, statistics are collected that summa- +rize the (detected) RFI occupancy and data quality. We have +collected these statistics for 304 of the currently processed ob- +servations. Averaged over all these observations and the full +bandwidth, the total detected RFI occupancy is 11.1% in the +cross-correlated baselines and 14.6% in auto-correlated base- +lines. Fig. 10 shows the detected spectral RFI occupancy for each +observation, as well as the occupancy averaged over all observa- +tions. Only cross-correlated data is included. At most frequen- +cies, the average loss of data due to RFI is about 10%, but with a +spread of approximately 0-15% between observations, and a few +larger outliers. +Article number, page 10 of 15 + +A. R. Offringa et al.: An interference detection strategy for Apertif based on AOFlagger 3 +Frequencies between 1400 and 1427 MHz are reserved for +radio astronomy. At these frequencies, the average RFI occu- +pancy is slightly lower (approximately 8%), but is evidently still +affected by instrumental effects (such as receiver saturation) or +natural and unintended RFI (such as lightning). Fig. 6 shows data +that is affected by such broadband artefacts. It is likely that the +∼10% base-level of occupancy is caused by such artefacts. +Some observations show a small excess RFI occupancy at +1420 MHz. This is caused by HI that is detected as RFI. The +methods to avoid flagging HI that are described in §2.8 were +implemented only halfway 2021. Some of the observations that +are flagged before that still show false-positive detections at HI +frequencies, but all observations after avoiding HI was imple- +mented show indeed no HI flagging. +The same base level of 10% is not visible at frequencies +above 1430 MHz. The reason for this difference is that only a +relative small number of observations cover frequencies above +1430 MHz. Frequencies between 1427 and 1492 MHz are allo- +cated to various services, including mobile communication and +fixed transmissions3. Some of these are satellite based. In 2020, +the 1452—1492 MHz band was auctioned in the Netherlands +and thereafter allocated for the use of 5G mobile phone down- +link. As shown in Fig. 10, the use of data above 1430 MHz is +limited. +Some channels between 1300–1400 MHz contain a few out- +lier RFI occupancies. These are caused by a nearby radar sta- +tion that is occasionally turned on. Frequencies between 1130 +and 1300 MHz are predominantly affected by RFI from Global +Navigation Satellite Systems (GNSS), such as the US GPS, Rus- +sian GLONASS, Chinese BeiDou, and European Galileo satel- +lite constellations. All these constellations use satellites in or- +bits at ∼2000 km and with high orbital inclinations (i = 54–65◦) +to provide global coverage. Frequencies for wide band trans- +missions are assigned to, and shared between, these systems at +1176.45, 1191.795, 1207.14, 1227.6, 1278.75 MHz (for GPS, +BeiDou, Galileo) and 1202.025 and 1242.9375–1251.6875 MHz +(for GLONASS). +Wide band signals are detected at these frequencies through- +out the entire observation of Fig. 8 covering the band down to +1130 MHz. Using orbital ephemerides of these satellite constel- +lations, we find that the strong temporal RFI observed in Fig. 8 at +13:06, 14:46, 16:29, 18:13 and 19:54UTC is caused by BeiDou +satellites passing within 5◦ from the pointing of the APERTIF +compound beam. The pass of 18:13UTC had a minimum sepa- +ration of 0◦.31 and led to saturation of the receiver, affecting the +entire observing band. Two GPS satellites passed at 1◦.47 and +2◦.30 separation from the beam pointing at 22:02 and 23:02UTC, +and one Galileo satellite at 3◦.72 at 22:59UTC, and coincident +increases of the RFI levels are observed, but not as strong as +with the passes of BeiDou satellites. The GNSS signals observed +away from these passes near the primary APERTIF beam are +likely due to far sidelobes or multi-path reflections of GNSS sig- +nals from the WSRT focus structure or other nearby structures +directly into the receiver. +3.3. Computational requirements +In this section we summarize the computational requirements +of the Apertif RFI detection strategy, with the aim of making +it possible to approximate the computational requirements for +other telescopes when a similar flagging strategy is used. Since +the total throughput is depending on many complex factors of +3 See https://www.agentschaptelecom.nl/ +the computing platform (e.g. clock speed, cores, memory band- +width, instruction set, vectorization), we aim at giving a first- +order estimate only. +We measure the performance of flagging a set with visibil- +ities from a single observation. We use an Apertif observation +with 1346 timesteps, 24572 channels and 4 polarizations, for a +total of 132M visibilities. This makes the visibility data, which +consists of 4-byte single-precision real and imaginary values, +1.1 GB in size. +We perform our test on a desktop machine with an AMD +Ryzen 7 2700X 8-Core processor and 64 GB of memory. This +processor can perform hyper-threading, and thus we run 16 de- +tections in parallel. We load the data in memory before detec- +tion and do not store the results, to avoid any disk access. Av- +eraged over 10 runs, it takes 46 seconds to run 16 detections, +which amounts to a throughput of 370 MB/s (or 46M visibili- +ties/second). At the time of writing, a typical fast spinning disk +achieves a sustained reading throughput of a few hundred MB/s. +Hence, disk access can be a significant cost of a stand-alone +RFI detection step. This can be problematic for supercomput- +ers, because they have high computing power, but not a high I/O +throughput. +3.4. Comparison against a machine learning approach +Some studies have found that machine learning can improve the +accuracy of RFI detection. In Yang et al. (2020), the authors +test their own sumthreshold implementation against a machine +learning approach, using a ground truth flag mask that is man- +ually determined by an engineer. Such a ground truth mask is +difficult to make in general, including for Apertif data, where +broadband RFI tapers off and it is unclear from which points +samples are truly unaffected by RFI. We can however conclude +that, after our pipeline, all visibly affected samples have been +identified. Moreover, imaging results have achieved the thermal +noise of the instrument, thereby indicating that the accuracy of +interference detection is not a limitation. +This conflicts somewhat with the conclusions made by Yang +et al. (2020). The sumthreshold implementation that is used +there to compare their results with, does not achieve the pub- +lished accuracy of aoflagger, because residual interference is vi- +sually present. Potential explanations for these differences could +be i) that Yang et al. train their network for a specific scenario but +did not optimize their sumthreshold approach; or ii) that they do +not use a full (i.e. aoflagger-like) sumthreshold-based pipeline +that includes the sir operation and that is similarly optimized +for their instrument. An important consideration is that morpho- +logical operations are aimed at detecting RFI that is below the +noise, therefore invisible to scientists that manually classify RFI. +In the comparisons done in Yang et al. 2020, samples detected +by the morphological operator would all be counted as false pos- +itives, whereas this operator has been shown to improve the final +science results (Offringa et al. 2012). It can therefore not yet +be stated that, based on accuracy, machine learning methods are +outperforming traditional based methods. Rather, it is clear that +both methods are competitive and are accurate enough to largely +mitigate the problem of interference in radio data. +There are differences in the computational performance +though. In Xiao et al. (2022), machine learning methods flag a +one-hour FAST observation of 67 GB in 61% of the observing +time using 8 computing nodes (Xiao et al. 2022). This amounts +to a single-node computational performance of 14 GB/hour. On +the other hand, the single-node performance of the aoflagger +approach listed in §3.3 is 370 MB/s, or 1.3 TB/hour, and aoflag- +Article number, page 11 of 15 + +A&A proofs: manuscript no. apertif-rfi +ger is therefore almost two orders of magnitude faster. While +the performance of the computing nodes used for the compu- +tational performance analyses may differ somewhat, and it is +therefore not a direct comparison, it is evident that the aoflag- +ger approach is significantly faster. In Sun et al. (2022), authors +compare the run-time of aoflagger to their convolutional neu- +ral network (CNN) approach and find that aoflagger is two to +four times faster. However, the authors measured the total run- +time of the aoflagger executable, which would include disk ac- +cess, start-up overhead and time spent in the casacore library to +transfer the measurement set data. Because the flagging speed is +near the disk access speed, this overhead can be substantial. A +better benchmark is possible by using the C++ or Python API +of aoflagger directly. On their Sim_RFI-1 dataset, they reach +an aoflagger speed of 250 GB/hour, while in this work, with a +more advanced strategy, we reach 1.3 TB/hour on similar hard- +ware. Their CNN method reaches a speed of 145 GB/hour, which +is an order of magnitude faster than what is reached by Xiao et al. +(2022), but is an order of magnitude below what we reach with +our aoflagger approach. +4. Discussion & conclusions +We have described and demonstrated an automated RFI detec- +tion strategy aimed at flagging Apertif data. Our detection strat- +egy implements novel sumthreshold and sir-operator algorithms +that take prior information about invalid data into account. It also +avoids the flagging of HI emission, works on auto-correlations, +corrects the sub-band band-pass and contains some further pa- +rameter optimizations for Apertif. The change from the AOFlag- +ger XML strategies towards fully scripted strategies provides flex- +ibility that made these changes quite easy to implement and sup- +ports flexibility during experimentation. Besides making the pro- +cess easier and faster, an automated RFI detection strategy also +makes the results reproducible, compared to when RFI is flagged +manually, and it allows reducing the data size by averaging early +on in the data reduction processing. +We expect that our RFI detection strategy will work for data +from other instruments, in particular those with a frequency cov- +erage comparable to Apertif, such as MeerKAT, ASKAP, JVLA +and future SKA-mid observations around 1.0 – 1.5 GHz. Differ- +ent bands might require some changes to the strategy parameters, +but should be able to reuse a large part of the approach. +While machine learning techniques may compete with the +accuracy of AOFlagger, they do not compete with its speed. +Moreover, we have shown it is possible to add new features +to AOFlagger, such as avoiding the 21-cm HI signal, accurate +detection in the presence of invalid data and flagging of auto- +correlations. None of the current available machine learning +techniques support these scenarios. Most parameters, such as +the sensitivity towards broadband and line RFI, or the expected +smoothness of the data, are intuitive and easy to tweak for sci- +ence cases that e.g. require that transients do not get flagged, or +that require a difference balance between taking out all visible +RFI on one hand, and keeping as much data available for further +processing on the other hand. This will be challenging, if at all +possible, to implement in a machine learning framework. +In this work, we have not made use of the multi-beaming +capabilities of Apertif: beam are flagged independently. While +some first-order testing indicates that using data integrated over +all beams does not improve flagging accuracy, it can be expected +that RFI does correlate somewhat over beams. A strategy where +the integrated data is searched for RFI, and where this is used +as additional input for the flagging of individual beams, might +be effective for detecting RFI that is below the noise for a single +beam. +Acknowledgements. This work makes use of data from the Apertif system in- +stalled at the Westerbork Synthesis Radio Telescope owned by ASTRON. AS- +TRON, the Netherlands Institute for Radio Astronomy, is an institute of the +Dutch Research Council (de Nederlandse Organisatie voor Wetenschappelijk +Onderzoek, NWO). BA acknowledges funding from the German Science Foun- +dation DFG, within the Collaborative Research Center SFB1491 ”Cosmic In- +teracting Matters - From Source to Signal”. EAKA is supported by the WISE +research programme, which is financed by NWO. JMvdH and KMH, acknowl- +edge funding from the European Research Council under the European Union’s +Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No. +291531 (‘HIStoryNU’). JvL, YM and LCO acknowledge funding from the Eu- +ropean Research Council under the European Union’s Seventh Framework Pro- +gramme (FP/2007-2013)/ERC Grant Agreement No. 617199 (‘ALERT’; PI: +JvL). KMH further acknowledges financial support from the State Agency +for Research of the Spanish Ministry of Science, Innovation and Universities +through the “Center of Excellence Severo Ochoa” awarded to the Instituto de +Astrofísica de Andalucía (SEV-2017-0709) from the coordination of the par- +ticipation in SKA-SPAIN, funded by the Ministry of Science and innovation +(MICIN) and grant RTI2018-096228-B-C31 (MCIU/AEI/FEDER,UE). JvL fur- +ther acknowledges funding from Vici research programme ‘ARGO’ with project +number 639.043.815, financed by NWO. DV acknowledges support from the +Netherlands eScience Center (NLeSC) under grant ASDI.15.406. +References +Adams, E. A. K., Adebahr, B., de Blok, W. J. G., et al. in press, A&A +Adebahr, B., Schulz, R., Dijkema, T., et al. 2022, Astronomy and Computing, +38, 100514 +Fridman, P. A. 2008, AJ, 35, 1810 +Gary, D. E., Liu, Z., & Nita, G. M. 2010, in Proc. of Science, RFI2010, +van Haarlem, M. P., Wise, M. W., Gunst, A. 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Offringa et al.: An interference detection strategy for Apertif based on AOFlagger 3 +250 +300 +350 +400 +450 +500 +550 +600 +650 +700 +750 +800 +850 +900 +950 +1000 +Visibility (Jy) +1280 +1290 +1300 +1310 +1320 +1330 +1340 +1350 +1360 +1370 +1380 +1390 +1400 +1410 +1420 +Frequency (MHz) +19:00 +20:00 +21:00 +22:00 +23:00 +0:00 +1:00 +2:00 +3:00 +4:00 +5:00 +Time (UTC, hh:mm) +Observation 200804041, band 33, RT7 x RT8 +250 +300 +350 +400 +450 +500 +550 +600 +650 +700 +750 +800 +850 +900 +950 +1000 +Visibility (Jy) +1280 +1290 +1300 +1310 +1320 +1330 +1340 +1350 +1360 +1370 +1380 +1390 +1400 +1410 +1420 +Frequency (MHz) +19:00 +20:00 +21:00 +22:00 +23:00 +0:00 +1:00 +2:00 +3:00 +4:00 +5:00 +Time (UTC, hh:mm) +Observation 200804041, band 33, RT7 x RT8 +Fig. 7. Typical flagging result for a single baseline in a wideband observation. The top panel shows the input visibilities, and the bottom panel +shows the visibilities overlaid with the detection result in white. These plots show the Stokes I visibilities. Some interference features are only +visible in Stokes Q, U or V, such as the vertical features around midnight. All interference features have successfully been detected, and no obvious +undesirable detections are visible, with the exception of horizontal flagged features every 200 kHz, caused by the sub-band bandpass (see Fig. 4). +18% of the data gets flagged for the baseline in this observation. +Article number, page 13 of 15 + +A&A proofs: manuscript no. apertif-rfi +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +2400 +2600 +2800 +3000 +Visibility (Jy) +1140 +1160 +1180 +1200 +1220 +1240 +1260 +1280 +1300 +1320 +1340 +1360 +1380 +1400 +1420 +Frequency (MHz) +12:00 +13:00 +14:00 +15:00 +16:00 +17:00 +18:00 +19:00 +20:00 +21:00 +22:00 +23:00 +Time (UTC, hh:mm) +Observation 200125001, Band 6, RT4 x RT6 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +2400 +2600 +2800 +3000 +Visibility (Jy) +1140 +1160 +1180 +1200 +1220 +1240 +1260 +1280 +1300 +1320 +1340 +1360 +1380 +1400 +1420 +Frequency (MHz) +12:00 +13:00 +14:00 +15:00 +16:00 +17:00 +18:00 +19:00 +20:00 +21:00 +22:00 +23:00 +Time (UTC, hh:mm) +Observation 200125001, Band 6, RT4 x RT6 +Fig. 8. Detection result for a full 300-MHz bandwidth observation. +Article number, page 14 of 15 + +A. R. Offringa et al.: An interference detection strategy for Apertif based on AOFlagger 3 +0 +200 +400 +600 +800 +1000 +1200 +1400 +1600 +1800 +2000 +2200 +2400 +2600 +2800 +3000 +Visibility (Jy) +1140 +1160 +1180 +1200 +1220 +1240 +1260 +1280 +1300 +1320 +1340 +1360 +1380 +1400 +1420 +Frequency (MHz) +12:00 +13:00 +14:00 +15:00 +16:00 +17:00 +18:00 +19:00 +20:00 +21:00 +22:00 +23:00 +Time (UTC, hh:mm) +Observation 200125001, Band 6, RT4 x RT6 +Fig. 9. Same as Fig. 8, but flagged with 3× lower sensitivity. + 0 + 20 + 40 + 60 + 80 + 100 + 1300 + 1350 + 1400 + 1450 + 1500 +RFI (%) +Frequency (MHz) +Observations +Average +Fig. 10. Percentage of RFI over frequency detected in 304 Apertif observations, excluding auto-correlations. +Article number, page 15 of 15 + diff --git a/W9AzT4oBgHgl3EQfmf0I/content/tmp_files/load_file.txt b/W9AzT4oBgHgl3EQfmf0I/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39aac54af1c721eb39829a45cf7dc4c10a904755 --- /dev/null +++ b/W9AzT4oBgHgl3EQfmf0I/content/tmp_files/load_file.txt @@ -0,0 +1,1051 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf,len=1050 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi ©ESO 2023 January 5, 2023 An interference detection strategy for Apertif based on AOFlagger 3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Vohl8, 1, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Ziemke1, 10 1 ASTRON, the Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4,7991 PD Dwingeloo, The Netherlands e-mail: offringa@astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='nl 2 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands 3 Astronomisches Institut der Ruhr-Universität Bochum (AIRUB), Universitätsstrasse 150, 44780 Bochum, Germany 4 Department of Physics, Virginia Polytechnic Institute and State University, 50 West Campus Drive, Blacksburg, VA 24061, USA 5 Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' of Astronomy, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' of Cape Town,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Private Bag X3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Rondebosch 7701,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' South Africa 6 Instituto de Astrofísica de Andalucía (CSIC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Glorieta de la Astronomía s/n,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' University of Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Postbus 94249,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 1090 GE Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The Netherlands 9 Netherlands eScience Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Science Park 402,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 1098 XH Amsterdam,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The Netherlands 10 University of Oslo Center for Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Box 1059, 0316 Oslo, Norway Received September 20, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' accepted January 3, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Apertif is a multi-beam receiver system for the Westerbork Synthesis Radio Telescope that operates at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 GHz, which overlaps with various radio services, resulting in contamination of astronomical signals with radio-frequency interference (RFI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We analyze approaches to mitigate Apertif interference and design an automated detection procedure for its imaging mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Using this approach, we present long-term RFI detection results of over 300 Apertif observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Our approach is based on the AOFlagger detection approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We introduce several new features, including ways to deal with ranges of invalid data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' caused by shadowing) in both the SumThreshold and scale-invariant rank operator steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' pre-calibration bandpass calibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' auto-correlation flagging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' and HI flagging avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' These methods are implemented in a new framework that uses the Lua language for scripting, which is new in AOFlagger version 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Our approach removes RFI fully automatically, and is robust and effective enough for further calibration and (continuum) imaging of these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Analysis of 304 observations show an average of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1% of lost data due to RFI with a large spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We observe 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='6% RFI in auto-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Computationally, AOFlagger achieves a throughput of 370 MB/s on a single computing node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Compared to published machine learning results, the method is one to two orders of magnitude faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Instrumentation: interferometers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Methods: observational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Techniques: interferometric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Surveys;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Radio continuum: general 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Introduction Technical advancement of mankind is driving an increase of man-made radio-frequency transmitters, both terrestrial and in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This raises the bar for radio astronomical studies that try to detect sky signals that are many orders of magnitude fainter than man-made transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Now that radio-astronomy is evolving into a science where it is the norm to measure data volumes in petabytes, mitigation of radio-frequency interference (RFI) needs to be computationally efficient and fully automated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Apertif is a receiver system upgrade for the Westerbork Syn- thesis Radio Telescope (WSRT) that makes use of phased-array feeds to allow for 40 simultaneous adjacent beams on the sky (Van Cappellen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Observations are performed at a central frequency of 1280 or 1370 MHz with an instantaneous bandwidth of 300 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The data volume produced by Apertif is considerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Volt- ages from the 12 dishes with Apertif receivers are correlated for all beams, typically integrated for 30 seconds and recorded with four polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The bandwidth of 300 MHz is split into 384 sub-bands, each with 64 channels of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='2 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because of the large bandwidth, it overlaps with various services, includ- ing GPS and air-traffic communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Although the WSRT resides in a radio protected zone, it is not shielded from satel- lites and air-traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Moreover, starting 2020, 5G transmissions make use of the 1452 – 1492 MHz bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For these reasons, Apertif requires an efficient approach to deal with RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Due to the large amount of data, such an approach has to work fully automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The most common method to deal with RFI, is to detect data samples that have a significant contribution of RFI and ignore affected data in the processing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Winkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Middel- berg 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2010a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Prasad & Chengalur 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Peck & Fenech 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This process is referred to as data flagging, and is also our method of choice for dealing with RFI in Apertif data in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Our detec- tion methodology builds upon the RFI detection pipelines for the Low-Frequency Array (LOFAR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Van Haarlem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Of- fringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2010b) and the Murchison Widefield Array (MWA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 1 of 15 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='01562v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='IM] 4 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi Tingay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Those pipelines inte- grate an aoflagger flagging strategy, which combines filtering, sumthresholding, morphological operations and heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' De- tails of the aoflagger approach will be discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Apertif supports a transient (beam-formed) mode and an imaging mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The RFI detection approach for these two modes are fundamentally different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this work we aim at RFI detection in imaging mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=', after having correlated and integrated the voltages from all the antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' See Sclocco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2019) for an approach to mitigate RFI in beam-forming mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Our approach is part of a fully automated Apertif imaging pipeline called aper- cal (Adebahr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A multi-beam receiver makes it possible to perform spatial filtering techniques to suppress interference (Kocz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2010, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Hellbourg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This requires fast dedicated com- puting hardware that processes the raw signals from all the beams, which for Apertif is not available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Spatial filtering is also mainly used to filter out a limited number of known transmit- ters, which for Apertif is likely not sufficient by itself, although it might save some part of the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Another approach to detect interference is by using the spec- tral kurtosis statistic (Gary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Purver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This has shown results that are competitive with amplitude-based detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' However, this requires a specialized correlator and a doubling of the data volume to be able to calcu- late the kurtosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Recently, machine learning has been used to address the is- sue of RFI detection (Harrison & Mishra 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2020) argue that convolutional neural networks can achieve an accuracy that is higher than that of their sumthreshold implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For this comparison, the authors use their own customized implementa- tion of the sumthreshold method, whereas in platforms such as aoflagger the method is typically applied iteratively and com- bined with filters (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2010a,b) and morphological operators (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Van de Gronde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2016) to en- hance the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' With these additions, it has been shown that pipelines such as aoflagger typically detect all interference that astronomers would manually flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this work, we will show- case what can be achieved with traditional methods — including their computational requirements — thereby providing an up- dated base-line to compare against.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this paper, we introduce a flagging strategy for Apertif data using the AOFlagger framework, and demonstrate our de- signed strategy on Apertif data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In §2, we will start by introduc- ing the AOFlagger steps used to construct the Apertif approach, and introduce several new operations that are integrated into the Apertif flagging strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In §3, results of applying this strat- egy are presented, including long-term statistics and the compu- tational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Finally, in §4 we discuss the results and draw conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Method For this work, we have designed an interference detection ap- proach for Apertif based on the existing aoflagger approach and integrated this into the apercal pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apercal is an automated processing pipeline for Apertif imaging observations (Adebahr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2022), consisting of common steps such as data formatting, interference detection, calibration and imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Interference de- tection is one of the first steps during data reduction and is funda- mental for achieving a good and persistent calibration and image quality and later steps of the processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' To improve the detection quality, several modifications to aoflagger are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This consists of extensions of existing algorithms and optimizing parameters for apertif, which we will discuss in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We will start with an overview of the de- tection approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Overview Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 1 shows an overview of the steps that the default aoflagger strategy performs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The aoflagger approach to RFI detection in a subset can be summarized as i) estimation and subtraction of the sky signal by applying a Gaussian high-pass filter in time- frequency space (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' and ii) detection of excessive values, with increased sensitivity towards spectral-lines and broadband features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The detection is performed with the sumthreshold al- gorithm (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Steps i) and ii) are typically iter- ated three times with increased sensitivity to make sure that the final sky signal estimate is minimally biased by interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As a final step, the flags from different polarizations are combined and are extended in time and frequency, using the scale-invariant rank (SIR) operator (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Van de Gronde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This latter step improves detection of interference that ta- pers off below the noise floor and fills gaps in the flag mask when a persistent transmitter is not fully detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' With aoflagger, detection of interference is performed inde- pendently on subsets of the data, and the pipeline of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 1 runs independently for each subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For Apertif, such a subset was chosen to contain the data from all four linearly polarized cor- relations (XX, XY, YX, YY), the full bandwidth (300 MHz), an interval of typically half an hour for a single beam and a single correlated baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Hence, the detection of interference for dif- ferent beams, baselines and time intervals is independently per- formed, even though these are part of the same observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The motivation for flagging these subsets independently is two fold: – It improves performance: it allows parallel and distributed detection of subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The independent flagging of beams and time intervals matches with the format of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Despite this, data access is still not ideal, because the data for one baseline is stored dispersed over the time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' – Combined detection does not significantly improve detec- tion: the added value of detection on combined subsets of data is small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=', one subset contains little information about the RFI in another subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This is because the impact of RFI can vary greatly between different beams and different base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Furthermore, it rarely occurs that RFI which affects image quality is not detectable in half an hour of data, but is detectable when multiple half hour intervals are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Performing detection on integrated baselines has, in some cases, been shown to make faint RFI detectable (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Wilensky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Early tests with Apertif data, how- ever, indicated that there is no gain in combining baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We have also performed tests that flag after integrating over multiple beams, but again found no improvement in doing so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' These tests were not exhaustive and it could be that combined detection on baselines or beams could still improve the accuracy somewhat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' aoflagger aims to take out RFI that requires raw, high- resolution data flagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because of the high resolution of pro- cessed data, the computational performance of detection is crit- ical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It is important to perform high-resolution flagging early, because it results in the highest accuracy and the impact of flag- ging is reduced compared to low-resolution flagging (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' On the other hand, some phenomena cause the loss Article number, page 2 of 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' : An interference detection strategy for Apertif based on AOFlagger 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The default aoflagger strategy for RFI-detection (before modifications for Apertif).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' These steps are independently performed on smaller subsets of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The input data of one independent run through these steps typically consists of approximately an hour of correlations from a single pair of antennas and a single beam, with the full bandwidth and all four linearly polarized cross-correlations present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' of large time intervals or frequency ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Common instrumen- tal causes are correlator failures, temporary local RFI or strong broadband transmitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Detection of such issues does not require the high-resolution data, and it is therefore less critical to detect such issues in the first aoflagger detection run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Such issues can be found in post-processing of lower-resolution data for which the performance is less critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Invalid data There are several instrumental issues that may result in data with invalid values that interrupt the data in time or frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A few examples of such issues are correlator malfunctions, dish shad- owing, incorrectly set sub-band gains, network failures (between stations and the correlator) or data corruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Such instrumen- tal issues result in visibilities that may have non-physical values for certain times, frequencies, feeds or antennas, or could lead to visibilities with a not-a-number (NaN) value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We will refer to such data as invalid data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In most cases, invalid data can be detected and flagged early in the processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For example, shadowing can be determined from the target direction and the layout of the array, and miss- ing sub-band data caused by network congestion can be detected by the correlator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this paper, we consider the detection of such issues outside the context of interference detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It does, how- ever, make it necessary for the detector to continue to work in the presence of (pre-detected) invalid data, which may affect only specific times, frequencies or some other selection of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Making the aoflagger algorithm aware of invalid data is one of the changes that was required for Apertif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The aoflagger al- gorithm was originally designed to work on raw high-resolution single-subband LOFAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It rarely happens that such a span of data is partially invalid, and initially aoflagger algorithms therefore do not take invalid data into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In the case of Apertif, the full bandwidth is offered to aoflagger, and the loss or corruption of a single subband causes therefore gaps in the bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Being a different instrument, Apertif is also affected by different issues that may not affect LOFAR, such as shadow- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For these reasons, we have extended the aoflagger algo- rithm to take invalid data into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This requires changes to the sumthreshold and sir-operator steps of the algorithm, which we will discuss in the next two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Extension of the sumthreshold algorithm The sumthreshold algorithm is a combinatorial thresholding method that detects line-like structures in the time-frequency data (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2010a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This method is effective for the de- tection of RFI, because most RFI raises the amplitude of consec- utive time or frequency samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The method iteratively thresh- olds the average over an increasing number of neighbouring samples with a decreasing threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' With i the zero-indexed it- eration number, Mi the number of samples, χi the threshold and ρ a constant normally chosen to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5, Mi = 2i (1) χi = χ0 ρ− log2 Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2) χ0 is a user parameter that controls the total sensitivity of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The various default aoflagger algorithms use values of χ0 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The mode of the noise σ is determined from the data that is (at that point in the detection) determined to be RFI free, and is estimated by calculating the truncated mode of the RFI free data, skipping 20% of the outlier values (the 10% min- imum and maximum values), thereby assuming that the inner 80% follow a Rayleigh distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Assuming that the contribu- tion of the noise is Gaussian distributed in the real and imaginary components of the visibilities, this results in a stable estimate of its standard deviation (Fridman 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A single iteration consists of thresholding all sequences of size Mi in both the time and the frequency direction (unless Mi = 1), possibly with different thresholds for the two dimen- sions, to separately control the sensitivity towards spectral line RFI and transient broadband RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Typically, a total of 9 of these iterations are performed, giving a maximum size of M8 = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A sample that is flagged in an earlier iteration or direction, is (temporarily) replaced by the mean of the non-flagged samples in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The following description demonstrates the first three iterations, using χ0 = 6 and ρ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Flag samples with an absolute value ≥ 6σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (a) Flag every sequence of 2 consecutive samples in time with an absolute average ≥ 4σ (because χ2 = 6σ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5− log2(22) = 4σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (b) Flag every sequence of 2 consecutive samples in fre- quency with an absolute average ≥ 4σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Repeat 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (a) and (b) with 4 samples and a threshold of χ4 = 6σ × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5− log2(23) = 2 2 3σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 3 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 0 10 20 30 40 50 60 70 80 90 Frequency index 0 20 40 60 80 100 120 140 160 180 Time index 0 10 20 30 40 50 60 70 80 90 Frequency index 40 45 50 55 Time index Include bad data 0 10 20 30 40 50 60 70 80 90 Frequency index 40 45 50 55 Time index Set bad data to zero 0 10 20 30 40 50 60 70 80 90 Frequency index 40 45 50 55 Time index Exclude bad data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Three methods of handling invalid data in the sumthreshold step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The top image shows the simulated input data, which consists of Gaussian complex noise, spectral line RFI every 10 channels that increases in strength in frequency direction, and a block of invalid data (time indices 50– 100), simulating e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' a temporary correlator failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The bottom images show a zoom in on the left edge of the invalid data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Flagged data is marked in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Bottom-left: normal sumthreshold without using knowledge of the invalid data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' bottom-centre: invalid samples are set to zero before sumthreshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' bottom-right: invalid samples are removed before sumthreshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Subsequent iterations will threshold sequences of 8, 16, 32, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' samples with an average above χ8 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='8σ, χ16 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='2σ, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In the form described by Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2010a), pre-existing classification of invalid data is not taken into account in the sumthreshold method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An example of such a case is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2, which considers a simulated observation with 200 timesteps and 100 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The observation contains spectral- line interference that affects one channel out of every ten chan- nels and increases power at higher frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Timesteps 50— 100 are known to be invalid data, and are set to high values by raising them with 10 times the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2 zooms in on time indices 30-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The first image of the second row shows the result of a basic application of sumthreshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For this result, the knowledge that some data was invalid is not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As a result, the invalid data is considered to be RFI, and samples before and after the block of invalid data are flagged with an increased sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As a result, the false-positive rate is clearly increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A simple approach to mitigate this is to consider invalid val- ues to be zero when applying the sumthreshold method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This re- sults in the plot shown in the middle of the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This result does not show increased false positives because of the invalid data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' With this approach, information about flagged samples on either side (before/after) of the missing data does not (significantly) aid detection, because the invalid data is consid- ered to be zero, and this lowers the average absolute sum in the iterations of the sumthreshold method that consider longer con- secutive ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This results in a higher false-negative rate than would theoretically be possible if the information on both sides of the invalid data would have been used together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In particu- lar, the faintest interfering line at channel index 5 is no longer detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While the loss in accuracy is minimal, there is a simple method to aid the detection of interference on one side of the block of invalid data with information from the other block: by completely skipping data in the summed direction (time or fre- quency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In other words, samples that are directly before and after a block of invalid data are treated as if they are consec- utive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The result of this is shown in the third column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2, which indeed shows a lower false-negative rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In particular, the faintest spectral line at channel 5 is now fully detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' When comparing these two approaches to deal with invalid data, the approach to exclude the invalid data leads to a small increase in false-positive detections when the RFI is not con- sistently present in time or frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' when it is present on one side of the invalid data block and not present on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This should be weighted against the increased sensitivity when the RFI is consistently present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The optimal choice there- fore depends on the behaviour of the RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because persistent RFI is common, and because it is more important to avoid false negatives in persistent RFI (which might negatively affect later processing steps) over avoiding false negatives in transient RFI (which would lead to a small loss of data), we use the method of excluding invalid data in our Apertif strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We have implemented this in two ways: i) stack all valid data into a temporary storage, run the normal sumthreshold algorithm on these data and reverse the stacking operation on the resulting mask;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' and ii) skip over the invalid data inside the sumthreshold method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We have timed these two implementa- tions on simulated complex Gaussian data with 10,000 timesteps × 256 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Each run is repeated 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The first im- plementation runs about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5× faster (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='18 s per data set) com- pared to the second implementation (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='45 s per data set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The first method is still 6× slower compared to the regular algorithm Article number, page 4 of 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' : An interference detection strategy for Apertif based on AOFlagger 3 (which takes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='03 s per data set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This can be explained by the extra copying of data that is required in each iteration (both for the time and for the frequency direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Extension of the scale-invariant rank operator The SIR-operator is a morphological operation that is used in aoflagger to extend the detected RFI mask in the time and fre- quency direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It is an effective step to follow threshold-based methods to detect faint RFI based on the morphology of detected flags (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Van de Gronde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It is scale invariant, which implies that the fractional increase in flags in one dimension is constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=', independent of the scale of that feature in that dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The SIR-operator is essentially a one-dimensional operator that can be applied to a sequence of flag values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' To apply it to radio interferometric data, Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2012) apply the op- erator in both the time and frequency dimensions: in time it is separately applied to all the channels, and in frequency it is ap- plied separately to all timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The union of these to steps is taken as the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Assume that X is a single sequence of flag values, such that X[i] holds a Boolean value that represents the state of the flag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The output ρ(X) of the SIR-operator applied to X, is defined as the union of all subsequences of the input X, for which #i: j F ≥ (1 − η) ( j − i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (3) Here, #i: j F is brief for #F (X[i : j]), which is the count-operator that returns the number of flagged samples in a sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' X[i : j] is the subsequence of samples consisting of all elements X[k] for i ≤ k < j and η ∈ [0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 1] is a tunable parameter that sets the aggressiveness of the operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (3) implies that a sequence of flags caused by invalid data is extended on both sides by a ratio of η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An example of this is given in the centre-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This behaviour is undesir- able because, unlike most RFI signals, invalid data typically has a sharp boundary and should be flagged like that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The extension of invalid data causes a high number of false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A simple solution is to count invalid data as unflagged data in the SIR operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This implies that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (3) is modified so that the count operator only counts the number of flags corresponding to valid data: #i: j F V ≥ (1 − η) ( j − i) , (4) where #F V is the number of valid samples that are flagged in the interval X[i : j] (as opposed to #F , which counts flagged val- ues that can both be valid or invalid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because the right side is unchanged and the left side remains equal or is decreased com- pared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (3), this modification always flags an equal or fewer amount of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An application of this approach is demon- strated in the centre-right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This approach reme- dies the extending of flags around invalid data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The downside of the approach of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (5) is that a continu- ous transmitter is assumed not to be present in the invalid data range, causing flags on either side to have a decreased proba- bility of getting flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For example, in case a correlator fails for a minute during which a transmitter remains present in one channel with decreasing power, the transmitter is less likely to be flagged after the correlator failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' To address this, we further modify Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (3) to: #i: j F ≥ (1 − η) #i: j V, (5) where #i: j V is the number of valid (flagged or unflagged) samples in interval X[i : j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This approach is effectively the same as re- moving the invalid samples from the sequence before applying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Therefore, a transmitter that gets interrupted by invalid data receives a higher probability to get flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An example of this approach is given in the bottom-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Invalid samples are skipped in this approach, and flagged samples on one side of a sequence of invalid samples may increase the prob- ability of samples on the other side of the sequence, irregardless of the size of the invalid sample sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The approach of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (5) can overstep its goal of using in- formation from before and after a sequence of invalid data, in particular in the case of very long sequences of invalid samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For example, when considering a transmitter that is active for one minute before the receiving antenna is shadowed for 6 hours (causing invalid data), it is undesirable that samples after shad- owing receive higher detection probability because of what hap- pened 6 hours ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A final modification to the SIR operator we consider is therefore to introduce a penalty parameter ρ that can balance between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (4) and (5): #i: j F ≥ (1 − η) � ( j − i)ρ + #i: j V(1 − ρ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (6) With ρ = 0, invalid samples are skipped, making the method equal to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (5) and with ρ = 1, invalid samples are counted as unflagged samples, making the method equal to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A value of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='2 implies that five invalid samples count as one unflagged sample, thereby lowering the probability of flagging through a block of invalid data, but still transferring some of the flag information from before to after the invalid data and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This method is demonstrated with a setting of ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1 in the bottom-right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Considering the results of all approaches in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3, it is clearly undesirable to generally extend invalid data using the tra- ditional SIR-operator defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Any of the three different variations of the algorithm (Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4, 5 and 6), which can be de- scribed by choosing different ρ-values in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (6), solve this is- sue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The different values of ρ do not cause significant changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We have tested values of ρ on a few observations, some with ar- tificially added invalid data, and visually compared the flagging results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Based on these results and the arguments given earlier about finding a balance between Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (4) and (5), we use ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Introducing the parameter for invalid-data weighting ρ has no significant effect on the speed of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The original algorithm can be implemented with a computational complex- ity of O(N) (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2012), and the same holds for the algorithm that includes the invalid-data penalty parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' High-pass filtering The high-pass filter that is applied to remove astronomical source contribution before thresholding is, for computational reasons, implemented as a Gaussian low-pass filter followed by subtracting the difference between the input and the low-pass fil- tered result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The high frequency resolution of Apertif makes it necessary to use a large filtering kernel in the frequency direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Effectively, a kernel with a Gaussian sigma of 875 channels and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 timesteps is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Before filtering, the data is averaged in the frequency direction by a factor of 175, and after low-pass filtering, the result is upscaled to the original resolution using nearest neighbour resampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This allows a convolution with a much smaller kernel, improving the speed of this operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The result is an approximate of a Gaussian high-pass filter, but for the purpose of removing the sky signal, this is sufficiently accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 5 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 6 0 10 20 30 40 50 60 70 80 90 Channel 0 20 40 60 80 100 120 140 160 180 Time Input 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 6 0 10 20 30 40 50 60 70 80 90 Channel 0 20 40 60 80 100 120 140 160 180 Time Flag invalid data before SIR operator 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 6 0 10 20 30 40 50 60 70 80 90 Channel 0 20 40 60 80 100 120 140 160 180 Time Unflag invalid data before SIR operator 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 6 0 10 20 30 40 50 60 70 80 90 Channel 0 20 40 60 80 100 120 140 160 180 Time Exclude invalid data before SIR operator Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Different ways of handling invalid data in the sir-operator step on a simulated data set with a Gaussian burst of interference in a few channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Purple marks invalid data, yellow is detected as interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The SIR-operator operates on the flag mask, hence the visibility values are not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Top: input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Centre-left: Invalid data is counted as flagged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Centre-right: Invalid data is counted as unflagged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Bottom-left: Invalid data is removed before applying the SIR-operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Bottom-right: Invalid data is penalized with ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Bandpass correction In the Apercal Apertif processing pipeline, the entire bandwidth of Apertif is used at once during RFI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This is different from the original LOFAR strategy, that flagged small (200 KHz) subbands independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Using the entire bandwidth has the ben- efit that broadband RFI that covers several sub-bands can be de- tected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This is relevant for Apertif observations, which are af- fected by broadband transmitting satellites and radar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because the bandwidth of Apertif is subdivided into sub- bands using a poly-phase filter bank, the band shape of the poly- phase filter is imprinted on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An example of this is shown in the top-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This is corrected for during cali- bration, but during flagging (which needs to be done before cal- ibration) the shape is still present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Performing detection using the entire bandwidth but without correcting for the poly-phase filter bank causes sub-band edge channels to be flagged, because the edges cause sharp transi- tions that trigger the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Moreover, the deviations in the data caused by the band-edges decrease the sensitivity of the de- tection toward actual RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The top-right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4 shows an example of flagging without bandpass correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' To remedy this, we implement a sub-band band-pass correc- tion step in the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This step corrects the poly-phase filter shape using a static, observation-independent correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We de- termine the shape by performing gain-calibration on a clean re- gion of the band, and average the solutions over the subbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The bottom-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4 shows the resulting corrected data set, and the bottom-right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4 shows the result of flagging the bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As can be seen, the band-pass cor- Article number, page 6 of 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' : An interference detection strategy for Apertif based on AOFlagger 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Static sub-band band-pass correction before flagging with the Apertif flagging strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Top-left: input before correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' top-right: flagged without correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' bottom-left: input after correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' bottom-right: flagged with correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' rection has decreased the number of false detections consider- ably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Some edge channels are still flagged, even after correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This is caused by aliasing in the sub-band edge channels, which change the statistics of those edge channels slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This can lead to artefacts which are very similar to RFI, hence they are occasionally flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This flagging is normally of limited con- cern, because those sub-band edge channels that are flagged are of lower quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because of this, they are often discarded during imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Flagging of auto-correlations Given the output voltage of the two feeds of the same antenna, e = � ex, ey � , auto-correlated visibilities are formed by taking the product eHe (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=', the outer product e ⊗ e) and integrating, re- sulting in XX, XY, YX and YY visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While auto-correlations are not often used for scientific data products, they are useful for system monitoring and quantifying the system noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For such analyses, it is desirable to flag RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Compared to cross-correlated visibilities, auto-correlated visibilities have different properties: in the XX and YY corre- lations, system noise and RFI will not decorrelate, and auto- correlated visibilities are sensitive to the global sky signal in- stead of fluctuations in the sky signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An example of auto-correlated dynamic spectrum from Apertif is shown in the top image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 5 (after sub-band band-pass correction as described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Compared to cross- correlations such as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4, the dynamic spectrum of auto-correlated visibilities appears much smoother, is system- atically offset from zero and contains stronger structure in the frequency direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The flagging strategy that was optimized for the cross- correlations detects RFI by comparing high-passed filtered am- plitudes of visibilities to the variance of these amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Be- cause the amplitude variance is much lower compared to cross- correlations, this results in flagging auto-correlations with in- creased sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' At the same time, the auto-correlations con- tain stronger instrumental frequency-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' These two effects combined causes the cross-correlation flagging strategy to flag all of the visibilities of the auto-correlations of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' To solve this, we use a different flagging configuration for the auto-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The difference with the cross-correlation strategy is as follows: The time-direction sumthreshold step (sensitive to consis- tently high values in the time direction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' band-pass struc- ture) is reduced in sensitivity by a factor of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The frequency-direction sumthreshold step (sensitive to consistently high values in the frequency direction, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' broad-band RFI) is reduced in sensitivity by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 7 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi 1280 1300 1320 1340 1360 1380 1400 1420 Frequency (MHz) 14:00 16:00 18:00 20:00 22:00 0:00 Time (UTC, hh:mm) 1280 1300 1320 1340 1360 1380 1400 1420 Frequency (MHz) 14:00 16:00 18:00 20:00 22:00 0:00 Time (UTC, hh:mm) 1280 1300 1320 1340 1360 1380 1400 1420 Frequency (MHz) 14:00 16:00 18:00 20:00 22:00 0:00 Time (UTC, hh:mm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Flagging of auto-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Top image: input after sub-band band-pass correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' centre image: same after iterative high-pass fil- tering and with 10x more sensitive colour scale;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' bottom image: af- ter flagging with the auto-correlation specific strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because auto- correlations have different properties compared to cross-correlations, they require a specialized flagging strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The size of the high-pass filter kernel is reduced by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 in the frequency direction, to filter out more of the spectral gain fluctuations of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The number of iterations is increased from 3 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This in- creases the required computations but improves robustness in the presence of a large dynamic range, as is the case for auto-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Only the XX, XY and YY correlations are used for detection, to reduce unnecessary computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' YX correlations are equal to the conjugated XY correlations, and using these for flag- ging does not provide additional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A result of this auto-correlations strategy is shown in the bot- tom image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Visual inspection shows that all visible RFI is indeed detected, and the number of false detections appears low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because we do not have a ground truth, we do not try to quantify these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Similar to the cross-correlation strategy, the auto-correlation strategy flags parts of the sub-band edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The centre image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 5 shows the high-pass filtered data of the final iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Avoiding HI removal In observations that cover bright nearby galaxies or the Galactic plane, the 1420 MHz HI line may be detectable in the visibil- ities from a single cross-correlated baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For example, the top-left image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6 shows one baseline from a M31 ob- servation, which clearly shows a contribution from HI-emission around 1420 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This poses a challenge for RFI detection, be- cause such a fine, spectrally-consistent signal is quite similar to RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As shown in the top-right image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6, when standard flagging is performed on these data, the HI emission is detected as RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We analyze different ways to mitigate this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In the Nether- lands, frequencies between 1400-1427 MHz are reserved for ra- dio astronomy and other forms of passive research1, and trans- mitting inside this band is not allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As a result, these frequen- cies are almost free of man-made emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A simple mitigation strategy is therefore to disable RFI detection inside this band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Unfortunately, the recorded visibilities do occasionally contain strong, non-astronomical values inside this band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The three ver- tical lines in the images of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6 are an example of such an observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Most frequently, these are caused by saturation of a receiver, causing a broadband-like signal in the recorded visibili- ties, although they might occasionally be caused by RFI emitted at these frequencies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' from a sparking device or lightning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Leaving these broadband contaminants in the data causes degra- dation of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In particular, they cause visible stripes in continuum, full bandwidth images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Another approach is to flag only based on Stokes Q, U and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Man-made RFI is often polarized, whereas the sky emission in these polarizations is generally much fainter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The result of this approach is shown in the bottom-left image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While a part of the HI emission has been left intact, it is still bright enough in these polarizations to get detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This is even the case when flagging on only one of these polarizations: the HI emission is present in all of the polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Moreover, we oc- casionally observe RFI that is only visible in Stokes I, and re- moving any of the polarizations decreases the effectiveness of RFI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6, the transmitter around 1425 MHz / 0:00 UTC is for example not as well detected in this approach com- pared to standard flagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because none of these approaches give good results, we con- sider another approach, and run the flagger twice: in run A) we flag the data with the normal detection strategy, and in run B) we run the detection with a strategy that is insensitive to spectral lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For frequencies outside the HI range we use the flags from run A), and inside the HI range (1418–1424 MHz) we use B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The result of this approach is shown in the bottom-right image of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' With this approach, broadband structures have been detected as RFI and HI emission is left in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' To avoid flagging spectral lines in run B), we adjust the fol- lowing flagging settings during this run): 1 The Dutch spectrum allocations can be found at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' agentschaptelecom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='nl/ Article number, page 8 of 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' : An interference detection strategy for Apertif based on AOFlagger 3 0 200 400 600 800 1000 1200 Visibility (Jy) 1405 1410 1415 1420 1425 Frequency (MHz) 20:00 22:00 0:00 2:00 4:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) 0 200 400 600 800 1000 1200 Visibility (Jy) 1405 1410 1415 1420 1425 Frequency (MHz) 20:00 22:00 0:00 2:00 4:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) 0 200 400 600 800 1000 1200 Visibility (Jy) 1405 1410 1415 1420 1425 Frequency (MHz) 20:00 22:00 0:00 2:00 4:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) 0 200 400 600 800 1000 1200 Visibility (Jy) 1405 1410 1415 1420 1425 Frequency (MHz) 20:00 22:00 0:00 2:00 4:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Band-pass corrected M31 data from WSRT RT9 × RTA with a strong HI signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Top-left image: input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The bright emission around 1420 MHz is from HI and should not be flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The vertical lines are instrument or RFI artefacts that should be flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Top-right image: after RFI detection without HI modifications, showing in pink what is flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Bottom-left image: after RFI detection using Stokes Q, U and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Bottom-right image: after RFI detection using a specialized strategy for 1418-1424 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The high-pass filter in frequency direction is set to have a kernel size of one channel, to filter out fluctuations in fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The sensitivity of the time-direction sumthreshold step is de- creased by a factor of 4, to reduce flagging of line-like struc- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The sensitivity of the frequency-direction sumthreshold step is decreased by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This reduces flagging of tem- poral fringes in HI emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The number of iterations is increased to remain robust in the presence of strong HI emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' On overall, the resulting strategy is almost entirely insensitive to spectral-line-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The sensitivity to broadband struc- tures will also be reduced because of these changes, but given that this strategy remains sensitive to faint broadband structures such as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6, we consider this tolerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because run B) requires only a small part of the full band- width, the second flagging run is relatively fast, hence the in- crease in computations caused by this is modest (about 20%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Reading overhead and memory considerations During the AOFlagger stage of the apercal pipeline, observa- tions are stored in the Casacore Measurement Set format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this format, the data of an observation is lexicographically sorted in time, and then in baseline and frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While this ordering is suitable for calibration, flagging requires the data baseline by baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Unfortunately, the data for a single baseline is spread throughout the file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Therefore, reading a baseline requires read- ing the file from beginning to end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because of the block size and caching of storage media, it is inefficient to read the baselines one by one with this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' AOFlagger supports three methods for accessing the data: Direct reading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this mode, the data is directly read from the measurement set just before they are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because multi- ple baselines are processed in parallel using multi-threading, a few baselines are read from the measurement set at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This mode results in scanning through the input data multiple times, which is computationally costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Reorder before processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this mode, the whole measure- ment set is reordered by baseline, frequency and then time and rewritten to disk in a binary, internal format before pro- cessing is started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This results in reading the data only twice and is generally faster than the direct reading mode, but re- quires disk space to store the copy of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In-memory data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this mode, the whole measurement set is read into memory before starting processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This results in reading the data only once and is generally the fastest mode, but requires a considerable amount of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Apertif data sets are large and expensive to read: reading the data more than once is undesirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As a result, the only acceptable reading mode is the in-memory mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In the particular comput- ing mode where Apercal runs, the amount of memory required by this mode is a considerable constraint, and requires a dedi- cated node for each flagging operation performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Other observatories have solved this issue by integrating aoflagger into a multi-step preprocessing pipeline that stream through the data, split the data in time for flagging and hand these data over part by part to AOFlagger via its application pro- gramming interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Examples of such pipelines are cotter (Of- fringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2015) and DP3 (Van Diepen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2018), which are preprocessing pipelines for the Murchison Widefield Array and the Low-Frequency Array, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this approach, sev- eral tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' conversion, phase rotation, flagging, averaging, compression) can be applied with a single read through the data, thereby reducing the read overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In the case of Apertif, such Article number, page 9 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi a streaming pipeline does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Instead, aoflagger runs as a stand-alone tool inside Apercal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' To solve the memory and reading issue for Apertif, we imple- mented a time-chunking approach into aoflagger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this mode, aoflagger reads small chunks in time and flags these indepen- dently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This makes it possible to use the memory reading mode, because the data for individual chunks is small enough to fit in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It does imply that the algorithm has less information available to do its RFI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Therefore, it is important to let time chunks still have a significant size, because AOFlagger would otherwise not be able to find faint RFI, that is persistent in time, but not detectable in a small chunk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For Apertif, we use a chunk size corresponding to about half an hour of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Use of Lua Before AOFlagger version 3, AOFlagger strategies were written in the extensible markup language (XML).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An XML file speci- fies a sequence of steps and is interpreted by AOFlagger, and this sequence is executed separately for the data from every baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The sequences run multi-threaded, and reading and writing of data is done outside of the strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Examples of XML steps are to calculate visibility amplitudes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' running sumthreshold or sir operations on the data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' or to combine the flags of all polariza- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Over the years, the use of AOFlagger extended to more and more use-cases: different telescopes, flagging after calibration, high-resolution flagging, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It became desirable to make the strategies more flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In particularly, it became desirable to support standard scripting structures such as loops, condition- als, variables and to provide standardized documentation of the steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The idea was therefore formed to embed a standard in- terpreter into AOFlagger and provide a function interface for each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The data-intensive computations are still performed by high-performance precompiled C++ code, while these are glued together using an interpreted script, thereby combining flexibility with high performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Our first approach was to embed it into Python, because of its popularity in astronomical data science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' After having im- plemented a prototype that embeds the Python interpreter into AOFlagger, it turns out some of the features of the Python inter- preter conflict with how AOFlagger runs these scripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Particular challenges were to deal with the global interpret lock;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' memory management;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' and fast restarts of the interpreter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While there are various ways to work around these issues, the design goals of the Python language and interpreters do not focus specifically to make the language embeddable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Lua2 is a scripting language that is widely used for em- bedding scripts in applications, notably in computer games to implement scripted game sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This scenario is close to the AOFlagger use-case: the interpreter is integrated into such games, called many times and supports multi-threaded script ex- ecution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Algorithmic code that requires high performance can be implemented in compiled languages (C++ in the AOFlagger case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' With this idea in mind, we decided to integrate the Lua interpreter into AOFlagger and implement all steps as Lua func- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The use of a full scripting language has increased the pos- sibilities inside the flagging strategies considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For exam- ple, it is now possible to adapt the strategy based on properties such as the baseline length, frequency, auto- or cross-correlation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A consequence of the new interface is that existing strate- 2 https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='lua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='org/ gies need to be rewritten, which can not be done automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' All default strategies have been rewritten to use Lua, which cur- rently includes specialized scripts for 11 observatories (Aartfaac, APERTIF, Arecibo, ATCA, Bighorns, JVLA, MWA, WSRT, LOFAR, NenuFAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' These have all been verified to produce the same result as the old XML-based strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because the new function interface gives better control over what steps need to be run, the speed of the new strategies is slightly higher (several percent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We do not notice any significant overhead from using Lua: the computational time is dominated by the computations inside the function calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Results Apertif observations are processed by the automated Apercal pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This pipeline includes the flagging strategy as de- scribed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this section, we present results of the full flagging step on Apertif observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The data that we look at has been recorded between 2019 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Science products from the first year of observing have been described in the first Apertif data release (Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' in press;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Kutkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' in press).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' RFI detection examples The detection strategy described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2 runs fully automated, and does not require further flagging before calibration and con- tinuum imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In general, manual inspection of data after RFI detection shows no residual RFI and few false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 7 shows the 1280–1430 MHz range of a typical observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The top plot shows the data before RFI detection, and the bottom plot shows in white what has been detected as RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 8 shows a challenging case with wider bandwidth, with a moderate amount of RFI, missing data (1200–1220 MHz) and strong fringes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Top and bottom plots show again before and after detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This also demonstrates the challenging situation for radio astronomi- cal science between 1150 and 1300 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For continuum imaging, it is often useful (or at least prag- matic) to take out any visibility that appears to have a contribu- tion from RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For spectral imaging, a flagging result such as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 8 is problematic, because many channels are fully removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In those cases, it is possible to reduce the sensitivity of the RFI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The sensitivity is specified as a variable in the script.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' For the detection result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 9, the sensitivity was decreased by a factor of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Compared with the result in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 8, this reduced the flagging from 49% to 33%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This takes out the strongest RFI, but leaves weak (but visible) RFI in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' De- creasing the sensitivity further continues to trade the availability of visibilities with a lower quality of those visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' RFI characteristics and long-term statistics During the flagging step, statistics are collected that summa- rize the (detected) RFI occupancy and data quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We have collected these statistics for 304 of the currently processed ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Averaged over all these observations and the full bandwidth, the total detected RFI occupancy is 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1% in the cross-correlated baselines and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='6% in auto-correlated base- lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 10 shows the detected spectral RFI occupancy for each observation, as well as the occupancy averaged over all observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Only cross-correlated data is included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' At most frequen- cies, the average loss of data due to RFI is about 10%, but with a spread of approximately 0-15% between observations, and a few larger outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 10 of 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' : An interference detection strategy for Apertif based on AOFlagger 3 Frequencies between 1400 and 1427 MHz are reserved for radio astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' At these frequencies, the average RFI occu- pancy is slightly lower (approximately 8%), but is evidently still affected by instrumental effects (such as receiver saturation) or natural and unintended RFI (such as lightning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 6 shows data that is affected by such broadband artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It is likely that the ∼10% base-level of occupancy is caused by such artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Some observations show a small excess RFI occupancy at 1420 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This is caused by HI that is detected as RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The methods to avoid flagging HI that are described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='8 were implemented only halfway 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Some of the observations that are flagged before that still show false-positive detections at HI frequencies, but all observations after avoiding HI was imple- mented show indeed no HI flagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The same base level of 10% is not visible at frequencies above 1430 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The reason for this difference is that only a relative small number of observations cover frequencies above 1430 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Frequencies between 1427 and 1492 MHz are allo- cated to various services, including mobile communication and fixed transmissions3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Some of these are satellite based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In 2020, the 1452—1492 MHz band was auctioned in the Netherlands and thereafter allocated for the use of 5G mobile phone down- link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 10, the use of data above 1430 MHz is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Some channels between 1300–1400 MHz contain a few out- lier RFI occupancies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' These are caused by a nearby radar sta- tion that is occasionally turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Frequencies between 1130 and 1300 MHz are predominantly affected by RFI from Global Navigation Satellite Systems (GNSS), such as the US GPS, Rus- sian GLONASS, Chinese BeiDou, and European Galileo satel- lite constellations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' All these constellations use satellites in or- bits at ∼2000 km and with high orbital inclinations (i = 54–65◦) to provide global coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Frequencies for wide band trans- missions are assigned to, and shared between, these systems at 1176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='45, 1191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='795, 1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='14, 1227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='6, 1278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='75 MHz (for GPS, BeiDou, Galileo) and 1202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='025 and 1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='9375–1251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='6875 MHz (for GLONASS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Wide band signals are detected at these frequencies through- out the entire observation of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 8 covering the band down to 1130 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Using orbital ephemerides of these satellite constel- lations, we find that the strong temporal RFI observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 8 at 13:06, 14:46, 16:29, 18:13 and 19:54UTC is caused by BeiDou satellites passing within 5◦ from the pointing of the APERTIF compound beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The pass of 18:13UTC had a minimum sepa- ration of 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='31 and led to saturation of the receiver, affecting the entire observing band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Two GPS satellites passed at 1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='47 and 2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='30 separation from the beam pointing at 22:02 and 23:02UTC, and one Galileo satellite at 3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='72 at 22:59UTC, and coincident increases of the RFI levels are observed, but not as strong as with the passes of BeiDou satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The GNSS signals observed away from these passes near the primary APERTIF beam are likely due to far sidelobes or multi-path reflections of GNSS sig- nals from the WSRT focus structure or other nearby structures directly into the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Computational requirements In this section we summarize the computational requirements of the Apertif RFI detection strategy, with the aim of making it possible to approximate the computational requirements for other telescopes when a similar flagging strategy is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Since the total throughput is depending on many complex factors of 3 See https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='agentschaptelecom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='nl/ the computing platform (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' clock speed, cores, memory band- width, instruction set, vectorization), we aim at giving a first- order estimate only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We measure the performance of flagging a set with visibil- ities from a single observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We use an Apertif observation with 1346 timesteps, 24572 channels and 4 polarizations, for a total of 132M visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This makes the visibility data, which consists of 4-byte single-precision real and imaginary values, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='1 GB in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We perform our test on a desktop machine with an AMD Ryzen 7 2700X 8-Core processor and 64 GB of memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This processor can perform hyper-threading, and thus we run 16 de- tections in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We load the data in memory before detec- tion and do not store the results, to avoid any disk access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Av- eraged over 10 runs, it takes 46 seconds to run 16 detections, which amounts to a throughput of 370 MB/s (or 46M visibili- ties/second).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' At the time of writing, a typical fast spinning disk achieves a sustained reading throughput of a few hundred MB/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Hence, disk access can be a significant cost of a stand-alone RFI detection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This can be problematic for supercomput- ers, because they have high computing power, but not a high I/O throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Comparison against a machine learning approach Some studies have found that machine learning can improve the accuracy of RFI detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2020), the authors test their own sumthreshold implementation against a machine learning approach, using a ground truth flag mask that is man- ually determined by an engineer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Such a ground truth mask is difficult to make in general, including for Apertif data, where broadband RFI tapers off and it is unclear from which points samples are truly unaffected by RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We can however conclude that, after our pipeline, all visibly affected samples have been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Moreover, imaging results have achieved the thermal noise of the instrument, thereby indicating that the accuracy of interference detection is not a limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This conflicts somewhat with the conclusions made by Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The sumthreshold implementation that is used there to compare their results with, does not achieve the pub- lished accuracy of aoflagger, because residual interference is vi- sually present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Potential explanations for these differences could be i) that Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' train their network for a specific scenario but did not optimize their sumthreshold approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' or ii) that they do not use a full (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' aoflagger-like) sumthreshold-based pipeline that includes the sir operation and that is similarly optimized for their instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' An important consideration is that morpho- logical operations are aimed at detecting RFI that is below the noise, therefore invisible to scientists that manually classify RFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In the comparisons done in Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2020, samples detected by the morphological operator would all be counted as false pos- itives, whereas this operator has been shown to improve the final science results (Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It can therefore not yet be stated that, based on accuracy, machine learning methods are outperforming traditional based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Rather, it is clear that both methods are competitive and are accurate enough to largely mitigate the problem of interference in radio data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' There are differences in the computational performance though.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2022), machine learning methods flag a one-hour FAST observation of 67 GB in 61% of the observing time using 8 computing nodes (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This amounts to a single-node computational performance of 14 GB/hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' On the other hand, the single-node performance of the aoflagger approach listed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='3 is 370 MB/s, or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='3 TB/hour, and aoflag- Article number, page 11 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi ger is therefore almost two orders of magnitude faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While the performance of the computing nodes used for the compu- tational performance analyses may differ somewhat, and it is therefore not a direct comparison, it is evident that the aoflag- ger approach is significantly faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2022), authors compare the run-time of aoflagger to their convolutional neu- ral network (CNN) approach and find that aoflagger is two to four times faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' However, the authors measured the total run- time of the aoflagger executable, which would include disk ac- cess, start-up overhead and time spent in the casacore library to transfer the measurement set data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Because the flagging speed is near the disk access speed, this overhead can be substantial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A better benchmark is possible by using the C++ or Python API of aoflagger directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' On their Sim_RFI-1 dataset, they reach an aoflagger speed of 250 GB/hour, while in this work, with a more advanced strategy, we reach 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='3 TB/hour on similar hard- ware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Their CNN method reaches a speed of 145 GB/hour, which is an order of magnitude faster than what is reached by Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' (2022), but is an order of magnitude below what we reach with our aoflagger approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Discussion & conclusions We have described and demonstrated an automated RFI detec- tion strategy aimed at flagging Apertif data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Our detection strat- egy implements novel sumthreshold and sir-operator algorithms that take prior information about invalid data into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' It also avoids the flagging of HI emission, works on auto-correlations, corrects the sub-band band-pass and contains some further pa- rameter optimizations for Apertif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The change from the AOFlag- ger XML strategies towards fully scripted strategies provides flex- ibility that made these changes quite easy to implement and sup- ports flexibility during experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Besides making the pro- cess easier and faster, an automated RFI detection strategy also makes the results reproducible, compared to when RFI is flagged manually, and it allows reducing the data size by averaging early on in the data reduction processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' We expect that our RFI detection strategy will work for data from other instruments, in particular those with a frequency cov- erage comparable to Apertif, such as MeerKAT, ASKAP, JVLA and future SKA-mid observations around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='0 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='5 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Differ- ent bands might require some changes to the strategy parameters, but should be able to reuse a large part of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While machine learning techniques may compete with the accuracy of AOFlagger, they do not compete with its speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Moreover, we have shown it is possible to add new features to AOFlagger, such as avoiding the 21-cm HI signal, accurate detection in the presence of invalid data and flagging of auto- correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' None of the current available machine learning techniques support these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Most parameters, such as the sensitivity towards broadband and line RFI, or the expected smoothness of the data, are intuitive and easy to tweak for sci- ence cases that e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' require that transients do not get flagged, or that require a difference balance between taking out all visible RFI on one hand, and keeping as much data available for further processing on the other hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This will be challenging, if at all possible, to implement in a machine learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' In this work, we have not made use of the multi-beaming capabilities of Apertif: beam are flagged independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' While some first-order testing indicates that using data integrated over all beams does not improve flagging accuracy, it can be expected that RFI does correlate somewhat over beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' A strategy where the integrated data is searched for RFI, and where this is used as additional input for the flagging of individual beams, might be effective for detecting RFI that is below the noise for a single beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' This work makes use of data from the Apertif system in- stalled at the Westerbork Synthesis Radio Telescope owned by ASTRON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' AS- TRON, the Netherlands Institute for Radio Astronomy, is an institute of the Dutch Research Council (de Nederlandse Organisatie voor Wetenschappelijk Onderzoek, NWO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' BA acknowledges funding from the German Science Foun- dation DFG, within the Collaborative Research Center SFB1491 ”Cosmic In- teracting Matters - From Source to Signal”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' EAKA is supported by the WISE research programme, which is financed by NWO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' JMvdH and KMH, acknowl- edge funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 291531 (‘HIStoryNU’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' JvL, YM and LCO acknowledge funding from the Eu- ropean Research Council under the European Union’s Seventh Framework Pro- gramme (FP/2007-2013)/ERC Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 617199 (‘ALERT’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' PI: JvL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' KMH further acknowledges financial support from the State Agency for Research of the Spanish Ministry of Science, Innovation and Universities through the “Center of Excellence Severo Ochoa” awarded to the Instituto de Astrofísica de Andalucía (SEV-2017-0709) from the coordination of the par- ticipation in SKA-SPAIN, funded by the Ministry of Science and innovation (MICIN) and grant RTI2018-096228-B-C31 (MCIU/AEI/FEDER,UE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' JvL fur- ther acknowledges funding from Vici research programme ‘ARGO’ with project number 639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content='815, financed by NWO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' DV acknowledges support from the Netherlands eScience Center (NLeSC) under grant ASDI.' metadata={'source': 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+page_content=', & Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 2020, Monthly Notices of the Royal Astronomical Society, 492, 1421 Article number, page 12 of 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' : An interference detection strategy for Apertif based on AOFlagger 3 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Visibility (Jy) 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 Frequency (MHz) 19:00 20:00 21:00 22:00 23:00 0:00 1:00 2:00 3:00 4:00 5:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) Observation 200804041,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' band 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' RT7 x RT8 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 Visibility (Jy) 1280 1290 1300 1310 1320 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 Frequency (MHz) 19:00 20:00 21:00 22:00 23:00 0:00 1:00 2:00 3:00 4:00 5:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) Observation 200804041,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' band 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' RT7 x RT8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Typical flagging result for a single baseline in a wideband observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' The top panel shows the input visibilities, and the bottom panel shows the visibilities overlaid with the detection result in white.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' These plots show the Stokes I visibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Some interference features are only visible in Stokes Q, U or V, such as the vertical features around midnight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' All interference features have successfully been detected, and no obvious undesirable detections are visible, with the exception of horizontal flagged features every 200 kHz, caused by the sub-band bandpass (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 18% of the data gets flagged for the baseline in this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 13 of 15 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' apertif-rfi 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Visibility (Jy) 1140 1160 1180 1200 1220 1240 1260 1280 1300 1320 1340 1360 1380 1400 1420 Frequency (MHz) 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) Observation 200125001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Band 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' RT4 x RT6 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Visibility (Jy) 1140 1160 1180 1200 1220 1240 1260 1280 1300 1320 1340 1360 1380 1400 1420 Frequency (MHz) 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 Time (UTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' hh:mm) Observation 200125001,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Band 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' RT4 x RT6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Detection result for a full 300-MHz bandwidth observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 14 of 15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Offringa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' : An interference detection strategy for Apertif based on AOFlagger 3 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Visibility (Jy) 1140 1160 1180 1200 1220 1240 1260 1280 1300 1320 1340 1360 1380 1400 1420 Frequency (MHz) 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 Time (UTC, hh:mm) Observation 200125001, Band 6, RT4 x RT6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 8, but flagged with 3× lower sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 0 20 40 60 80 100 1300 1350 1400 1450 1500 RFI (%) Frequency (MHz) Observations Average Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Percentage of RFI over frequency detected in 304 Apertif observations, excluding auto-correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} +page_content=' Article number, page 15 of 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AzT4oBgHgl3EQfmf0I/content/2301.01562v1.pdf'} diff --git a/WNAzT4oBgHgl3EQfKPsc/vector_store/index.faiss b/WNAzT4oBgHgl3EQfKPsc/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f5e04db9bebfb9697c1a257a326282f49f6e4b81 --- /dev/null +++ b/WNAzT4oBgHgl3EQfKPsc/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:19cb6b82ea0b15c83e23bc786899a0a5b23ca1d505c9d0a129898fb3116dbdb8 +size 10158125 diff --git a/XNAyT4oBgHgl3EQfh_h5/content/tmp_files/2301.00387v1.pdf.txt b/XNAyT4oBgHgl3EQfh_h5/content/tmp_files/2301.00387v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6459f4ad5515fed9069c09bf46ec03c40bf035fb --- /dev/null +++ b/XNAyT4oBgHgl3EQfh_h5/content/tmp_files/2301.00387v1.pdf.txt @@ -0,0 +1,897 @@ +arXiv:2301.00387v1 [cs.DS] 1 Jan 2023 +Exactly Hittable Interval Graphs +S.M. Dhannya, N.S. Narayanaswamy, K.K. Nisha +Department of Computer Science and Engineering +Indian Institute of Technology Madras, Chennai, India. +Abstract. Given a set system X = {U, S}, where U is a set of elements +and S is a set of subsets of U, an exact hitting set U′ is a subset of U such +that each subset in S contains exactly one element in U′. We refer to a +set system as exactly hittable if it has an exact hitting set. In this paper, +we study interval graphs which have intersection models that are exactly +hittable. We refer to these interval graphs as exactly hittable interval +graphs (EHIG). We present a forbidden structure characterization for +EHIG. We also show that the class of proper interval graphs is a strict +subclass of EHIG. Finally, we give an algorithm that runs in polynomial +time to recognize graphs belonging to the class of EHIG. +1 +Introduction +We study classes of simple graphs which are intersection graphs of set systems +that have exact hitting sets. In particular, we introduce a class of interval graphs +which can be represented as intersection graphs of intervals that have exact hit- +ting sets. We refer to this class as Exactly Hittable Interval Graphs (EHIG). We +also present an infinite family of forbidden structures for EHIG. In the following, +we introduce a setting of exact hitting sets and intersection graphs, before pre- +senting our results. +Exact Hitting Sets: Set systems are synonymous with hypergraphs. A hitting +set of a hypergraph H is a subset T of the vertex set of H such that T has +at least one vertex from every hyperedge. If every hyperedge has exactly one +element from T , then T is called an exact hitting set. Also known as the Unique +Hitting Set problem, the Exact Hitting Set problem is a well-studied de- +cision problem that aims to find if a given hypergraph has an exact hitting set. +It finds applications in combinatorial cryptosystems [4] and computational biol- +ogy among many others. Exact Hitting Set problem is the dual of Exact +Cover problem which, in turn, seeks to find a set cover that covers all vertices +of a hypergraph such that the number of occurrences each vertex has in the cover +is exactly one. Some famous examples of Exact Cover problem are sudoku, +tiling dominoes, and n-queens problem. Exact Cover problem is a special case +of Minimum Membership Set Cover problem (MMSC) [12]. While the classic +Set Cover problem seeks to find a set cover of minimum cardinality, MMSC +aims to find a set cover that minimizes the maximum number of occurrences +each vertex has in the cover. MMSC is known to be NP-complete on arbitrary + +set systems [13]. However, for interval hypergraphs, MMSC has been shown to +be solvable in polynomial time by Dom et al. [3]. If a hypergraph H has an +exact hitting set, we refer to H as an exactly hittable hypergraph. It is, indeed, a +natural question to ask if a given hypergraph is exactly hittable. The algorithm +by Dom et al. [3] for MMSC answers this question, but not efficiently for gen- +eral hypergraphs (due to inherent NP-hardness for general cases). However, for +special kinds of hypergraphs like interval hypergraphs, an exact solution can be +obtained in polynomial time. In our work, we address a natural extension to this +question and study simple graphs which are intersection graphs (defined in Sec- +tion 1.1) of exactly hittable hypergraphs, with a focus on interval hypergraphs. +Intersection Graphs: The theory of graphs and hypergraphs are connected +by a very well-studied notion of intersection graphs [5]. It is well-known that +every graph G is an intersection graph of some hypergraph H [11]. H is referred +to as an intersection model or a set representation of G [10], [11]. Interestingly, +certain special classes of graphs are characterized by the structure of their inter- +section models. For instance, Gavril has shown that the class of chordal graphs +are the intersection graphs of subtrees of a tree [6]. When the hyperedges are +restricted to be paths on a tree, the resulting intersection graph class is that of +path chordal graphs which is a proper subclass of the class of chordal graphs +[1],[7],[15],[17]. +Forbidden Structure Characterizations: While a graph G may be identified +as an intersection graph of a structured hypergraph, characterization of G based +on forbidden structures has also been equally well-studied. For instance, the +class of chordal graphs are characterized by the absence of induced cycles of +size 4 or more [10]. Similarly, by the celebrated theorem of Kuratowski [19], the +class of planar graphs must not have subgraphs that are subdivisions of K5 and +K3,3. Interval graphs are known to be the class of chordal graphs without an +asteroidal triple as induced subgraph [14]. The class of proper interval graphs is +a subclass of interval graphs that do not have a K1,3 as an induced subgraph +[18]. Refer to Table 1 for a summary of these examples. Clearly, characterization +of simple graphs based on their intersection models and forbidden structures are +extremely well-studied notions in defining graph classes. +Our results +1. We begin our set of results with a simple extension to a well-known theorem +by Harary that every graph G is the intersection graph of some hypergraph +H [11]. +Observation 1 Every simple undirected graph is the intersection graph of +an exactly hittable hypergraph. Further, if G is a connected chordal graph, +then it is the intersection graph of an exactly hittable set of subtrees of a +tree. +2 + +Table 1. Intersection models and forbidden structures for well-known graph classes +Graph Class +Intersection Model +Forbidden Structures +Simple +An exactly hittable hypergraph +NIL +Planar +Segments on a plane +Subdivisions of K5 and K3,3 [19] +Chordal +Subtrees of a tree +Ck, for k ≥ 4 [10] +Path chordal +Paths on a tree +List given in [16] +Interval +Subpaths on a path +Ck, for k ≥ 4 and asteroidal triple +[14] +Proper interval +Sets of intervals not properly +contained in each other +Ck, for k ≥ 4, asteroidal triple and +K1,3 [18] +Exactly +hittable +interval +graphs (New +graph class) +Exactly hittable sets of intervals Ck, for k ≥ 4, asteroidal triple, K1,3 +and induced path Pk which has, in its +open neighbourhood, an independent +set of at least k + 3 vertices +We present proof of this observation in Section 2. Further to this observa- +tion, we look at a subclass of chordal graphs namely interval graphs, which +are intersection graphs of subpaths on a path. We ask if there is an exactly +hittable intersection model for every interval graph, as in the case of arbi- +trary graphs and arbitrary chordal graphs. Interestingly, the answer is no. +2. We introduce the class of Exactly Hittable Interval Graphs (EHIG), which is +the set of interval graphs that have an exactly hittable interval representa- +tion. We say that an interval graph is exactly hittable if and only if it has +at least one exactly hittable interval representation. We present a forbidden +structure characterization for EHIG. First, we define a family F of simple +graphs as follows: +Definition 1. Let F be the set of an infinite family of interval graphs with +the following structure: each graph has an induced path P consisting of k +vertices, for some k ≥ 1, and the open neighbourhood of P contains an +independent set of size at least k + 3. +Our main contribution in this paper is to prove that every graph in F is a +forbidden structure for EHIG. See Fig 1 for examples of forbidden structures. +In Fig 1(i), u is the induced path P consisting of one vertex with an inde- +pendent set of four vertices {a, b, c, d} in its neighbourhood. Similarly, in Fig +1(ii), a-b is the induced path P consisting of two vertices and {c, d, u, e, f} +is the independent set of five vertices in the neighbourhood of P. +u +a +b +c +d +(i) +c +d +a +b +e +f +u +(ii) +Fig. 1. Two simple instances of forbidden structures +3 + +Theorem 2. An interval graph G is exactly hittable if and only if it does +not contain any graph from the set F as an induced subgraph. +This theorem has been proved in Section 3. We believe that this result is an +interesting addition to the existing graph characterizations primarily because +we could not find such an equivalence elsewhere in the literature, including +graph classes repositories like graphclasses.org. +3. In Section 2, we introduce, what we refer to as, a canonical interval represen- +tation for an interval graph. Given interval graph G, we start with a maximal +clique ordering of G and construct an interval representation from the or- +dering such that the intersection graph of this representation is isomorphic +to G. By construction, there exists exactly one canonical interval represen- +tation for every interval graph. While the canonical representation may be +of independent interest, this representation is crucial in proving Theorem 2 +in this paper. +In Section 2, we prove the following theorem. +Theorem 3. Let G be an interval graph. Let HG be its canonical interval +representation constructed as described in Section 2. Then, G is exactly hit- +table if and only if HG is exactly hittable. +4. Given an interval graph G and its canonical interval representation HG, we +show that the algorithm by Dom et al. [3] to solve the MMSC problem in +interval hypergraphs can be used to recognize EHIG. We present the details +in Section 3.1. +5. We show that the class of EHIG is positioned between the class of proper +interval graphs and the class of interval graphs in the containment hierarchy +of graph classes. +Theorem 4. Proper interval graphs ⊂ EHIG ⊂ Interval Graphs. +The proof of the second part of the above theorem follows from the definition +of EHIG. The first part of the containment relationship has been proven in +Lemma 7. Interestingly, the smallest forbidden structure of EHIG is K1,4 +whereas the forbidden structure for the class of proper interval graphs is +K1,3. +1.1 +Preliminaries +Given a set system X = (U, S), the intersection graph G(X) of sets in X is +the simple graph obtained as follows. For every set S ∈ S, there exists a vertex +vS ∈ G. An edge (vSi, vSj) occurs in G if and only if there exists two sets +Si, Sj ∈ F such that Si ∩ Sj ̸= 0. The family S is called a set representation of +the graph G. A set representation is also referred to as an intersection model [10], +[11]. A hypergraph H = (V, E) is a graph theoretic representation of a set system +4 + +X = (U, S), where the set V corresponds to U and the set E corresponds to S. +The set V contains vertices of hypergraph H and the set E contains hyperedges. +In the intersection graph G, for every hyperedge E ∈ E, there exists a vertex +vE ∈ G. An edge (vEi, vEj) occurs in G if and only if the hyperedges Ei and +Ej have a non-empty intersection. A graph G = (V, E) is an interval graph if +an interval I(v) can be associated to a vertex v ∈ V (G) such that there exists +an edge (u, v) in G and the associated intervals I(u) and I(v) have a non-empty +intersection. The set of intervals {I(v)}v∈V (G) is an interval representation or +intersection model of G. When E is a family of intervals on a line, then G is an +interval graph and H is an interval hypergraph [9]. +Definition 2. [2] The hypergraph H = ([n], I), where [n] = {1, . . ., n} and +I ⊆ {{i, i + 1, . . . , j} | i ≤ j, i, j ∈ [n]} is known as an interval hypergraph. +Each hyperedge in I is a set of consecutive integers, which we call an interval. +In an interval I = {i, i + 1, . . . , j}, i and j are the left and right endpoints of I +respectively, which we denote by l(I) and r(I), respectively. We use V(H) (or +simply V) and I(H) (or simply I) to denote the vertex set and the hyperedge +set, respectively, of an interval hypergraph H. An interval hypergraph is said +to be proper if no interval is contained in another interval. If, for an interval +graph G, there exists an interval representation in which no interval is properly +contained inside another interval, then G is a proper interval graph. +An interval graph is characterized by the existence of a linear ordering of +its maximal cliques. In Section 3, we use the following characterization to ob- +tain an exactly hittable interval representation for an interval graph, if such a +representation exists. +Theorem 5 (Gilmore and Hoffman, 1964 [8]). The maximal cliques of an +interval graph G can be linearly ordered such that, for every vertex x of G, the +maximal cliques containing x occur consecutively. +The class of interval graphs is a subfamily of the class of chordal graphs. A +chordal graph is a simple graph that does not contain any induced cycle of size +≥ 4 [10]. Chordal graphs are known to be intersection graphs of subtrees of a +tree [6]. +Note: We draw the reader’s attention to the distinction between interval hyper- +graphs and interval graphs, and proper interval hypergraphs and proper interval +graphs, as these are used extensively throughout the paper. We also use the +adjective exactly hittable to both simple graphs and hypergraphs. Recall that +a interval graph is an Exactly Hittable Interval Graph if it has an intersection +model that has an exact hitting set. On the other hand, an Exactly Hittable +Hypergraph is one that has an exact hitting set. +Observation 6 Since our goal is to characterize interval graphs that have an +exactly hittable interval representation, we assume without loss of generality that, +in the graph G, for every sequence of consecutive maximal cliques in a linear +ordering, there is at most one vertex which starts and ends in this sequence. +5 + +Indeed if a given graph violates this property and there are two or more vertices +that start and end in a sequence, then we retain only one of those vertices. +The justification for this assertion is that if the resulting graph has an exactly +hittable representation, so does the original graph. +Notations: All standard definitions and notations from West [19] have been +used throughout the paper. +2 +A Canonical Interval Representation +In this section, we obtain a canonical interval representation HG of a given in- +terval graph G. The canonical interval representation is nothing but a special +intersection model of G. Consequently, the intersection graph of intervals in HG +is isomorphic to G. The construction follows a well-defined set of steps with +the result that every interval graph has a unique canonical interval representa- +tion. The canonical representation HG is obtained by stretching intervals so that +all intervals have distinct left endpoints and distinct right endpoints. In other +words, no pair of intervals start at the same point or end at the same point. +The canonical interval representation is crucial to the proof of our main result +in Section 3. +Outline: The starting point of this construction is to use the well known lin- +ear ordering of maximal cliques associated with an interval graph [10] (Refer +Theorem 5). Figure 2 gives an illustration of how to obtain the canonical in- +terval representation of an interval graph. Let G = (V, E) be the given in- +terval graph. Let O = {Q1, Q2 . . . Qr} be an ordering of the maximal cliques +in G. For each v ∈ V (G), let the interval representation of G obtained from +O be I(v) = [l(v), r(v)], where l(v) is the index of the leftmost clique in O +that contains v, and r(v) is the index of the rightmost clique containing v. Let +I′ = {I(v) | v ∈ V (G)}. To construct the canonical interval representation, we +associate a gadget Di with maximal clique Qi, for 1 ≤ i ≤ r. For every maximal +clique Qi, we look at Di and stretch those intervals in I′ that either start at +i or end at i. Intuitively, we can think of I(v) as being stretched to the left if +l(v) = i and as being stretched to the right if r(v) = i. Inside gadget Di, there is a +point, which we denote by zi, with the following property: any interval for which +l(v) = i, starts at zi or to the left of zi and any interval for which r(v) = i, ends +at zi or to the right of zi. We refer to zi as the zero point of gadget Di. The exact +construction of stretched intervals is detailed in the subsequent paragraphs. +The gadgets D1, D2 . . . , Dr are arranged in the same order as that of the +maximal cliques in O. Further, for each v ∈ V (G), the stretched interval as- +sociated with I(v) has Dl(v) as its left-most gadget and Dr(v) as its rightmost +gadget. To complete the construction, between each pair of consecutive gadgets +we add an additional point. We show later that this additional point is crucial in +obtaining the exact hitting set of special cases of exactly hittable interval graphs. +The stretched interval of I(v) contains all these additional points between con- +secutive gadgets in the ordered set {Dl(v), Dl(v)+1, . . . , Dr(v)}. Let HG = (V, I) +6 + +denote the canonical interval hypergraph thus obtained. V is the set of all points +internal to the gadgets (defined below) and the r − 1 additional points between +consecutive gadgets (as described above). The hyperedges in I are the stretched +intervals corresponding to each interval in I′. We now describe the gadget Di +associated with maximal clique Qi, 1 ≤ i ≤ r. +u +a +d +b +c +e +Fig (i) +Q1 +Q2 +Q3 +Q4 +u +u +u +u +a +b +b +c +d +d +e +e +Fig (ii) +Q1 +Q2 +Q3 +Q4 +Iu +Ia +Ib +Ic +Id +Ie +z1 +z2 +z3 +z4 +Fig (iii) +Iu +Id +D1 +D2 +D3 +D4 +Iu +Iu +Iu +Iu +Iu +Iu +Ia +Ia +Ia +Id +Id +Id +Ib +Ib +Ib +Ie +Ie +Ie +Ie +Ic +Ic +Ic +1 +2 +3 +z1 +4 +5 +z2 +6 +7 +z3 +8 +9 +z4 +10 +11 +Fig (iv) +Ia +Id +Iu +Ib +Ie +Ic +1 +2 +3 +z1 +4 +5 +z2 +6 +7 +z3 +8 +9 +z4 +10 +11 +Fig (v) +Fig. 2. Construction of Canonical Interval Representation (i) Interval Graph G with its maximal +cliques Q1, Q2, Q3, Q4 (ii) Linear ordering of maximal cliques O = {Q1, Q2, Q3, Q4} (iii) Interval +representation of G obtained from O (iv) Gadgets D1 to D4 (v) Canonical interval representation +for G +Construction of the gadget Di for maximal clique Qi: Let {I(v1), I(v2), +. . . , I(vt)} be the ordered set of intervals such that for each 1 ≤ k ≤ t, l(vk) = i +and r(vk) > r(vj) whenever 1 ≤ k < j ≤ t. In other words, the ordered set +considers the intervals whose left endpoint is i in descending order of their right +endpoints. Then, for each 1 ≤ k ≤ t, the left endpoint of the stretched interval of +7 + +I(vk) is −k +1: this can be viewed as stretching I(vk) to the left. On the integer +line, the left endpoint of the stretched interval of I(vk) is zi − k + 1. We next +consider those intervals I(v) such that r(v) = i. Let {I(v1), I(v2), . . . , I(vt)} +be the ordered set of intervals such that for each 1 ≤ k ≤ t, r(vk) = i and +l(vk) < l(vj) whenever 1 ≤ k < j ≤ t. In other words, the ordered set considers +the intervals whose right endpoint is i in ascending order of their left endpoints. +Then, for each 1 ≤ k ≤ t, the right endpoint of the stretched interval of I(vk) +is k − 1: this can be viewed as stretching I(vk) to the right. On the integer line, +the right endpoint of the stretched interval of I(vk) would be zi + k − 1. This +completes the description of the gadget Di. Note that for I(v) in I, the stretched +interval is stretched to the left only in the leftmost gadget in which it is present, +and it is stretched to the right in the rightmost gadget in which it is present. By +construction, no two intervals share the same left endpoint and the same right +endpoint. +Lemma 1. Let HG be the canonical interval representation of graph G as con- +structed using the above procedure. Then, G is isomorphic to the intersection +graph of intervals in HG. +Proof. The gadgets D1, . . . , Dr are arranged according to the same order as +the maximal cliques in the ordered set O = {Q1, Q2 . . . Qr}. For each v ∈ G, +the starting gadget (and the ending gadget) of interval I(v) and the starting +maximal clique (and the ending maximal clique) of vertex v in O are the same +by construction. Further, I(v) contains all the points in the intervening gadgets +between the starting and ending gadgets of I(v) just as v occurs in all the +intervening maximal cliques between the starting and ending maximal cliques +to which v belongs to. It follows that I(u) and I(v) intersect if and only if +the corresponding stretched intervals have a non-empty intersection. Thus the +intersection graph of intervals in HG is isomorphic to G. +⊓⊔ +3 +Exactly Hittable Interval Graphs +Characterizing simple graphs as intersection graphs is a well-pursued line of +study in graph theory. Harary had presented results on this problem in his book +[11]. We address the question of when a simple graph is the intersection graph +of an exactly hittable hypergraph. We modify the proof given by Harary to an- +swer this question. In addition, we present similar results for the class of chordal +graphs (refer to Section 1.1 for definition). We prove Observation 1 about arbi- +trary graphs and arbitrary chordal graphs. +Proof of Observation 1: +Proof. The proof of the first statement is based on a slight modification to the +intersection model constructed from G in Theorem 2.5 in the book by Harary +[11]. Let H = (V, E) be the intersection model constructed as follows. The uni- +verse V of the hypergraph is V (G) ∪ E(G). The set E contains a hyperedge Ev +8 + +for each vertex v ∈ V (G), and Ev contains all the edges incident on v and the +element v. Clearly, the intersection graph of H is isomorphic to G and V (G) is +an exact hitting set of H. +The proof of the second statement, which is for a chordal graph G, is similar and +is as follows. Since G is a chordal graph let it be isomorphic to the intersection +graph of some subtrees of a tree T . In particular, let T be the clique tree of the +chordal graph G [10]. Let {Tv | v ∈ V (G)} be the set of subtrees in T , where +Tv is the subtree associated with v and the tree nodes in Tv correspond to those +maximal cliques in G which contain the vertex v. We modify T to get T ′ by +adding n = |V (G)| new nodes, each corresponding to a vertex in V (G). For each +v ∈ V (G), the new node corresponding to v is made adjacent in T to some node +in Tv. The resulting tree is T ′ and T ′ +v is the subtree of T ′ consisting of Tv and +the new node corresponding to v. Clearly, the newly added nodes form an exact +hitting set of the set {T ′ +v | v ∈ V (G)} in T ′, and the intersection graph of the +subtrees {T ′ +v | v ∈ G} is same as G. +⊓⊔ +Interestingly, the property of being exactly hittable is not universal for the +class of interval graphs. There are interval graphs that do not have any ex- +actly hittable intersection model. In this paper, we present a forbidden structure +characterization for the class of interval graphs that have an exactly hittable in- +tersection model. In this section, we prove that every graph in F (see Definition +1) is a forbidden structure for EHIG. First, we state and prove one direction of +Theorem 2. +We use the following notations throughout the section. H′ denotes an interval +representation of G. We denote the open neighbourhood of vertex v by N(v). +N(P) denotes open neighbourhood of all vertices in path P. I(P) denotes the +set of intervals in H′ corresponding to vertices in path P, XN(P ) denotes the +set of independent vertices in N(P) and I(XN(P )) denotes set of intervals in H′ +corresponding to XN(P ). +Lemma 2. Let G be an interval graph. Let F ∈ F be any forbidden structure. If +G contains F as an induced subgraph, then G is not an Exactly Hittable Interval +Graph. +Proof. Our proof is by contradiction. Let H′ be any exactly hittable interval +representation of G. Let P be an induced path of length k in G that has an +independent set of at least k + 3 vertices in its neighbourhood. Let S be an +exact hitting set of H′. Recall that I(P) denotes the set of intervals in H′ +corresponding to vertices in path P. By our assumption that G contains F, the +number of intervals in I(XN(P )) is at least k + 3. Hence |I(XN(P )) ∩ S| ≥ +k + 3. Since XN(P ) is an independent set, there can be at most two intervals in +I(XN(P )) that have their endpoints outside the union of intervals in I(P) - one +on either side of P. Therefore, even if these two intervals in I(XN(P )) are hit +outside the intervals in I(P) at either ends, the remaining k + 1 independent +intervals have to be hit inside the union of intervals in I(P). Hence |I(P)∩S| ≥ +k + 1. But there are only k intervals inside I(P). Therefore, by the pigeonhole +principle, at least one interval among the intervals in I(P) has to be hit more +9 + +than once. Thus S cannot be an exact hitting set of H′. We have arrived at a +contradiction to the assumption that H′ is exactly hittable. Since we started +with an arbitrary exactly hittable representation and arrived at a contradiction, +we conclude that G is not exactly hittable. +⊓⊔ +Now, we prove the other direction of Theorem 2. Let O = {Q1, Q2 . . . Qt} be a +linear ordering of maximal cliques in G (Refer Theorem 5 and Section 2). Let +HG be the canonical interval representation of G obtained from O. We use the +following notations in this section. +We denote a minimum clique cover of neighbourhood of a vertex v which is +formed by the minimum number of maximal cliques in O by C(N[v]). Note that +such a clique cover exists. We prove a simple observation here. +Observation 7 If Qi . . . Qj, i, j ∈ [1, t], i ≤ j denote the maximal cliques con- +taining vertex v ∈ V , then Qj ∈ C(N[v]). +Proof. We prove this by contradiction. Let us assume that Qj /∈ C(N[v]). As +Qj ̸= Qj−1, there exists a vertex u in Qj which is not in Qj−1. It follows that +u is not contained in any maximal cliques previous to Qj−1 since the maximal +cliques containing a vertex occur consecutively in the linear ordering of maximal +cliques of an interval graph. Therefore, if Qj /∈ C(N[v]), then u is not covered. +It contradicts the fact that C(N[v]) is a clique cover of N[v]. It follows that +Qj ∈ C(N[v]). +⊓⊔ +From now on, when we refer to a minimum clique cover of the input graph, +we mean a minimum clique cover formed by the minimum number of maximal +cliques in O unless specified otherwise. Let |C(N[v])| denote the number of +cliques in C(N[v]). Similarly, we denote a minimum clique cover of vertices in +the maximal cliques Qi to Qj in the ordering O, i < j, by C(Qi, . . . , Qj). +Our proof is based on the structural properties of a path P in G, the construction +of which is presented in Algorithm 1. The structural properties of path P are +proved as lemmas later in the section. +Outline of Algorithm 1 : +We construct an induced path P which contains a minimal set of vertices from +graph G. The vertices in path P are selected such that every maximal clique +in O has a non-empty intersection with path P. Further, we incrementally con- +struct a clique cover of the given graph by taking the clique cover of the closed +neighborhood each of the individual vertices in P. +10 + +Qi−2 +r +Qi−1 +r +Qi−1 +r+1 +Qi +r +Qt +vi−1 +vi +Fig. 3. Construction of path P +Algorithm 1: Construction of path P and computation of clique cover +1: i = 1 +2: v1 ← Interval in Q1 with largest right endpoint +3: P ← v1 +4: Q1 +r = Maximal clique in which v1 ends +5: C(N[v1]) = Minimum clique cover of N[v1] +6: Q1 +r′ = Maximal clique previous to Q1 +r in C(N[v1]) +7: CC(N[v1]) = C(N[v1]) +8: while Qi +r ̸= Qt do +9: +i = i + 1 +10: +vi = Interval I ∈ Qi−1 +r +\ Qi−1 +r′ +which has largest right endpoint +11: +P ← P ∪ vi +12: +Qi +r = Maximal clique in which vi ends +13: +CC(N[v1, . . . , vi]) = CC(N[v1, . . . , vi−1]) ∪ C(Qi−1 +r+1, . . . , Qi +r) +14: +Qi +r′ = Maximal clique previous to Qi +r in CC(N[v1, . . . , vi]) +15: end while +16: K = CC(N[v1, . . . , vi]) +17: return P +Let {v1, v2, . . . , vp} be the ordered set of vertices in path P with respect to +the linear ordering O.Let vi, . . . , vj, 1 ≤ i ≤ j ≤ n be any subset of vertices +in path P. We use CC(N[vi, vi+1, . . . , vj]), i ≤ j to denote a clique cover of +(N[vi] ∪ N[vi+1] ∪ . . . ∪ N[vj]) and |CC(N[vi, vi+1, . . . , vj])| to denote the num- +ber of cliques in CC(N[vi, vi+1, . . . , vj]). Note that this is a clique cover of G. +However, there exists graphs for which it is not a minimum clique cover. The +clique cover of the closed neighborhood of vertices in path P is stored in K. We +denote the maximal cliques which constitute K in the order in which they appear +in CC(N[v1, . . . , vp]), by K1, K2, . . . , Kα′. +In any perfect graph, the size of a minimum clique cover equals the size of +a maximum independent set. Based on this, we state an observation that we +11 + +use in proving some important properties of the constructed clique cover of the +neighborhood of vertices in path P. +Observation 8 In any perfect graph G′, for each maximal clique K in a min- +imum clique cover K of G′, there exists a vertex u ∈ G′ such that u does not +belong to any other maximal clique in K. +Lemma 3. For 1 ≤ i ≤ p, |C(N[vi])| ≤ 3. +Proof. The proof is by contradiction. Let | C(N[vi]) |> 3. By definition, C(N[vi]) +contains only the maximal cliques in the linear ordering O. From Observation +8, it follows that for each maximal clique Q ∈ C(N[vi]), there exists a vertex +w which is unique to Q. Since |C(N[vi])| > 3, there exists at least 4 vertices +w1, w2, w3, w4 in N[vi] that forms an independent set. It follows that vi together +with w1, w2, w3, w4 form a forbidden structure K1,4 (refer Figure 1 (i)). This is +a contradiction to our premise that G does not contain any forbidden structure. +⊓⊔ +Note: You may refer Algorithm 1 for the notations used in the proofs of lemmas +given below: +Lemma 4. In the path P, if for any vertex vi, 1 ≤ i < p , |C(N[vi])| = 3, then +|C(N[vi+1])| ≤ 2. +Proof. The proof is by contradiction. Assume that there exists a vertex vi ∈ +P, 1 ≤ i ≤ p for which |C(N[vi])| = 3 and |C(N[vi+1])| ≥ 3. By Lemma 3, +|C(N[vi+1])| cannot exceed 3. Vertices vi and vi+1 form an edge in the path P. +Cr′ +Cr +vi +vi+1 +Fig. 4. Forbidden structure formation +We consider the following cases based on the cardinality of C(N[vi])∪C(N[vi+1]). +Case when | C(N[vi]) ∪ C(N[vi+1]) |= 4: Recall from Algorithm 1 that Qi−1 +r +and Qi +r are the maximal cliques in the ordering O which contains the right end- +points of the intervals corresponding to vi−1 and vi respectively. For any vertex +vi ∈ P, Qi−1 +r +, Qi +r ∈ C(N[vi]). By our choice of vi+1 in the construction of path +P, vi+1 ∈ {Qi +r \ Qi +r′}. Thus vi+1 is covered by Qi +r . As per our assumption, +12 + +| C(N[vi]) ∪ C(N[vi+1]) | += 4, and | C(N[vi]) | += 3 by our premise. There- +fore those vertices of N[vi+1] which are not covered by Qi +r have to be covered +by exactly one more clique. It follows that | C(N[vi+1]) | += 2, which is a +contradiction to our assumption that | C(N[vi+1]) | += 3. This, in turn, is a +contradiction to our initial premise that | C(N[vi]) ∪ C(N[vi+1]) |= 4. Thus +the only possibility is, | C(N[vi]) ∪ C(N[vi+1]) |= 5, which we discuss in the +next case. Observe that | C(N[vi])∪C(N[vi+1]) | cannot be greater than 5 since +vi+1 ∈ Qi +r and | C(N[vi+1]) | += 3. +Case when | C(N[vi]) ∪ C(N[vi+1]) |= 5: The proof is by contradiction to +our premise that G does not contain any forbidden structure. We first show +that C(N[vi])∪C(N[vi+1]) is indeed a minimum clique cover of N[vi]∪N[vi+1]. +Then, using Observation 8, we show that there exists a forbidden structure. +By definition, C(N[vi]) is a minimum clique cover of N[vi]. Therefore, each of +the three maximal cliques in C(N[vi]) has atleast one unique vertex which does +not belong to any other maximal clique. Since vi+1 ∈ Qi +r and Qi +r ∈ C(N[vi]), +Qi +r ∈ C(N[vi+1]). Let the other two maximal cliques in C(N[vi+1] be Qj and +Qk. By Observation 8, Qj and Qk contain a unique vertex each. It follows that +any minimum clique cover of N[vi] ∪ N[vi+1] contains all three maximal cliques +of C(N[vi]) along with Qj and Qk. Hence C(N[vi]) ∪ C(N[vi+1]) is a minimum +clique cover of N[vi]∪N[vi+1]. By Observation 8, on C(N[vi])∪ C(N[vi+1]), there +is a set V ′ of 5 vertices in C(N[vi]) ∪ C(N[vi+1]) that are mutually disjoint and +form an independent set of size five. The edge (vi, vi+1), together with V ′ form a +forbidden structure (see Definition 1). Thus we have arrived at a contradiction. +It follows that if | C(N[vi]) | += 3, then | C(N[vi+1]) | += 2. +⊓⊔ +Observation 9 For each vertex v ∈ P \ vp, | C(N[v]) |≥ 2. +Proof. By construction of path P, +CC(N[v1, . . . , vi]) = CC(N[v1, . . . , vi−1]) ∪ C(Qi−1 +r+1, . . . , Qi +r) +where Qi−1 +r +is the rightmost maximal clique in CC(N[v1, . . . , vi−1]) and it covers +vi. Qi +r is the rightmost maximal clique to which vi belongs to, in the ordering +O. Note that C(N[vi]) = Qi−1 +r +∪ C(Qi−1 +r+1, . . . , Qi +r). Let A = N[vi] ∩ (Qi−1 +r+1 ∪ +· · · ∪ Qi +r). Since vi ̸= vp, we know that there is vi+1 ∈ P which is chosen such +that vi+1 /∈ Qi−1 +r +and vi+1 ∈ N[vi]. It follows that A ̸= ∅. By choice of vi, it +has the rightmost right endpoint among that of all vertices in Qi−1 +r +\ Qi−1 +r′ +and +vi ̸= vp. Hence ∃u ∈ A which is not covered by Qi−1 +r +. Therefore, there exists at +least one Q ∈ C(Qi−1 +r+1, . . . , Qi +r) that covers all the vertices in A,. In other words, +C(Qi−1 +r+1, . . . , Qi +r) ̸= ∅. It follows that C(N[vi]) is of size at least 2. +⊓⊔ +Now we prove a stronger claim. +Lemma 5. In path P, there is at most one vertex v where | C(N[v]) |= 3. +Proof. The proof of this claim is again by contradiction. Assume that there is +more than one vertex with minimum clique cover equal to 3 in path P. Let vi be +13 + +the first such vertex in P in the increasing order of left endpoints. By Lemma +4, we know that the minimum clique cover of vi+1 is of size less than 3. By our +assumption, ∃j > i+ 1 such that minimum clique cover of vj is of size 3. Let the +number of vertices in the subpath of P from vi to vj (including both vi and vj) be +l. It follows from Observation 9 that for each vertex vk ∈ P, k ∈ [i+1, j −1], the +minimum clique cover is of size 2. We compute CC(N[vi, . . . , vk]) with respect +to CC(N[vi, . . . , vk−1]). Note that vk is already covered in CC(N[vi, . . . , vk−1]) +and CC(N[vi, . . . , vk]) additionally covers N[vk] \ N[vk−1]. Since |C(N[vk])| = +2, it adds just 1 to the number of cliques in CC(N[vi, . . . , vk−1]). i.e, +|CC(N[vi, . . . , vk])| = |CC(N[vi, . . . , vk−1])| + 1 +It follows that each of the l − 2 vertices in {vi+1, . . . , vj−1} increments the size +of the clique cover by 1. i.e, |CC(N[vi+1, . . . , vj−1])| = l − 2. Thus +|CC(N[vi, . . . , vj])| = |C(N[vi])| + |CC(N[vi+1, . . . , vj−1])| + |C(N[vj])| − 1 += 3 + (l − 2) + 3 − 1 += l + 3 +vi +vj +vi+1 +vi+2 +l − 2 +Fig. 5. Vertices vi and vj belong to three consecutive cliques +Note that we deduct 1 from | C(N[vj]) | since vj is already covered by +CC(N[vi+1], . . . , N[vj−1]). We can see that the vertices from vi to vj form a +path of length l which has an independent set of size l + 3 in its neighbourhood. +The vertices from vi to vj, together with the independent set of size l + 3 in its +neighbourhood forms a forbidden structure. We have arrived at a contradiction +to our premise that G does not contain any forbidden structures. Therefore it is +proved that in path P, there is at most one vertex which has a minimum clique +cover of size 3. +⊓⊔ +By Algorithm 1, K = {K1, K2 . . . Kα′} is the clique cover of path P. Note that +it forms a minimal clique cover of input graph G. +Claim. | Ki−1 ∩ Ki ∩ Ki+1 |≤ 1, 2 ≤ i ≤ α′ − 1. +14 + +Proof. By Lemma 5, there is at most one vertex with minimum clique cover of +size 3 in P. If such a vertex exists, it would belong to three consecutive maximal +cliques in K, and let us denote them by Ka−1, Ka and Ka+1, a ∈ [2, α′ − 1]. +For all other i ̸= a, i ∈ [2, α′ − 1], | Ki−1 ∩ Ki ∩ Ki+1 |= 0. It follows that +| Ki−1 ∩ Ki ∩ Ki+1 |≤ 1 for 2 ≤ i ≤ α′ − 1. +⊓⊔ +We now use the properties of P proved in above lemmas to construct a clique +cover of G with the property that the cliques are all vertex disjoint. The clique +cover is denoted by B = {B1, B2 . . . Bα′}. We outline the steps in the procedure +below. +Let vl, 1 ≤ l ≤ p be the vertex in P such that |C(N[vl])| = 3. If no such +vertex exists in P, then let l = p + 1, and let Kp+2 be the empty set. Further, +Kp+3 is the empty set. We know that there is at most one such vertex, and the +construction below will also take care of the case when for all vi, 1 ≤ i ≤ p, +|C(N[vi])| = 2. +1. For 1 ≤ i ≤ l − 1, Bi = Ki \ Ki+1. +2. For i = l ≤ p, we define Bl, Bl+1, Bl+2. +– Bl = Kl \ Kl+1. +– Bl+1 = Kl+1 \ (Kl ∪ Kl+2) +– Bl+2 = Kl+2 \ Bl+1. +3. For i ≥ l + 1, Bi+2 = Ki+2 \ Ki+3. +Since G does not have the forbidden structure, it follows that B is a clique cover, +and by construction, it is a parition of the vertex set. Further, the number of +cliques in B is α′. +To complete the proof of Theorem 2, we use the crucial property of the +canonical representation HG that no two intervals in HG have the same left end +point or the same right end point. Using this property, we show that for each +Bi there is a point pi in HG such that the intervals in HG which contain pi are +exactly those which correspond to the vertices in Bi. These points pi, 1 ≤ i ≤ α′ +form an exact hitting set of HG. This completes the characterization of EHIG. +3.1 +Algorithm to recognize exactly hittable interval graphs +In this section, we present an algorithm to recognize an exactly hittable interval +graph. This algorithm makes use of the canonical interval representation in Sec- +tion 2 and the result by Dom et al. for MMSC problem on interval hypergraphs +[3]. In their paper, Dom et al. showed that an integer linear programming (ILP) +formulation, say L, for MMSC problem on interval hypergraphs can be solved in +polynomial time. The coefficients of inequalities in L results in a totally unimod- +ular matrix and the polyhedron corresponding to L is an integer polyhedron. If +the input instance to ILP is an exactly hittable instance, then the solution re- +turned is 1. We use this algorithm below to test if a given interval hypergraph +instance is exactly hittable. +15 + +Algorithm isEHIG: Given an interval graph G, construct the canonical inter- +val representation as described in Section 2. Let HG be the resulting interval +representation. Run MMSC algorithm by Dom et al. [3] on HG as input. If the +algorithm returns value 1, then return yes. Else return no. +Lemma 6. Algorithm isEHIG(G) outputs yes if and only if G is exactly hittable. +Proof. The proof follows from Lemma 1, Theorem 3 and the correctness of al- +gorithm for MMSC problem on interval hypergraphs. +⊓⊔ +3.2 +Proper Interval Graphs is a subclass of EHIG +The following lemma proves our Theorem 4. +Lemma 7. The set of Proper Interval Graphs are strictly contained in the set +of EHIG. +Proof. Let G be a proper interval graph and let it be the intersection graph of +the interval hypergraph H = (V, I) in which no interval properly contains an- +other. Since H is a proper interval hypergraph, no two intervals in I) can have +the same left endpoint. Hence order intervals in I according to increasing order +of their left endpoints. Let this ordering be I1 < I2 < . . . < Im. Add r(I1) (which +is the smallest right endpoint among all intervals) to set S. Remove all intervals +hit by r(I1). Recurse on the remaining set of intervals until all the intervals are +hit by S. Clearly, S is an exact hitting set. +To show the strict containment, we show that the graph K1,3 which is a for- +bidden structure [18] for Proper Interval Graphs has an exactly hittable in- +terval representation. Let the vertices of the K1,3 be {u, a, b, c} and edges be +{(u, a), (u, b), (u, c)}. The intervals assigned to the vertices a, b, c and u are shown +in Figure 6. Hence the lemma. +⊓⊔ +u +a +b +c +a +b +c +u +1 +2 +3 +4 +5 +Fig. 6. Exactly hittable interval representation of K1,3 +This completes the proof of Theorem 4. +References +1. J´er´emie Chalopin and Daniel Gon¸calves. Every planar graph is the intersection +graph of segments in the plane. +In Proceedings of the forty-first annual ACM +symposium on Theory of computing, pages 631–638. ACM, 2009. +16 + +2. Panagiotis Cheilaris and Shakhar Smorodinsky. Conflict-free coloring with respect +to a subset of intervals. arXiv preprint arXiv:1204.6422, 2012. +3. Michael Dom, Jiong Guo, Rolf Niedermeier, and Sebastian Wernicke. Minimum +Membership Set Covering and the Consecutive Ones Property, pages 339–350. +Springer Berlin Heidelberg, Berlin, Heidelberg, 2006. +4. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Com- +plexity. Springer Publishing Company, Incorporated, 2013. +5. Paul Erd˝os, Adolph W Goodman, and Lajos P´osa. The representation of a graph +by set intersections. Canad. J. Math, 18(106-112):86, 1966. +6. F Gavril. +The intersection graphs of subtrees in trees are exactly the chordal +graphs. Journal of Combinatorial Theory, Series B, 16(1):47–56, 1974. +7. F Gavril. A recognition algorithm for the intersection graphs of paths in trees. +Discrete Mathematics, 23(3):211–227, 1978. +8. P. C. GILMORE and A. J. HOFFMAN. A CHARACTERIZATION OF COM- +PARABILITY GRAPHS AND OF INTERVAL GRAPHS, pages 65–74. WORLD +SCIENTIFIC, 2011. +9. Martin Charles Golumbic. Interval graphs and related topics. Discrete Mathemat- +ics, 55(2):113 – 121, 1985. +10. Martin Charles Golumbic. Algorithmic Graph Theory and Perfect Graphs (Annals +of Discrete Mathematics, Vol 57). North-Holland Publishing Co., Amsterdam, The +Netherlands, The Netherlands, 2004. +11. Frank Harary. Graph Theory. Addison-Wesley Series in Mathematics. Addison +Wesley, 1969. +12. Richard M. Karp. +Reducibility among Combinatorial Problems, pages 85–103. +Springer US, Boston, MA, 1972. +13. Fabian Kuhn, Pascal von Rickenbach, Roger Wattenhofer, Emo Welzl, and Aaron +Zollinger. Interference in cellular networks: The minimum membership set cover +problem. In Proceedings of the 11th Annual International Conference on Com- +puting and Combinatorics, COCOON’05, pages 188–198, Berlin, Heidelberg, 2005. +Springer-Verlag. +14. C Lekkeikerker and J Boland. Representation of a finite graph by a set of intervals +on the real line. Fundamenta Mathematicae, 51(1):45–64, 1962. +15. Benjamin L´evˆeque, Fr´ed´eric Maffray, and Myriam Preissmann. Characterizing path +graphs by forbidden induced subgraphs. Journal of Graph Theory, 62(4):369–384, +2009. +16. Benjamin L´evˆeque, Fr´ed´eric Maffray, and Myriam Preissmann. Characterizing path +graphs by forbidden induced subgraphs. Journal of Graph Theory, 62(4):369–384, +2009. +17. Clyde L Monma and Victor K Wei. Intersection graphs of paths in a tree. Journal +of Combinatorial Theory, Series B, 41(2):141–181, 1986. +18. Fred S Roberts. Graph theory and its applications to problems of society, pages +27–31. SIAM, 1978. +19. Douglas B. West. Introduction to Graph Theory. Prentice Hall, 2 edition, Septem- +ber 2000. +17 + diff --git a/XNAyT4oBgHgl3EQfh_h5/content/tmp_files/load_file.txt b/XNAyT4oBgHgl3EQfh_h5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..aa718bed1c2fbbad38d412a8860bd859097b5a3e --- /dev/null +++ b/XNAyT4oBgHgl3EQfh_h5/content/tmp_files/load_file.txt @@ -0,0 +1,705 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf,len=704 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='00387v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='DS] 1 Jan 2023 Exactly Hittable Interval Graphs S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Dhannya, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Narayanaswamy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Nisha Department of Computer Science and Engineering Indian Institute of Technology Madras, Chennai, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Given a set system X = {U, S}, where U is a set of elements and S is a set of subsets of U, an exact hitting set U′ is a subset of U such that each subset in S contains exactly one element in U′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We refer to a set system as exactly hittable if it has an exact hitting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In this paper, we study interval graphs which have intersection models that are exactly hittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We refer to these interval graphs as exactly hittable interval graphs (EHIG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We present a forbidden structure characterization for EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We also show that the class of proper interval graphs is a strict subclass of EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 1 Introduction We study classes of simple graphs which are intersection graphs of set systems that have exact hitting sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In particular, we introduce a class of interval graphs which can be represented as intersection graphs of intervals that have exact hit- ting sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We refer to this class as Exactly Hittable Interval Graphs (EHIG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We also present an infinite family of forbidden structures for EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In the following, we introduce a setting of exact hitting sets and intersection graphs, before pre- senting our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Exact Hitting Sets: Set systems are synonymous with hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' A hitting set of a hypergraph H is a subset T of the vertex set of H such that T has at least one vertex from every hyperedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If every hyperedge has exactly one element from T , then T is called an exact hitting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Also known as the Unique Hitting Set problem, the Exact Hitting Set problem is a well-studied de- cision problem that aims to find if a given hypergraph has an exact hitting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It finds applications in combinatorial cryptosystems [4] and computational biol- ogy among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Exact Hitting Set problem is the dual of Exact Cover problem which, in turn, seeks to find a set cover that covers all vertices of a hypergraph such that the number of occurrences each vertex has in the cover is exactly one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Some famous examples of Exact Cover problem are sudoku, tiling dominoes, and n-queens problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Exact Cover problem is a special case of Minimum Membership Set Cover problem (MMSC) [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' While the classic Set Cover problem seeks to find a set cover of minimum cardinality, MMSC aims to find a set cover that minimizes the maximum number of occurrences each vertex has in the cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' MMSC is known to be NP-complete on arbitrary set systems [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' However, for interval hypergraphs, MMSC has been shown to be solvable in polynomial time by Dom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If a hypergraph H has an exact hitting set, we refer to H as an exactly hittable hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It is, indeed, a natural question to ask if a given hypergraph is exactly hittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The algorithm by Dom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' [3] for MMSC answers this question, but not efficiently for gen- eral hypergraphs (due to inherent NP-hardness for general cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' However, for special kinds of hypergraphs like interval hypergraphs, an exact solution can be obtained in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In our work, we address a natural extension to this question and study simple graphs which are intersection graphs (defined in Sec- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='1) of exactly hittable hypergraphs, with a focus on interval hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Intersection Graphs: The theory of graphs and hypergraphs are connected by a very well-studied notion of intersection graphs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It is well-known that every graph G is an intersection graph of some hypergraph H [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' H is referred to as an intersection model or a set representation of G [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Interestingly, certain special classes of graphs are characterized by the structure of their inter- section models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For instance, Gavril has shown that the class of chordal graphs are the intersection graphs of subtrees of a tree [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' When the hyperedges are restricted to be paths on a tree, the resulting intersection graph class is that of path chordal graphs which is a proper subclass of the class of chordal graphs [1],[7],[15],[17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Forbidden Structure Characterizations: While a graph G may be identified as an intersection graph of a structured hypergraph, characterization of G based on forbidden structures has also been equally well-studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For instance, the class of chordal graphs are characterized by the absence of induced cycles of size 4 or more [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Similarly, by the celebrated theorem of Kuratowski [19], the class of planar graphs must not have subgraphs that are subdivisions of K5 and K3,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Interval graphs are known to be the class of chordal graphs without an asteroidal triple as induced subgraph [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The class of proper interval graphs is a subclass of interval graphs that do not have a K1,3 as an induced subgraph [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Refer to Table 1 for a summary of these examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Clearly, characterization of simple graphs based on their intersection models and forbidden structures are extremely well-studied notions in defining graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Our results 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We begin our set of results with a simple extension to a well-known theorem by Harary that every graph G is the intersection graph of some hypergraph H [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Observation 1 Every simple undirected graph is the intersection graph of an exactly hittable hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Further, if G is a connected chordal graph, then it is the intersection graph of an exactly hittable set of subtrees of a tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 2 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Intersection models and forbidden structures for well-known graph classes Graph Class Intersection Model Forbidden Structures Simple An exactly hittable hypergraph NIL Planar Segments on a plane Subdivisions of K5 and K3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='3 [19] Chordal Subtrees of a tree Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' for k ≥ 4 [10] Path chordal Paths on a tree List given in [16] Interval Subpaths on a path Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' for k ≥ 4 and asteroidal triple [14] Proper interval Sets of intervals not properly contained in each other Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' for k ≥ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' asteroidal triple and K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='3 [18] Exactly hittable interval graphs (New graph class) Exactly hittable sets of intervals Ck,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' for k ≥ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' asteroidal triple,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='3 and induced path Pk which has,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' in its open neighbourhood,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' an independent set of at least k + 3 vertices We present proof of this observation in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Further to this observa- tion, we look at a subclass of chordal graphs namely interval graphs, which are intersection graphs of subpaths on a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We ask if there is an exactly hittable intersection model for every interval graph, as in the case of arbi- trary graphs and arbitrary chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Interestingly, the answer is no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We introduce the class of Exactly Hittable Interval Graphs (EHIG), which is the set of interval graphs that have an exactly hittable interval representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We say that an interval graph is exactly hittable if and only if it has at least one exactly hittable interval representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We present a forbidden structure characterization for EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' First, we define a family F of simple graphs as follows: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let F be the set of an infinite family of interval graphs with the following structure: each graph has an induced path P consisting of k vertices, for some k ≥ 1, and the open neighbourhood of P contains an independent set of size at least k + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Our main contribution in this paper is to prove that every graph in F is a forbidden structure for EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' See Fig 1 for examples of forbidden structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In Fig 1(i), u is the induced path P consisting of one vertex with an inde- pendent set of four vertices {a, b, c, d} in its neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Similarly, in Fig 1(ii), a-b is the induced path P consisting of two vertices and {c, d, u, e, f} is the independent set of five vertices in the neighbourhood of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' u a b c d (i) c d a b e f u (ii) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Two simple instances of forbidden structures 3 Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' An interval graph G is exactly hittable if and only if it does not contain any graph from the set F as an induced subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' This theorem has been proved in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We believe that this result is an interesting addition to the existing graph characterizations primarily because we could not find such an equivalence elsewhere in the literature, including graph classes repositories like graphclasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In Section 2, we introduce, what we refer to as, a canonical interval represen- tation for an interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Given interval graph G, we start with a maximal clique ordering of G and construct an interval representation from the or- dering such that the intersection graph of this representation is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By construction, there exists exactly one canonical interval represen- tation for every interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' While the canonical representation may be of independent interest, this representation is crucial in proving Theorem 2 in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In Section 2, we prove the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let G be an interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let HG be its canonical interval representation constructed as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Then, G is exactly hit- table if and only if HG is exactly hittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Given an interval graph G and its canonical interval representation HG, we show that the algorithm by Dom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' [3] to solve the MMSC problem in interval hypergraphs can be used to recognize EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We present the details in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We show that the class of EHIG is positioned between the class of proper interval graphs and the class of interval graphs in the containment hierarchy of graph classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proper interval graphs ⊂ EHIG ⊂ Interval Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The proof of the second part of the above theorem follows from the definition of EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The first part of the containment relationship has been proven in Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Interestingly, the smallest forbidden structure of EHIG is K1,4 whereas the forbidden structure for the class of proper interval graphs is K1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='1 Preliminaries Given a set system X = (U, S), the intersection graph G(X) of sets in X is the simple graph obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For every set S ∈ S, there exists a vertex vS ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' An edge (vSi, vSj) occurs in G if and only if there exists two sets Si, Sj ∈ F such that Si ∩ Sj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The family S is called a set representation of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' A set representation is also referred to as an intersection model [10], [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' A hypergraph H = (V, E) is a graph theoretic representation of a set system 4 X = (U, S), where the set V corresponds to U and the set E corresponds to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The set V contains vertices of hypergraph H and the set E contains hyperedges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In the intersection graph G, for every hyperedge E ∈ E, there exists a vertex vE ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' An edge (vEi, vEj) occurs in G if and only if the hyperedges Ei and Ej have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' A graph G = (V, E) is an interval graph if an interval I(v) can be associated to a vertex v ∈ V (G) such that there exists an edge (u, v) in G and the associated intervals I(u) and I(v) have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The set of intervals {I(v)}v∈V (G) is an interval representation or intersection model of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' When E is a family of intervals on a line, then G is an interval graph and H is an interval hypergraph [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' [2] The hypergraph H = ([n], I), where [n] = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=', n} and I ⊆ {{i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , j} | i ≤ j, i, j ∈ [n]} is known as an interval hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Each hyperedge in I is a set of consecutive integers, which we call an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In an interval I = {i, i + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , j}, i and j are the left and right endpoints of I respectively, which we denote by l(I) and r(I), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We use V(H) (or simply V) and I(H) (or simply I) to denote the vertex set and the hyperedge set, respectively, of an interval hypergraph H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' An interval hypergraph is said to be proper if no interval is contained in another interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If, for an interval graph G, there exists an interval representation in which no interval is properly contained inside another interval, then G is a proper interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' An interval graph is characterized by the existence of a linear ordering of its maximal cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In Section 3, we use the following characterization to ob- tain an exactly hittable interval representation for an interval graph, if such a representation exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Theorem 5 (Gilmore and Hoffman, 1964 [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The maximal cliques of an interval graph G can be linearly ordered such that, for every vertex x of G, the maximal cliques containing x occur consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The class of interval graphs is a subfamily of the class of chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' A chordal graph is a simple graph that does not contain any induced cycle of size ≥ 4 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Chordal graphs are known to be intersection graphs of subtrees of a tree [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Note: We draw the reader’s attention to the distinction between interval hyper- graphs and interval graphs, and proper interval hypergraphs and proper interval graphs, as these are used extensively throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We also use the adjective exactly hittable to both simple graphs and hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Recall that a interval graph is an Exactly Hittable Interval Graph if it has an intersection model that has an exact hitting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' On the other hand, an Exactly Hittable Hypergraph is one that has an exact hitting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Observation 6 Since our goal is to characterize interval graphs that have an exactly hittable interval representation, we assume without loss of generality that, in the graph G, for every sequence of consecutive maximal cliques in a linear ordering, there is at most one vertex which starts and ends in this sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 5 Indeed if a given graph violates this property and there are two or more vertices that start and end in a sequence, then we retain only one of those vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The justification for this assertion is that if the resulting graph has an exactly hittable representation, so does the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Notations: All standard definitions and notations from West [19] have been used throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 2 A Canonical Interval Representation In this section, we obtain a canonical interval representation HG of a given in- terval graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The canonical interval representation is nothing but a special intersection model of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Consequently, the intersection graph of intervals in HG is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The construction follows a well-defined set of steps with the result that every interval graph has a unique canonical interval representa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The canonical representation HG is obtained by stretching intervals so that all intervals have distinct left endpoints and distinct right endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In other words, no pair of intervals start at the same point or end at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The canonical interval representation is crucial to the proof of our main result in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Outline: The starting point of this construction is to use the well known lin- ear ordering of maximal cliques associated with an interval graph [10] (Refer Theorem 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Figure 2 gives an illustration of how to obtain the canonical in- terval representation of an interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let G = (V, E) be the given in- terval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let O = {Q1, Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Qr} be an ordering of the maximal cliques in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For each v ∈ V (G), let the interval representation of G obtained from O be I(v) = [l(v), r(v)], where l(v) is the index of the leftmost clique in O that contains v, and r(v) is the index of the rightmost clique containing v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let I′ = {I(v) | v ∈ V (G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' To construct the canonical interval representation, we associate a gadget Di with maximal clique Qi, for 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For every maximal clique Qi, we look at Di and stretch those intervals in I′ that either start at i or end at i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Intuitively, we can think of I(v) as being stretched to the left if l(v) = i and as being stretched to the right if r(v) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Inside gadget Di, there is a point, which we denote by zi, with the following property: any interval for which l(v) = i, starts at zi or to the left of zi and any interval for which r(v) = i, ends at zi or to the right of zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We refer to zi as the zero point of gadget Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The exact construction of stretched intervals is detailed in the subsequent paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The gadgets D1, D2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Dr are arranged in the same order as that of the maximal cliques in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Further, for each v ∈ V (G), the stretched interval as- sociated with I(v) has Dl(v) as its left-most gadget and Dr(v) as its rightmost gadget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' To complete the construction, between each pair of consecutive gadgets we add an additional point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We show later that this additional point is crucial in obtaining the exact hitting set of special cases of exactly hittable interval graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The stretched interval of I(v) contains all these additional points between con- secutive gadgets in the ordered set {Dl(v), Dl(v)+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Dr(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let HG = (V, I) 6 denote the canonical interval hypergraph thus obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' V is the set of all points internal to the gadgets (defined below) and the r − 1 additional points between consecutive gadgets (as described above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The hyperedges in I are the stretched intervals corresponding to each interval in I′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We now describe the gadget Di associated with maximal clique Qi, 1 ≤ i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' u a d b c e Fig (i) Q1 Q2 Q3 Q4 u u u u a b b c d d e e Fig (ii) Q1 Q2 Q3 Q4 Iu Ia Ib Ic Id Ie z1 z2 z3 z4 Fig (iii) Iu Id D1 D2 D3 D4 Iu Iu Iu Iu Iu Iu Ia Ia Ia Id Id Id Ib Ib Ib Ie Ie Ie Ie Ic Ic Ic 1 2 3 z1 4 5 z2 6 7 z3 8 9 z4 10 11 Fig (iv) Ia Id Iu Ib Ie Ic 1 2 3 z1 4 5 z2 6 7 z3 8 9 z4 10 11 Fig (v) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Construction of Canonical Interval Representation (i) Interval Graph G with its maximal cliques Q1, Q2, Q3, Q4 (ii) Linear ordering of maximal cliques O = {Q1, Q2, Q3, Q4} (iii) Interval representation of G obtained from O (iv) Gadgets D1 to D4 (v) Canonical interval representation for G Construction of the gadget Di for maximal clique Qi: Let {I(v1), I(v2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , I(vt)} be the ordered set of intervals such that for each 1 ≤ k ≤ t, l(vk) = i and r(vk) > r(vj) whenever 1 ≤ k < j ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In other words, the ordered set considers the intervals whose left endpoint is i in descending order of their right endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Then, for each 1 ≤ k ≤ t, the left endpoint of the stretched interval of 7 I(vk) is −k +1: this can be viewed as stretching I(vk) to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' On the integer line, the left endpoint of the stretched interval of I(vk) is zi − k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We next consider those intervals I(v) such that r(v) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let {I(v1), I(v2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , I(vt)} be the ordered set of intervals such that for each 1 ≤ k ≤ t, r(vk) = i and l(vk) < l(vj) whenever 1 ≤ k < j ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In other words, the ordered set considers the intervals whose right endpoint is i in ascending order of their left endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Then, for each 1 ≤ k ≤ t, the right endpoint of the stretched interval of I(vk) is k − 1: this can be viewed as stretching I(vk) to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' On the integer line, the right endpoint of the stretched interval of I(vk) would be zi + k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' This completes the description of the gadget Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Note that for I(v) in I, the stretched interval is stretched to the left only in the leftmost gadget in which it is present, and it is stretched to the right in the rightmost gadget in which it is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By construction, no two intervals share the same left endpoint and the same right endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let HG be the canonical interval representation of graph G as con- structed using the above procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Then, G is isomorphic to the intersection graph of intervals in HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The gadgets D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Dr are arranged according to the same order as the maximal cliques in the ordered set O = {Q1, Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Qr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For each v ∈ G, the starting gadget (and the ending gadget) of interval I(v) and the starting maximal clique (and the ending maximal clique) of vertex v in O are the same by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Further, I(v) contains all the points in the intervening gadgets between the starting and ending gadgets of I(v) just as v occurs in all the intervening maximal cliques between the starting and ending maximal cliques to which v belongs to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that I(u) and I(v) intersect if and only if the corresponding stretched intervals have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Thus the intersection graph of intervals in HG is isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ 3 Exactly Hittable Interval Graphs Characterizing simple graphs as intersection graphs is a well-pursued line of study in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Harary had presented results on this problem in his book [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We address the question of when a simple graph is the intersection graph of an exactly hittable hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We modify the proof given by Harary to an- swer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In addition, we present similar results for the class of chordal graphs (refer to Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='1 for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We prove Observation 1 about arbi- trary graphs and arbitrary chordal graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof of Observation 1: Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The proof of the first statement is based on a slight modification to the intersection model constructed from G in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='5 in the book by Harary [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let H = (V, E) be the intersection model constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The uni- verse V of the hypergraph is V (G) ∪ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The set E contains a hyperedge Ev 8 for each vertex v ∈ V (G), and Ev contains all the edges incident on v and the element v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Clearly, the intersection graph of H is isomorphic to G and V (G) is an exact hitting set of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The proof of the second statement, which is for a chordal graph G, is similar and is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since G is a chordal graph let it be isomorphic to the intersection graph of some subtrees of a tree T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In particular, let T be the clique tree of the chordal graph G [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let {Tv | v ∈ V (G)} be the set of subtrees in T , where Tv is the subtree associated with v and the tree nodes in Tv correspond to those maximal cliques in G which contain the vertex v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We modify T to get T ′ by adding n = |V (G)| new nodes, each corresponding to a vertex in V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For each v ∈ V (G), the new node corresponding to v is made adjacent in T to some node in Tv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The resulting tree is T ′ and T ′ v is the subtree of T ′ consisting of Tv and the new node corresponding to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Clearly, the newly added nodes form an exact hitting set of the set {T ′ v | v ∈ V (G)} in T ′, and the intersection graph of the subtrees {T ′ v | v ∈ G} is same as G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ Interestingly, the property of being exactly hittable is not universal for the class of interval graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' There are interval graphs that do not have any ex- actly hittable intersection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In this paper, we present a forbidden structure characterization for the class of interval graphs that have an exactly hittable in- tersection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In this section, we prove that every graph in F (see Definition 1) is a forbidden structure for EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' First, we state and prove one direction of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We use the following notations throughout the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' H′ denotes an interval representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We denote the open neighbourhood of vertex v by N(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' N(P) denotes open neighbourhood of all vertices in path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' I(P) denotes the set of intervals in H′ corresponding to vertices in path P, XN(P ) denotes the set of independent vertices in N(P) and I(XN(P )) denotes set of intervals in H′ corresponding to XN(P ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let G be an interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let F ∈ F be any forbidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If G contains F as an induced subgraph, then G is not an Exactly Hittable Interval Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Our proof is by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let H′ be any exactly hittable interval representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let P be an induced path of length k in G that has an independent set of at least k + 3 vertices in its neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let S be an exact hitting set of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Recall that I(P) denotes the set of intervals in H′ corresponding to vertices in path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By our assumption that G contains F, the number of intervals in I(XN(P )) is at least k + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Hence |I(XN(P )) ∩ S| ≥ k + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since XN(P ) is an independent set, there can be at most two intervals in I(XN(P )) that have their endpoints outside the union of intervals in I(P) - one on either side of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Therefore, even if these two intervals in I(XN(P )) are hit outside the intervals in I(P) at either ends, the remaining k + 1 independent intervals have to be hit inside the union of intervals in I(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Hence |I(P)∩S| ≥ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' But there are only k intervals inside I(P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Therefore, by the pigeonhole principle, at least one interval among the intervals in I(P) has to be hit more 9 than once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Thus S cannot be an exact hitting set of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We have arrived at a contradiction to the assumption that H′ is exactly hittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since we started with an arbitrary exactly hittable representation and arrived at a contradiction, we conclude that G is not exactly hittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ Now, we prove the other direction of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let O = {Q1, Q2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Qt} be a linear ordering of maximal cliques in G (Refer Theorem 5 and Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let HG be the canonical interval representation of G obtained from O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We use the following notations in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We denote a minimum clique cover of neighbourhood of a vertex v which is formed by the minimum number of maximal cliques in O by C(N[v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Note that such a clique cover exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We prove a simple observation here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Observation 7 If Qi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Qj, i, j ∈ [1, t], i ≤ j denote the maximal cliques con- taining vertex v ∈ V , then Qj ∈ C(N[v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We prove this by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let us assume that Qj /∈ C(N[v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' As Qj ̸= Qj−1, there exists a vertex u in Qj which is not in Qj−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that u is not contained in any maximal cliques previous to Qj−1 since the maximal cliques containing a vertex occur consecutively in the linear ordering of maximal cliques of an interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Therefore, if Qj /∈ C(N[v]), then u is not covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It contradicts the fact that C(N[v]) is a clique cover of N[v].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that Qj ∈ C(N[v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ From now on, when we refer to a minimum clique cover of the input graph, we mean a minimum clique cover formed by the minimum number of maximal cliques in O unless specified otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let |C(N[v])| denote the number of cliques in C(N[v]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Similarly, we denote a minimum clique cover of vertices in the maximal cliques Qi to Qj in the ordering O, i < j, by C(Qi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Qj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Our proof is based on the structural properties of a path P in G, the construction of which is presented in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The structural properties of path P are proved as lemmas later in the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Outline of Algorithm 1 : We construct an induced path P which contains a minimal set of vertices from graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The vertices in path P are selected such that every maximal clique in O has a non-empty intersection with path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Further, we incrementally con- struct a clique cover of the given graph by taking the clique cover of the closed neighborhood each of the individual vertices in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 10 Qi−2 r Qi−1 r Qi−1 r+1 Qi r Qt vi−1 vi Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Construction of path P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='Algorithm 1: Construction of path P and computation of clique cover ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='1: i = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='2: v1 ← Interval in Q1 with largest right endpoint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='3: P ← v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='4: Q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='r = Maximal clique in which v1 ends ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='5: C(N[v1]) = Minimum clique cover of N[v1] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='6: Q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='r′ = Maximal clique previous to Q1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='r in C(N[v1]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='7: CC(N[v1]) = C(N[v1]) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='8: while Qi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='r ̸= Qt do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='9: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='i = i + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='10: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='vi = Interval I ∈ Qi−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='\\ Qi−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='r′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='which has largest right endpoint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='11: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='P ← P ∪ vi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='12: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='Qi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='r = Maximal clique in which vi ends ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='13: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='CC(N[v1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vi]) = CC(N[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vi−1]) ∪ C(Qi−1 r+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Qi r) 14: Qi r′ = Maximal clique previous to Qi r in CC(N[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vi]) 15: end while 16: K = CC(N[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vi]) 17: return P Let {v1, v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vp} be the ordered set of vertices in path P with respect to the linear ordering O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='Let vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj, 1 ≤ i ≤ j ≤ n be any subset of vertices in path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We use CC(N[vi, vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj]), i ≤ j to denote a clique cover of (N[vi] ∪ N[vi+1] ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ∪ N[vj]) and |CC(N[vi, vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj])| to denote the num- ber of cliques in CC(N[vi, vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Note that this is a clique cover of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' However, there exists graphs for which it is not a minimum clique cover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The clique cover of the closed neighborhood of vertices in path P is stored in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We denote the maximal cliques which constitute K in the order in which they appear in CC(N[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vp]), by K1, K2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Kα′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In any perfect graph, the size of a minimum clique cover equals the size of a maximum independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Based on this, we state an observation that we 11 use in proving some important properties of the constructed clique cover of the neighborhood of vertices in path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Observation 8 In any perfect graph G′, for each maximal clique K in a min- imum clique cover K of G′, there exists a vertex u ∈ G′ such that u does not belong to any other maximal clique in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For 1 ≤ i ≤ p, |C(N[vi])| ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The proof is by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let | C(N[vi]) |> 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By definition, C(N[vi]) contains only the maximal cliques in the linear ordering O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' From Observation 8, it follows that for each maximal clique Q ∈ C(N[vi]), there exists a vertex w which is unique to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since |C(N[vi])| > 3, there exists at least 4 vertices w1, w2, w3, w4 in N[vi] that forms an independent set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that vi together with w1, w2, w3, w4 form a forbidden structure K1,4 (refer Figure 1 (i)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' This is a contradiction to our premise that G does not contain any forbidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ Note: You may refer Algorithm 1 for the notations used in the proofs of lemmas given below: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In the path P, if for any vertex vi, 1 ≤ i < p , |C(N[vi])| = 3, then |C(N[vi+1])| ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The proof is by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Assume that there exists a vertex vi ∈ P, 1 ≤ i ≤ p for which |C(N[vi])| = 3 and |C(N[vi+1])| ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By Lemma 3, |C(N[vi+1])| cannot exceed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Vertices vi and vi+1 form an edge in the path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Cr′ Cr vi vi+1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Forbidden structure formation We consider the following cases based on the cardinality of C(N[vi])∪C(N[vi+1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Case when | C(N[vi]) ∪ C(N[vi+1]) |= 4: Recall from Algorithm 1 that Qi−1 r and Qi r are the maximal cliques in the ordering O which contains the right end- points of the intervals corresponding to vi−1 and vi respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For any vertex vi ∈ P, Qi−1 r , Qi r ∈ C(N[vi]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By our choice of vi+1 in the construction of path P, vi+1 ∈ {Qi r \\ Qi r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Thus vi+1 is covered by Qi r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' As per our assumption, 12 | C(N[vi]) ∪ C(N[vi+1]) | = 4, and | C(N[vi]) | = 3 by our premise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' There- fore those vertices of N[vi+1] which are not covered by Qi r have to be covered by exactly one more clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that | C(N[vi+1]) | = 2, which is a contradiction to our assumption that | C(N[vi+1]) | = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' This, in turn, is a contradiction to our initial premise that | C(N[vi]) ∪ C(N[vi+1]) |= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Thus the only possibility is, | C(N[vi]) ∪ C(N[vi+1]) |= 5, which we discuss in the next case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Observe that | C(N[vi])∪C(N[vi+1]) | cannot be greater than 5 since vi+1 ∈ Qi r and | C(N[vi+1]) | = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Case when | C(N[vi]) ∪ C(N[vi+1]) |= 5: The proof is by contradiction to our premise that G does not contain any forbidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We first show that C(N[vi])∪C(N[vi+1]) is indeed a minimum clique cover of N[vi]∪N[vi+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Then, using Observation 8, we show that there exists a forbidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By definition, C(N[vi]) is a minimum clique cover of N[vi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Therefore, each of the three maximal cliques in C(N[vi]) has atleast one unique vertex which does not belong to any other maximal clique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since vi+1 ∈ Qi r and Qi r ∈ C(N[vi]), Qi r ∈ C(N[vi+1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let the other two maximal cliques in C(N[vi+1] be Qj and Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By Observation 8, Qj and Qk contain a unique vertex each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that any minimum clique cover of N[vi] ∪ N[vi+1] contains all three maximal cliques of C(N[vi]) along with Qj and Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Hence C(N[vi]) ∪ C(N[vi+1]) is a minimum clique cover of N[vi]∪N[vi+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By Observation 8, on C(N[vi])∪ C(N[vi+1]), there is a set V ′ of 5 vertices in C(N[vi]) ∪ C(N[vi+1]) that are mutually disjoint and form an independent set of size five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The edge (vi, vi+1), together with V ′ form a forbidden structure (see Definition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Thus we have arrived at a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that if | C(N[vi]) | = 3, then | C(N[vi+1]) | = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ Observation 9 For each vertex v ∈ P \\ vp, | C(N[v]) |≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By construction of path P, CC(N[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vi]) = CC(N[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vi−1]) ∪ C(Qi−1 r+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Qi r) where Qi−1 r is the rightmost maximal clique in CC(N[v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vi−1]) and it covers vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Qi r is the rightmost maximal clique to which vi belongs to, in the ordering O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Note that C(N[vi]) = Qi−1 r ∪ C(Qi−1 r+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Qi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let A = N[vi] ∩ (Qi−1 r+1 ∪ · · ∪ Qi r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since vi ̸= vp, we know that there is vi+1 ∈ P which is chosen such that vi+1 /∈ Qi−1 r and vi+1 ∈ N[vi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that A ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By choice of vi, it has the rightmost right endpoint among that of all vertices in Qi−1 r \\ Qi−1 r′ and vi ̸= vp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Hence ∃u ∈ A which is not covered by Qi−1 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Therefore, there exists at least one Q ∈ C(Qi−1 r+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Qi r) that covers all the vertices in A,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In other words, C(Qi−1 r+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , Qi r) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that C(N[vi]) is of size at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ Now we prove a stronger claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In path P, there is at most one vertex v where | C(N[v]) |= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The proof of this claim is again by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Assume that there is more than one vertex with minimum clique cover equal to 3 in path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let vi be 13 the first such vertex in P in the increasing order of left endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By Lemma 4, we know that the minimum clique cover of vi+1 is of size less than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By our assumption, ∃j > i+ 1 such that minimum clique cover of vj is of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let the number of vertices in the subpath of P from vi to vj (including both vi and vj) be l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows from Observation 9 that for each vertex vk ∈ P, k ∈ [i+1, j −1], the minimum clique cover is of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We compute CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vk]) with respect to CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vk−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Note that vk is already covered in CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vk−1]) and CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vk]) additionally covers N[vk] \\ N[vk−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since |C(N[vk])| = 2, it adds just 1 to the number of cliques in CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vk−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='e, |CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vk])| = |CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vk−1])| + 1 It follows that each of the l − 2 vertices in {vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj−1} increments the size of the clique cover by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='e, |CC(N[vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj−1])| = l − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Thus |CC(N[vi, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj])| = |C(N[vi])| + |CC(N[vi+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , vj−1])| + |C(N[vj])| − 1 = 3 + (l − 2) + 3 − 1 = l + 3 vi vj vi+1 vi+2 l − 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Vertices vi and vj belong to three consecutive cliques Note that we deduct 1 from | C(N[vj]) | since vj is already covered by CC(N[vi+1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' , N[vj−1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We can see that the vertices from vi to vj form a path of length l which has an independent set of size l + 3 in its neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The vertices from vi to vj, together with the independent set of size l + 3 in its neighbourhood forms a forbidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We have arrived at a contradiction to our premise that G does not contain any forbidden structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Therefore it is proved that in path P, there is at most one vertex which has a minimum clique cover of size 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ By Algorithm 1, K = {K1, K2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Kα′} is the clique cover of path P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Note that it forms a minimal clique cover of input graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' | Ki−1 ∩ Ki ∩ Ki+1 |≤ 1, 2 ≤ i ≤ α′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' By Lemma 5, there is at most one vertex with minimum clique cover of size 3 in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If such a vertex exists, it would belong to three consecutive maximal cliques in K, and let us denote them by Ka−1, Ka and Ka+1, a ∈ [2, α′ − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For all other i ̸= a, i ∈ [2, α′ − 1], | Ki−1 ∩ Ki ∩ Ki+1 |= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' It follows that | Ki−1 ∩ Ki ∩ Ki+1 |≤ 1 for 2 ≤ i ≤ α′ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ We now use the properties of P proved in above lemmas to construct a clique cover of G with the property that the cliques are all vertex disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The clique cover is denoted by B = {B1, B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Bα′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We outline the steps in the procedure below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let vl, 1 ≤ l ≤ p be the vertex in P such that |C(N[vl])| = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If no such vertex exists in P, then let l = p + 1, and let Kp+2 be the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Further, Kp+3 is the empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We know that there is at most one such vertex, and the construction below will also take care of the case when for all vi, 1 ≤ i ≤ p, |C(N[vi])| = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For 1 ≤ i ≤ l − 1, Bi = Ki \\ Ki+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For i = l ≤ p, we define Bl, Bl+1, Bl+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' – Bl = Kl \\ Kl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' – Bl+1 = Kl+1 \\ (Kl ∪ Kl+2) – Bl+2 = Kl+2 \\ Bl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' For i ≥ l + 1, Bi+2 = Ki+2 \\ Ki+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since G does not have the forbidden structure, it follows that B is a clique cover, and by construction, it is a parition of the vertex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Further, the number of cliques in B is α′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' To complete the proof of Theorem 2, we use the crucial property of the canonical representation HG that no two intervals in HG have the same left end point or the same right end point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Using this property, we show that for each Bi there is a point pi in HG such that the intervals in HG which contain pi are exactly those which correspond to the vertices in Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' These points pi, 1 ≤ i ≤ α′ form an exact hitting set of HG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' This completes the characterization of EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='1 Algorithm to recognize exactly hittable interval graphs In this section, we present an algorithm to recognize an exactly hittable interval graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' This algorithm makes use of the canonical interval representation in Sec- tion 2 and the result by Dom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' for MMSC problem on interval hypergraphs [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' In their paper, Dom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' showed that an integer linear programming (ILP) formulation, say L, for MMSC problem on interval hypergraphs can be solved in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The coefficients of inequalities in L results in a totally unimod- ular matrix and the polyhedron corresponding to L is an integer polyhedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If the input instance to ILP is an exactly hittable instance, then the solution re- turned is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' We use this algorithm below to test if a given interval hypergraph instance is exactly hittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 15 Algorithm isEHIG: Given an interval graph G, construct the canonical inter- val representation as described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let HG be the resulting interval representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Run MMSC algorithm by Dom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' [3] on HG as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' If the algorithm returns value 1, then return yes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Else return no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Algorithm isEHIG(G) outputs yes if and only if G is exactly hittable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The proof follows from Lemma 1, Theorem 3 and the correctness of al- gorithm for MMSC problem on interval hypergraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content='2 Proper Interval Graphs is a subclass of EHIG The following lemma proves our Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The set of Proper Interval Graphs are strictly contained in the set of EHIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let G be a proper interval graph and let it be the intersection graph of the interval hypergraph H = (V, I) in which no interval properly contains an- other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Since H is a proper interval hypergraph, no two intervals in I) can have the same left endpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Hence order intervals in I according to increasing order of their left endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let this ordering be I1 < I2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' < Im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Add r(I1) (which is the smallest right endpoint among all intervals) to set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Remove all intervals hit by r(I1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Recurse on the remaining set of intervals until all the intervals are hit by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Clearly, S is an exact hitting set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' To show the strict containment, we show that the graph K1,3 which is a for- bidden structure [18] for Proper Interval Graphs has an exactly hittable in- terval representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Let the vertices of the K1,3 be {u, a, b, c} and edges be {(u, a), (u, b), (u, c)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' The intervals assigned to the vertices a, b, c and u are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Hence the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' ⊓⊔ u a b c a b c u 1 2 3 4 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' Exactly hittable interval representation of K1,3 This completes the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf'} +page_content=' J´er´emie Chalopin and Daniel Gon¸calves.' 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Parameter SIRD Model +Cem C¸akmaklı ∗ ,1 and Yasin S¸im¸sek †,2 +1Ko¸c University +2Duke University +February 1, 2023 +Abstract +This paper extends the canonical model of epidemiology, the SIRD model, to allow +for time-varying parameters for real-time measurement and prediction of the trajectory +of the Covid-19 pandemic. Time variation in model parameters is captured using the +generalized autoregressive score modeling structure designed for the typical daily count +data related to the pandemic. The resulting specification permits a flexible yet parsi- +monious model with a low computational cost. The model is extended to allow for un- +reported cases using a mixed-frequency setting. Results suggest that these cases’ effects +on the parameter estimates might be sizeable. Full sample results show that the flex- +ible framework accurately captures the successive waves of the pandemic. A real-time +exercise indicates that the proposed structure delivers timely and precise information +on the pandemic’s current stance. This superior performance, in turn, transforms into +accurate predictions of the confirmed and death cases. +Keywords: Covid-19, SIRD, Observation driven models, Score models, Count data, Time- +varying parameters +JEL Classification: C13, C32, C51, I19 +∗Correspondence to: Cem C¸akmaklı, Ko¸c University, Rumelifeneri Yolu 34450 Sarıyer Istanbul Turkey, +e–mail: ccakmakli@ku.edu.tr. +†e–mail: yasin.simsek@duke.edu +1 +arXiv:2301.13692v1 [econ.EM] 31 Jan 2023 + +1 +Introduction +The outbreak of the new coronavirus, the Covid-19 pandemic, is one of the most severe +health crises the world has encountered in the last decades. Since the onset of the pan- +demic in early January 2020, it has exhibited a varying pattern for many reasons. First, +countries have repeatedly taken various measures to reduce the transmission of the virus. +These measures involve complete lockdowns, such as full closure of business and curfew, +and partial lockdown that implies a partial closure of daily routines. These interventions +seem to mitigate the spread of the virus at various stages of the pandemic to the extent +that people comply with the ’shelter-in-place’ orders. Second, mutations of the virus might +lead to changes in its main characteristics. Furthermore, the death rate seems to be par- +tially lowered thanks to vaccination campaigns, increasing medical knowledge, and ongoing +research on the virus. +While the pandemic evolves rapidly with successive waves of infections, efficient and +timely monitoring is crucial. Making prompt and effective decisions of imposing or relaxing +lockdown measures for policymakers and taking timely precautions for individuals critically +relies on knowledge about the pandemic. Therefore, epidemiological models for estimat- +ing and, perhaps even more crucially, for predicting the pandemic’s trajectory come to the +forefront. Conventional statistical epidemiology models mainly involve structural parame- +ters that remain constant throughout the pandemic. However, suppose these interventions +are effective and cause changes in the pandemic’s natural course. Essentially, even if the +underlying structural parameters related to the pandemic remain unaltered, there might +be various reasons for the resulting estimated parameter to be time-varying. These include +changes in the people’s attitude towards the disease1. Besides, the virus might undergo +some mutations which alter the contagiousness and fatality of the virus; see, for example, +Karim and Karim (2021) for the recent Omicron variant of the virus. Hence, these muta- +tions translate into changes in key structural model parameters that alter smoothly with +the changing weight of infections of each virus variant. These observations are the depar- +ture point of this paper. Specifically, we develop a computationally simple and statistically +1Arias et al. (Forthcoming) uses the terminology of behavioral parameters to refer to the behavioral +aspects that lead to time variation in the structural parameters. Similarly, Avery et al. (2020) refers to the +changes in the parameters as potential endogeneity of these parameters as the precautions taken could be a +function of current cases. We thoroughly discuss these issues and the inability to measure the observations +of units/compartments, such as the number of susceptible people required for estimating the models in +Section 2.3. +1 + +coherent model that allows for time variation in the epidemiological model parameters. +We start our analysis by confronting a simple version of the workhorse epidemiological +model with the existing data. From the perspective of econometrics, we specify a counting +process for modeling the course of the Covid-19 pandemic for the US based on the SIRD +model, which is an abbreviation representing the four identified states of the pandemic +as Susceptible, Infected, Recovered, and Death. +The SIRD model depicts these states’ +evolution depending on the number of infected individuals; see Kermack and McKendrick +(1927); Allen (2008). Provided that these daily counts of susceptible, infected, recovered, +and death cases are available, the model is well-identified conditional on the infected cases’ +initial value. +The pandemic’s course is determined by the contestation of these forces, +i.e., the parameters governing infection and resolution (either recovery or death) rates. +As a result, if the rate of infection (multiplied by the share of susceptible people in the +population) is larger than the resolution rate, the number of infections evolves according to +a nonstationary process representing the virus’s increasing spread. In contrast, the opposite +case results in a stationary process. Therefore, we opt for a Bayesian estimation strategy +of the parameters for computing reliable credible intervals for inference conditional on the +available data rather than utilizing asymptotic analysis. +Equipped with these tools, we extend the econometric model by allowing for time vari- +ation in the structural parameters by resorting to the Generalized Autoregressive Score +(henceforth GAS) modeling framework, which is a class of observation-driven models. The +proposed model permits a flexible yet feasible framework to track structural parameters’ +evolution timely and accurate. A relevant aspect of our specification is its relatively low +computational cost. The computational cost might be crucial, especially at the beginning +of the pandemic, when the data is scarce, plaguing the inference of flexible models, and the +uncertainty is overwhelming. Furthermore, since the typical Covid-19 dataset exhibits a +sizeable seasonal pattern, the model is extended to capture these stark seasonal variations +using a seasonal component for each parameter by resorting to the frequency domain and +GAS framework model structures. We extend the model further, taking undocumented +cases (as these infected individuals do not show symptoms) into consideration by exploit- +ing the information on testing for infection following Grewelle and De Leo (2020) and on +the number of excess deaths, i.e., the total number of deaths caused by the pandemic +that is in excess of the reported death cases. Since the typical excess death data is at the +weekly frequency, unlike the remaining daily counts, the resulting extension permits a mixed +2 + +frequency time-varying parameter epidemiological model blending datasets with mixed fre- +quency. Finally, we also consider the model in a multi-country setting. In particular, we +put forward the factor TVP-SIRD model, where we consider four countries jointly, focusing +on the common and idiosyncratic patterns of the model parameters. The resulting model +can efficiently capture the common behavior in the diffusion of the pandemic throughout +the world for monitoring the stance of the pandemic from a global perspective. +We consider the US dataset related to the pandemic to demonstrate the proposed frame- +work’s efficacy. The US has ample experience in containing the virus with differential mo- +mentum in fighting the disease at various pandemic stages. While the initial US response +to the pandemic in terms of imposing restrictions and establishing a containment infras- +tructure such as mass testing was not fast, the US was among the first countries to start +a massive inoculation campaign at the start of 2021. These distinct experiences provide a +testing ground with various patterns to examine the proposed model’s efficacy. Our results +indicate that the model parameters exhibit substantial changes over time. The virus’ trans- +mission had reduced considerably with the start of 2021 thanks to the massive vaccination +campaign in the US. However, this pattern is interrupted by the two successive waves of +the Delta variant and then the Omicron variant, with a rapid increase in virus transmission +and a relatively low death rate captured by our model. The mixed frequency extension of +the model that also captures unreported cases suggests that the actual number of infected +cases is potentially as high as three times more than the reported cases for some periods. +Provided by the wide 95% credibility set ranging from two to five, this finding is consistent +with the estimates around four provided by the Center for Diseases Control and Prevention +(CDC) for the period until the end of 2020, see Reese et al. (2020). The multi-country +analysis involving Germany, Italy, and Brazil on top of the US using the factor TVP-SIRD +model indicates that the pandemic diffusion shares a sizable common pattern with Euro- +pean countries, including Germany and Italy. However, the diffusion of the pandemic is +relatively more idiosyncratic in Brazil. +We examine the model’s performance in real-time by conducting a recursive estima- +tion and forecasting exercise with real-time datasets predicting 1- to 30-day ahead number +of confirmed and death cases. Results indicate that the proposed model with time-varying +parameters provides timely information on the pandemic’s current stance ahead of the com- +peting models. While the results are relatively mixed for long horizons from 2-week up to +a 1-month ahead, our model yields superior forecasting performance up to 2-week horizon +3 + +against many competitors, including a linear Gaussian state-space model and a subclass +of our model framework. Therefore, the resulting model is instrumental in providing cru- +cial information on the stance of the weeks ahead of the pandemic. We further confront +the weekly predictions using our model with those of the models that are included in the +Forecast Hub2, which is a forecast data repository with the predictions created by dozens +of leading infectious disease modeling teams from around the globe, in coordination with +the CDC. Our comparison indicates that the proposed TVP-SIRD model outperforms more +than half of those leading models when 1-week ahead forecasts are considered. However, this +outperformance reduces gradually with the increasing horizon. The Forecast Hub addition- +ally provides an ensemble model; a forecast combination scheme generated using individual +models built on different assumptions. The results show that our model provides superior +forecasts, specifically at the onset of the pandemic when the data is scarce, reflecting our +proposed framework’s flexible yet parsimonious model structure. +The literature on estimating the SIRD model (with fixed parameters) and variants to +evaluate the current stance of the Covid-19 pandemic has exploded since its outbreak. Rela- +tively earlier analyses include Read et al. (2020) and Lourenco et al. (2020), which estimate +a SIRD-based model with the data from China for the former and from the UK and Italy for +the latter using a likelihood-based inference strategy. Wu et al. (2020) blend data related +to Covid-19 for China with mobility data and estimate the epidemiological model using +Bayesian inference to predict the spread of the infection domestically and internationally. +Li et al. (2020) conduct a similar analysis employing a modified SIRD model together with +a network structure and mobility data to uncover the size of the undocumented cases; see +also Horta¸csu et al. (2021). Zhang et al. (2020) extend the standard SIRD model with +many additional compartments and estimate some parameters using Bayesian inference. +Identification of the model parameters in these models hinges upon the data availability for +each compartment. Otherwise, parameter values are set based on the pandemic’s stylized +facts; see Manski and Molinari (2020); Atkeson (2020); Korolev (2020). +Several factors might lead to the time variation in the parameters of the epidemiological +model. On the one hand, lockdown measures implemented by the policymakers isolate the +infected from the susceptible individuals. Therefore, the parameter governing the infection +rate, that is, the average number of contacts of an individual, is likely to alter with lockdown +conduct, see Hale et al. (2020) for example. On the other hand, advancements in the fight +2https://covid19forecasthub.org/ +4 + +against Covid-19, including the recovery of drugs and vaccination, could effectively mitigate +the course of the disease. In addition, the installment or the lack of medical equipment +such as ventilators might alter the rate of recovery or, in other words, the duration of +the state of being infected, see for example Greenhalgh and Day (2017) on time variation +in recovery rates. +Accordingly, Anastassopoulou et al. (2020) use a least-squares-based +approach on a rolling window of daily observations. They document the time variation +of parameters in the SIRD-based model using Chinese data. Tan and Chen (2020) also +employ a similar but more articulated rolling window strategy to capture the time variation +in the model parameters. Other frameworks with time-varying model parameters almost +exclusively allow for the time variation only in the infection rate. An application before +the Covid-19 outbreak includes, for example, Xu et al. (2016) among others, who utilize a +Gaussian process prior to the incidence rate involving the infection rate using a Bayesian +nonparametric structure. In the context of the Covid-19 pandemic, Kucharski et al. (2020) +estimate a modified SIRD model using a parameter-driven model framework allowing the +infection rate to follow a geometric random walk with the remaining parameters kept as +constant; see Arroyo-Marioli et al. (2021) for a similar approach. Similarly, Yang et al. +(2020) and Fern´andez-Villaverde and Jones (2022) allow for time variation in the infection +rate, keeping the remaining parameters constant. Arias et al. (Forthcoming), on the other +hand, extends the model with time variation in the remaining parameters and provides a +Bayesian inference methodology of the resulting model using a particle filter. Liu et al. +(2020) proposes an econometric specification where the growth rate of infections follows an +autoregressive process around a deterministic trend with a structural break. +In this paper, we propose an alternative modeling strategy to capture the time vari- +ation in the structural parameters of the SIRD model. On the one hand, our modeling +framework is statistically consistent with the typical count data structure related to the +pandemic, unlike the models that either employ least-squares or likelihood-based inference +using Gaussian distribution, that is, the Kalman filter. On the other hand, our framework is +computationally inexpensive, unlike the models that are statistically consistent but compu- +tationally costly such as the particle filter. This computational efficiency might be crucial, +most notably when the data is scarce, and uncertainty about the pandemic is abounding +at the start of the pandemic. Our framework belongs to the observation-driven models +class, specifically the GAS models proposed by Creal et al. (2013). GAS models involve +many celebrated econometric models like the Generalized Autoregressive Heteroskedasticity +5 + +(GARCH) model and various variants as a specific case, and thus, they proved to be useful +in both fitting and prediction. Koopman et al. (2016) provide a comprehensive analysis +of these models’ predictive power compared to parameter-driven models in many settings, +including models with count data. +Observation-driven models for count data are considered, in many cases, independent +of the analysis of the Covid-19 pandemic. +Davis et al. (2003) provide a comprehensive +analysis of observation-driven models with a particular focus on data with (conditional) +Poisson distributions. Ferland et al. (2006) derive an integer-valued analog of the GARCH +model (IN-GARCH) using Poisson distribution instead of Gaussian distribution. Fokianos +et al. (2009) consider the Poisson autoregression of linear and nonlinear forms like the IN- +GARCH model as a specific case. Chen and Lee (2016) extend the Poisson autoregression +to allow for smooth regime switches in parameters. Our framework naturally extends these +approaches to the epidemiological model framework for each of the core compartments of +the SIRD model using a multivariate structure. +The remainder of the paper is organized as follows. Section 2 describes the canonical +SIRD model and introduces the SIRD model with time-varying parameters. Section 3 dis- +cusses econometric issues, including identifying model parameters and how to account for +sample selection. This section further elaborates on our estimation strategy and the result- +ing simulation scheme. Section 4 presents estimation results using full sample data from the +US. In Section 5, we evaluate our model framework’s real-time performance in capturing +the pandemic’s current stance and forecasting compared to frequently used competitors. +Section 6 discusses potential extensions of the model. Finally, we conclude in Section 7. +2 +Model specification +2.1 +The canonical model of the pandemic, the SIRD model +We start our analysis by discussing the epidemiological model denoted as the SIRD model +of Kermack and McKendrick (1927). Specifically, the SIRD model categorizes a population +into four classes of individuals representing four distinct states of the pandemic; Susceptible +(S(t)), Infected (I(t)), Recovered (Rc(t)) and Death (D(t)) in period t. The susceptible +group does not yet have immunity to disease, and individuals in this group have the pos- +sibility of getting infected. On the other hand, the recovered group consists of individuals +who are immune to the disease, and finally, D(t) represents individuals who have suc- +6 + +cumbed to the disease. The Susceptible-Infected-Recovered-Death (SIRD) model builds on +the principle that a fraction of the infected individuals in the population, I(t) +N , can transmit +the disease to susceptible ones, S(t), with a (structural) infection rate of β by assuming a +quadratic matching in the spirit of gravity law, see Acemoglu et al. (2021) for details on +alternative matching structures. Therefore, the number of newly infected individuals in the +current period is βS(t) I(t) +N . The newly infected individuals, that is, confirmed cases, C(t), +should be deducted from the susceptible individuals in the current period. Meanwhile, in +each period, a fraction γ of the infected people recover from the disease, which reduces the +number of actively infected individuals. Similarly, a fraction ν of the infected people have +succumbed to the disease, further reducing the number of actively infected individuals3. +Hence, a fraction γ + ν of the infections are ‘resolved’ in total. This structure leads to the +following sets of equations: +˙S(t) += +−βS(t) I(t) +N +˙Rc(t) += +γI(t) +˙D(t) += +νI(t) +˙I(t) += +˙S(t) + ˙Rc(t) + ˙D(t) +(1) +where ˙x corresponds to dx/dt, and we assume that the population remains constant.4 +2.2 +Econometric analysis of the SIRD model with fixed parameters +The parameters of interest are the structural parameters β, γ, and ν that provide informa- +tion on the transmission and resolution rates of the Covid-19 pandemic. A central metric +that characterizes the course of the pandemic is the effective reproduction number, eR(t). +The effective reproduction number refers to the speed of the diffusion, which can be com- +puted by the ratio of newly confirmed cases, denoted as ˙C(t), to the resolved cases, that is, +˙C(t)/( ˙Rc(t) + ˙D(t)). Therefore, it serves as a threshold parameter of many epidemiological +models for examining whether the disease will be extinct or spread further. Accordingly, +using (1) eR(t) is identical to β S(t) +N /(γ + ν) and when t = 0, it is identical to β/(γ + ν), +3We note the difference between the term death rate and the terms case fatality ratio or mortality rate. +While the case fatality ratio refers to the ratio of the (cumulative) number of deaths to the (cumulative) +number of the infected individuals, the mortality rate measures the proportion of deaths due to a specific +disease among the entire population for a given period. On the other hand, the death rate, νt, measures the +portion of the actively infected population who succumbed to Covid-19 for a given period. +4In fact, the number of deaths reduces the total population. We assume that the total number of deaths +is negligible compared to the population for the tractability of the resulting SIRD model. +7 + +in which case it is denoted as the basic reproduction rate. In this sense, a value of eR(t) +being less than unity indicates that the pandemic is contained, and if it exceeds unity, this +implies that the spread of the pandemic continues. Our primary motivation for employing +the model from the econometrics perspective is to conform to this canonical epidemiological +model with the existing datasets and pinpoint the pandemic’s stance timely. For that pur- +pose, we first discretize (1) as the typical Covid-19 dataset involves daily observations on +the counts of individuals belonging to these states of health. Motivated by this, we specify +a counting process for the states using the Poisson distribution conditional on past cases +of active infections implying a nonhomogenous Poisson process for all the counts see, for +example, Allen (2008); Yan (2008); Rizoiu et al. (2018) for earlier examples, and Li et al. +(2020) in the Covid-19 context for a similar approach. We specify the following for the +stochastic evolution of the counts of these states; +∆Ct|Ωt−1 +∼ +Poisson(β St−1 +N It−1) +∆Rct|Ωt−1 +∼ +Poisson(γIt−1) +∆Dt|Ωt−1 +∼ +Poisson(νIt−1) +∆It += +∆Ct − ∆Rct − ∆Dt, +(2) +where Ωt stands for information set that is available up to time t. We assume that ∆Ct, +∆Rct, and ∆Dt, representing the daily counts of the pandemic states, are independent +conditional on Ωt−1. The final identity leads to an autoregressive process for the number +of active infections, It. The resulting distribution for the number of active infections is a +Skellam distribution (conditional on Ωt−1) with the mean πt−1It−1, where πt−1 = (1+β(1− +eR−1 +t−1)) and the variance as β(1+eRt−1)It−1. Here, we use the identity in the last equation +of (2) together with the definition of eRt. Therefore, the stationarity of the resulting process +depends on whether eRt−1 < 1 or eRt−1 ≥ 1, i.e., whether the pandemic is taken under +control or not. In addition, in case St−1 +N +≈ 1 or in other words, eRt−1 ≈ R0, the first and +second unconditional moments are as follows, +E[It] += +πtI0 +V ar(It) += +β(1 + R−1 +0 ) πt−1(1−πt) +1−π +I0, +(3) +where we assume that the initial condition, I0, is known. If the initial condition is considered +a parameter to be estimated, then the variance is further amplified with a factor in the +8 + +initial condition’s variance. Accordingly, the unconditional moments of the pandemic states +are linear functions of these unconditional moments of It. We refer to Section A of the +supplementary material for details. +2.3 +Motivation for time variation in parameters +The canonical model’s structural parameters represent the characteristics of the Covid-19 +virus in terms of infectiousness and effectiveness. Therefore, unless there is a change in +these characteristics, such as the emergence of a virus variant, the pandemic’s underlying +structural parameters remain unaffected. However, when confronting the epidemiological +model with the real datasets of the pandemic, often, it is not feasible to observe/measure +all the categories/compartments of the pandemic precisely. These measurement problems +might arise from various reasons. For example, non-pharmaceutical interventions might +be one of the underlying reasons, as identifying the number of perfectly isolated people +who comply with stay-at-home orders is not easy. Even in a full lockdown scenario, some +citizens might either break the rules or work in essential sectors that are always kept open. In +addition, some behavioral shifts might be reflected in these parameters because, confronted +with a high number of infections or alerted by the strict public measures, people would +self-isolate even more with the fear of getting infected. These swings in attitudes are also +reflected as time variations in parameters. Therefore, some authors refer to this distinction +by rephrasing these as ’behavioral’ parameters, see for example Arias et al. (Forthcoming) +or ’potential endogeneity of parameters’, see for example Avery et al. (2020). +Here we +refer to a broader stance and refer to these parameters as ’implied’ parameters in the sense +that throughout the paper, the term ’parameter’ refers to ’implied parameters’. Another +motivation for the time-variation is the emergence of the new variant(s) of the virus with +altered characteristics, such as the Delta and Omicron variants, for example, see Karim and +Karim (2021). With the increasing number of infections by the new variant, the number +of confirmed cases would correspond to a mixture of the prevailing variants. The weights +of this mixture gradually evolve depending on the prevalence of more infectious variants. +To elaborate further, we reconsider the first equation of the SIRD model in (2), i.e., the +equation concerning the number of confirmed cases, this time considering the measurement +error and the virus variants explicitly as +∆Ct +∝ +S∗ +t−1 +N (β1I1,t−1 + β2I2,t−1) +(4) +9 + +where v = 1, 2 denotes the vth variant of the virus and Iv,t−1 are the number of people +infected with this variant. For ease of demonstration, we assume that in a given period t, +only two variants can be spread in the population. S∗ +t−1 indicates the number of susceptible +people. However, we observe the number of susceptible people only with a measurement +error that we denote with Ξt. +Therefore, the observed number of susceptible people is +St−1 = S∗ +t−1 + Ξt. Moreover, the total number of active infections is the sum of the number +of variant-specific infections over the type of infections, i.e., It = I1,t + I2,t. Considering the +following decomposition, we can define the time-varying parameter as +βt +St−1 +N It−1 +≈ +S∗ +t−1 +N (β1I1,t−1 + β2I2,t−1) += +S∗ +t−1 +N +((β2 − β1)I2,t−1 + β1It−1) += +St−1−Ξt−1 +N +((β2 − β1)I2,t−1 + β1It−1) += +St−1 +N β1It−1 + St−1 +N (β2 − β1)I2,t−1 − Ξt−1 +N β1It−1 − Ξt−1 +N (β2 − β1)I2,t−1 +(5) +The last three terms on the right-hand side represent various sources of factors that can +lead to time variation in β. First, if in society only the first variant of the virus with the +infection rate β1 prevails and susceptible people are counted perfectly, then these terms +drop from the expression and βt = β1. However, if a second variant of the virus emerges +with the infection rate β2 then we would observe a smooth change in the parameter with +the increasing number of infections I2,t−1 relative to I1,t−1. If the dominance of the second +variant does not materialize immediately, then the changes will be smooth until βt = β2, +where the prevailing variant will be the second variant. On the other hand, if the number +of susceptible could not be efficiently measured, then we would have a nonzero Ξt−1. This +measurement error would magnify these changes further as, in this case, the third and fourth +terms would contribute to the changes in the implied parameter. We provide a detailed +analysis of these underlying causes of time variation related to measurement errors especially +using the vaccination dataset and additionally in case of the probability of reinfection in +Section B of the supplementary material. +We refer to that section for a more detailed +analysis. In this analytical demonstration, we only focus on the rate of infection for the +discussion’s compactness but modeling the underlying drivers of the rate of recovery and +death could be conducted similarly. +Essentially, for the remaining parameters, i.e., the +rates of recovery and death, these arguments related to the difficulties in measurement and +time-varying weight of the mixture of virus variants in the society might play an integral +10 + +role in the time variation in parameters. In addition, advancements in the fight against +Covid-19, including recovery of drugs and vaccination, could effectively mitigate the course +of the disease. +2.4 +SIRD model with time-varying parameters - the TVP-SIRD model +In this section, we put forward the SIRD model with time-varying parameters. We use the +framework of the Generalized Autoregressive Score model for modeling the time variation +in parameters. This framework encompasses a wide range of celebrated models in econo- +metrics, including the GARCH model and its variants. Briefly, the GAS model relies on +the intuitive principle of modeling the time variation in key parameters in an autoregressive +manner which evolves in the direction implied by the score function and thereby improving +the (local) likelihood; see Creal et al. (2013) for a detailed analysis of the GAS model. As +in the case of the GARCH model, it effectively captures the time dependence in long lags +in a parsimonious yet quite flexible structure. Consider the SIRD model with time-varying +parameters as βt, γt, and νt. While the parameter for the rate of infection βt is positive, +the parameters γt and νt are on the unit line. Therefore, we transform βt using logarithmic +transformation and γt and νt using logit transformation.5 +Let the parameter with a ˜(.) +denote the corresponding transformations as ˜βt = ln(βt), ˜γt = logit(γt) and ˜νt = logit(νt) +where logit refers to the inverse of the logistic transformation as logit(x) = log( +x +1−x). The +resulting Time-Varying Parameters - SIRD (TVP-SIRD) model is as follows +∆Ct|Ωt−1 +∼ +Poisson(βt +St−1 +N It−1) +∆Rct|Ωt−1 +∼ +Poisson(γtIt−1) +∆Dt|Ωt−1 +∼ +Poisson(νtIt−1) +∆It += +∆Ct − ∆Rct − ∆Dt. +(6) +We decompose the transformed parameters further into a smooth level component and a +high-frequency seasonal component. Because the typical daily Covid-19 dataset exhibits +an immense daily seasonal pattern potentially due to frictions in reporting, we also put a +5Essentially, we require an additional requirement that γt + νt ∈ [0, 1]. +We go through a detailed +specification for the convenient transformation of the parameters and consider three cases. In the first case, +all three parameters are subject to only logarithmic transformation. In the second case, the parameters +γt, νt ∈ [0, 1] using logistic transformation, while in the third case, we impose an additional restriction of +γt + νt ∈ [0, 1]. Results show that the final restriction γt + νt ∈ [0, 1] is not binding. However, it imposes +important challenges on the feasibility of the estimation, especially when we enhance the model to allow for +seasonality. Therefore, we proceed with the second case throughout the paper. The specification search and +the findings are displayed in Section C of the supplementary material. +11 + +particular emphasis on the seasonal component. Specifically, consider the decomposition as +θt += +θl,t + θs,t +(7) +where for parameter θt = ˜βt, ˜γt and ˜νt, respectively, θl,t and θs,t are the level and seasonal +components, respectively.6 +For the level parameter, the evolution of the parameters is +specified as +θl,t += +ωθ + βθθl,t−1 + αθsθ,t−1 +(8) +where sθ,t for θt = ˜β, ˜γt and ˜νt are the (scaled) score functions of the joint likelihood +which is identical to that for the level parameter due to the additive structure. Since the +SIRD model’s likelihood function is constituted by the (conditionally) independent Poisson +processes, each score function is derived using the corresponding compartment. Specifically, +let ∇˜β,t = ∂L(∆Ct;˜βt) +∂ ˜βt +, ∇˜γ,t = ∂L(∆Rct; ˜γt) +∂ ˜γt +and ∇˜ν,t = ∂L(∆Dt; ˜νt) +∂ ˜νt +denote the score functions +for period t observation. We specify sθ,t such that the score functions are scaled by their +variance as sθ,t = +∇θ,t +Var(∇θ,t) for θt = ˜β, ˜γt and ˜νt.7 In the specific case of the SIRD model, +this modeling strategy leads to the following specification for the (scaled score functions) +in terms of the corresponding link function +s˜β,t += +∆Ct−1−λ1,t−1 +λ1,t−1 +s˜γ,t += +∆Rt−1−λ2,t−1 +λ2,t−1 +1 +(1−γt) +s˜ν,t += +∆Dt−1−λ3,t−1 +λ3,t−1 +1 +(1−νt) +(9) +where λ1,t = βt +It−1St−1 +N +, λ2,t = γtIt−1 and λ3,t = νtIt−1. The resulting specification implies +an intuitive updating rule because the parameters (in the logarithmic form) are updated +using a combination of the previous parameter value and a function of the previous per- +centage deviation from the mean. We refer to Section D of the supplementary material +for the details on the derivation of (9). One drawback of the specification in (8) is that +when βθ is close to unity, identification of ωθ is cumbersome, see for example Kastner and +Fr¨uhwirth-Schnatter (2014). This is also our experience when estimating the model using +6Such additive structure in the transformed parameters leads to a multiplicative seasonal structure in +exponential form as a function of original parameters; see, for example, Hansen and Schmidtblaicher (2021) +for a similar approach. +7Alternative approaches for scaling the score function include the standard deviation rather than the +variance and the score function without scaling. Our findings suggest that using the variance as the scaling +function leads to smoother and more robust evolution of parameters over time. +12 + +real datasets. Therefore, we restrict the βθ parameter to be unity. In this case, we remove +the intercept parameter ωθ, but the level of the time-varying parameters is estimated as the +initial condition θl,0 along with other model parameters.8 +For the seasonal component, we specify a structure using frequency domain for capturing +the daily seasonality in a given week adequately departing from Hansen and Schmidtblaicher +(2021)9. Consider the model +θs,t += +�s/2 +j=1 θjs,t +(10) +with +θjs,t += +cos(Λj)θjs,t−1 + sin(Λj)θ∗ +js,t−1 + ψjsθ,t−1 +θ∗ +js,t += +− sin(Λj)θjs,t−1 + cos(Λj)θ∗ +js,t−1 + ψ∗ +Jsθ,t−1 +(11) +where Λj = +2πj +s +for j = 1, 2, 3 and θt = ˜β, ˜γt and ˜νt. +The structure in (10) and (11) +provides a quite flexible yet parsimonious model structure for capturing seasonal behavior. +Essentially, it can be shown that in case the score functions are zero, it reduces to +θjs,t += +− sin(Λjt)θjs,t−1 + cos(Λjt)θ∗ +js,t−1. +(12) +This structure implies that the cycle is captured by the three periodic series with frequencies +Λ1 = 2π +7 , Λ2 = 4π +7 and Λ3 = 6π +7 each with period 7, 3.5 and 2.3 days. While the first series +has the fundamental frequency, the remaining parts could be obtained by integrating the +fundamental frequency; see Proietti and Pedregal (2022) for further details. The sine and +cosine terms together function as two orthogonal bases generalizing the model in Hansen +and Schmidtblaicher (2021). +As in the level case, the score function is identical to the +general score function owing to the linear decomposition of the parameters into the level +and seasonal components. This structure facilitates the estimation substantially and enables +us to capture the potential link between level and seasonality. +The specification in (6)-(11) leads to quite rich dynamics both in terms of mean and the +variance of the resulting process. These rich dynamics enable us to accurately capture the +pandemic’s evolution reflected in the timely and prompt response of the parameters to the +8We also compute the Bayes factor of this model relative to the unrestricted model for formal model +comparison. We find that the Bayes factor is to be 0.99, indicating that the restriction is also supported by +the data, as expected. +9We also consider models for seasonality that exploits the time domain using daily components for +modeling seasonality where the corresponding coefficients are time-varying. Our findings suggest that models +in the frequency domain facilitate the estimation and performs better than their time domain counterpart. +We display these results in Section E of the supplementary material. +13 + +pandemic states’ data changes. To elaborate further, we also consider the implied moments +of the resulting process. +We display these findings in Section F of the supplementary +material. +3 +Econometric inference +Two major issues plague the inference of the SIRD model parameters. +First, from the +epidemiology point of view, the SIRD model could be extended in various directions by +incorporating other phases, or in other words, compartments of the disease. As this implies +additional parameters to be estimated, identifying these parameters is challenging. Second, +detection of the infected individuals might be burdensome as some of them do not show +symptoms, yet they are infectious. In this section, we first discuss the challenges of the +econometric inference, and second, we introduce the details of the simulation-based Bayesian +estimation strategy for the econometric inference of the TVP-SIRD model. +3.1 +Identification of model parameters +The pandemic’s course in the evolution of active cases depends on the structural parameters, +β, γ and ν that are used to construct π in (3). The initial condition I0 is also required +because the process might be nonstationary if the pandemic is not contained. We estimate +the models starting from the period when the number of cumulative confirmed cases exceeds +1000, and we use this first observation in our sample as the initial condition.10 The structural +parameters represent the compartments of the SIRD model where the compartments refer +to the specific phases of the disease as ’susceptible’, ’infected’, ’recovered’, or ’death’. Still, +it is possible to extend the model with additional compartments. For example, it is known +that the virus has an incubation period in which the susceptible person is ’exposed’ to the +virus but not yet affected by it. Nevertheless, she can transmit the virus to other people. +Departing from this point Korolev (2020), for example, discusses identification problems of +the structural parameters regarding the SIRD model with an additional compartment of +’exposed’ case. However, these compartments require additional parameters to be estimated; +see Lourenco et al. (2020) for details. If the specific compartments’ data, such as the number +of infected cases where the virus is in the incubation period, is available, these structural +10Different starting points (such as the periods when the number of cumulative confirmed cases exceeds +10000) yield very similar results. Results are available upon request by the authors. +14 + +parameters are well identified. +Therefore, while these additional compartments provide +further refinements to the SIRD model, these refinements plague the identification of the +structural parameters if additional data is missing; see, for example, Atkeson (2020) who +discusses the identification of the structural parameters regarding the SIRD model. He +demonstrates how different parameter setups might result in very similar initial phases of +the pandemic but result in divergent patterns in the long-run absence of the data on the +model’s compartments. +3.2 +Accounting for sample selection +A fundamental underlying assumption of the model specification in previous sections is that +the variables of the infected, recovered, and succumbed individuals represent the aggregate +numbers. However, one of the stylized facts related to the Covid-19 pandemic is the presence +of infected individuals who do not have any symptoms, denoted as asymptomatic. These +hard-to-detect cases complicate the analysis as it leads to a selection bias in econometric +inference, among other factors, see Manski and Molinari (2020). These unreported infection +cases prohibit the tests from being randomly assigned, plaguing econometric inference. This +section provides a model extension based on some assumptions on the model structure to +capture asymptomatic infected individuals. We use two sources of additional datasets to +extract the total number of active cases, including the reported and unreported ones. The +first additional data we use is the number of excess deaths. These include the estimates +of additional deaths directly or indirectly attributed to Covid-19 in excess of the published +number of deaths. This projection provides weekly estimates of excess deaths, and these +weekly counts of deaths are compared with historical trends. Accordingly, it provides an +essential source of identification for the unreported cases, as potentially a significant part +of the excess deaths might be attributed to this sample selection11. The second variable +we exploit is the number of positives in the tested individuals. To motivate the idea, let Pt +denote an indicator function that takes the value one if an individual is infected in period +t and 0 otherwise. Further, let Tt denote another indicator function, which takes the value +one if an individual is tested for the infection in period t and 0 otherwise. Using the Bayes +11Please +see, +https://www.cdc.gov/nchs/nvss/vsrr/covid19/excess_deaths.htm +and +https: +//ourworldindata.org/excess-mortality-covid for additional discussion and methodology on the +computation of excess death +15 + +rule, we can show that, +P(Pt = 1) = P(Pt = 1|Tt = 1)P(Tt = 1) +P(Tt = 1|Pt = 1) +, +(13) +see also Stock (2020). In case the assignment for testing of an individual is carried out +randomly, then P(Tt = 1) = P(Tt = 1|Pt = 1) and there is no identification problem due +to sample selection. Neither, P(Pt = 1) nor P(Tt = 1|Pt = 1) are observed. Nevertheless, +P(Tt = 1) could be computed as the fraction of tested individuals in the population in +period t. Furthermore, P(Pt = 1|Tt = 1) could be considered as the daily positive test rate. +Equipped with these, identification of P(Tt = 1|Pt = 1) boils down to the identification of +P(Pt = 1), the true prevalence of the infection, including asymptomatic cases. Departing +from Grewelle and De Leo (2020), we make use of a parametric identification strategy for +approximation of P(Tt = 1|Pt = 1), +P(Tt = 1|Pt = 1) = exp(−kρt) +(14) +where ρt is the fraction of positives in the tested individuals in period t, and k is a positive +constant. Briefly, the underlying idea stems from the fact that detecting infections, including +asymptomatic individuals, would improve with the increasing number of testing. In that +sense, the fraction of tested individuals in the population should be related to the ratio +of reported infections to the total number of infections. With the increasing number of +testing on the population, this fraction approaches one. +On the other hand, if testing +is concentrated only on symptomatic individuals, this fraction approaches a lower bound, +captured by the parameter exp(−k), where the functional form admits exponential decay. +Unlike the daily dataset we employ for estimating the TVP-SIRD model, the excess +death data is at the weekly frequency. Therefore, we extend our model framework to allow +for a mixed-frequency dataset. +Let I∗ +t be the number of infected individuals involving +asymptomatic and symptomatic cases, and let S∗ +t and Rc∗ +t denote the total number of +susceptible and recovered individuals, respectively. +Let δt = 1 − exp(−kρt) denote the +fraction of the unreported infection cases among all infection cases. Using (14), these could +be computed as +∆C∗ +t += +∆Ct +1−δt +∆Rc∗ +t += +∆Rct +1−δt for t = 1, 2, . . . , T. +(15) +16 + +Further denote the total number of weekly deaths as ∆ ¯Dw +t which is computed as, +∆ ¯Dw +t += +∆Dw +t + ∆EDt for t = 7k and k = 1, 2, . . . , +(16) +where Dw +t stands for the reported deaths at the weekly frequency and ∆EDt for the Excess +Death numbers estimated in period t. Finally, the mixed frequency TVP-SIRD (MF-TVP- +SIRD) model in terms of the total numbers can be written as +∆C∗ +t |Ωt−1 +∼ +Poisson(βt +S∗ +t−1 +N I∗ +t−1) +∆Rc∗ +t |Ωt−1 +∼ +Poisson(γtI∗ +t−1) for t = 1, 2, . . . , T +∆ ¯Dw +t |Ωt−1 +∼ +Poisson( +t� +s=t−6 +νsI∗ +s−1) for t = 7k and k = 1, 2, . . . +∆C∗ +t = −∆S∗ +t += +∆I∗ +t + ∆Rc∗ +t + ∆ ¯Dd +t +(17) +Here ∆ ¯Dd +t is computed as νtI∗ +t−1 +12. The evolution of the model parameters decomposed as +in (7) follow the recursions in (8) and (10) using the (scaled) score functions displayed in +(9) for the parameters βt and γt. For the score function of the νt, we have the following +expression due to the change in the frequency of the Poisson process +s˜ν,t += +∆ ¯Dt−¯λ3,t−1 +¯λ3,t−1 +1 +(1−νt) +(18) +with ¯λ3,t−1 = νt +t� +s=t−6 +I∗ +s−1 for t = 7k and k = 1, 2, . . . , that is for the periods where the +weekly data is released, i.e., observed. The score function takes the value 0 when the weekly +excess death variable is not observed, which completes the specification of the MF-TVP- +SIRD model. +3.3 +Estimation strategy and the simulation algorithm +3.3.1 +Bayesian inference +We use simulation-based Bayesian estimation techniques for inference on model param- +eters. +Bayesian inference involves updating the prior distributions of model parameters +with the data likelihood to form the parameters’ posterior distributions. Considering the +SIRD model, Bayesian inference is especially appealing since the inference is conditional +on the data at hand and does not require asymptotic analysis. Therefore, we can compute +12Notice that the sum of the independent Poisson processes is a Poisson process, which enables us to +switch between daily and weekly frequency when necessary. +17 + +the credible intervals when the underlying process of the number of infected cases, It, is +nonstationary. This property is especially reassuring in our case since, obviously, nonsta- +tionarity, or in other words, effective reproduction rate being greater than one, eRt > 1, is +an inevitable feature of the pandemic, at least locally. +Here we demonstrate the likelihood function and prior specifications for the TVP- +SIRD model because the SIRD model with fixed parameters boils down to a special case +of this model. +The likelihood function is based on the specification in (6) where we +specify conditionally independent Poisson distributions for each of the components of the +SIRD model. +Let yt = (∆Ct, ∆Rct, ∆Dt)′ be the vector of observations. +Notice that +It = It−1 + ∆Ct − ∆Rct − ∆Dt, and thus the number of active infections can be computed +using the information set Ωt = (y′ +t, It−1)′. Accordingly, we have the following likelihood +function +f(yt|Ωt−1) = +λ∆Ct +1,t +exp(−λ1,t) +Γ(∆Ct+1) +λ∆Rct +2,t +exp(−λ2,t) +Γ(∆Rct+1) +λ∆Dt +3,t +exp(−λ3,t) +Γ(∆Dt+1) +, +(19) +where λ1,t = βt +St−1It−1 +N +, λ2,t = γtIt−1 and λ3,t = νtIt−1 and Γ(.) is the Gamma function. +Note that the time-varying parameters decomposed as in (7) follow the recursions in (8) +and (10) using the (scaled) score functions displayed in (9). We want to obtain posterior +results driven by the data rather than the prior distributions. Therefore, we impose rather +diffuse prior specifications for the model parameters. This strategy implies that for the +representative model parameters φ, we specify the following improper prior specifications +f(φ) ∝ 1 +(20) +for φ ∈ Φ = (Θ′ +l,0, α′, ψ +′ +˜β, ψ∗′ +˜β , ψ +′ +˜γ, ψ∗′ +˜γ , ψ +′ +˜ν, ψ∗′ +˜ν )′ where Θl,0 = (βl,0, γl,0, νl,0)′, α = (αβ, αγ, αν)′, +ψθ = (ψ1,θ, ψ2,θ, ψ3,θ)′ and ψ∗ +θ = (ψ∗ +1,θ, ψ∗ +2,θ, ψ∗ +3,θ)′ for θt = ˜β, ˜γt and ˜νt. For the MF-TVP- +SIRD model, we specify the prior distribution for the additional k parameter noninformative +in the positive domain. +3.3.2 +Simulation scheme +For the SIRD model with fixed parameters, the likelihood with Poisson distributions as in +(19) with fixed parameters and noninformative or conjugate priors in the form of Gamma +distribution lead to a Gamma distribution for the posterior distributions of the model +parameters. Therefore, these can be sampled using the plain Gibbs sampler. For the TVP- +18 + +SIRD model, the fact that we have time-varying parameters with deterministic recursions +leads to nonstandard posterior distributions. Therefore, we cannot use standard distribu- +tions we can easily simulate for the inference, as is the case for the Gibbs sampler. Instead, +we resort to the (adaptive) random walk Metropolis-Hastings (MH) algorithm within the +Gibbs sampler; see Robert and Casella (2013) for details. The algorithm is as follows +1. Sample α from f(α|ST , IT , Φ−α) using MH step +2. Sample Θl,0 from f(Θl,0|RcT , IT , Φ−Θl,0) using MH step +3. Sample ψθ from f(ψθ|Y T , Φ−ψθ) using MH step +4. Sample ψ∗ +θ from f(ψ∗ +θ|Y T , Φ−ψ∗ +θ) using MH step +Here Y T = (Y1, . . . , Yt, . . . , YT )′ for Yt = St, It, Rct, Dt indicating the full sample of the +count data regarding the states of the pandemic, respectively. Φ−X indicates the vector +of parameters Φ excluding the parameters X. For the MH steps, the candidate generating +density is constructed using the random walk specification as +φm += +φm−1 + Σ1/2 +φ εm +(21) +where φm is the parameter draw depending on the step at the iteration m, and εm follows a +standard (multivariate) t−distribution with degrees of freedom 15. For the starting values +of the parameters to initialize the sampler, Φ0, and for the covariance matrix ΣΦ0, we use +the maximum likelihood estimate of the model parameters. Therefore, we use the mode and +the inverse Hessian of the likelihood function at the mode in (21). To improve the sampler’s +performance, we follow the adaptive scheme described in Haario et al. (2001). This scheme +involves replacing ΣΦ0 with χSM +ϵI, once we obtain a sufficient number of draws to replace +the inverse Hessian of the likelihood function with the simulated curvature of the posterior +distribution. Here Sm corresponds to the empirical covariance matrix computed using the +draws up to step M, I indicates the identity matrix, and ϵ is a small number. ϵI ensures a +nonsingular empirical covariance matrix. In addition, we use χ for optimizing the sampler’s +performance for the candidate generating density to be efficient enough to cover the tails +of the posterior distribution. Let φcand ∼ q(φcand|φm−1) be a draw from the candidate +generating density in iteration m of the sampler, the candidate is accepted with probability +π += +min +� +1, q(φm−1|φcand)p(φcand|Y T ) +q(φcand|φm−1)p(φm−1|Y T ) +� +. +(22) +19 + +Here p(.|Y T ) refers to the posterior distribution. Note that, due to the symmetry of the +random walk specification, q(φm−1|φcand) and q(φcand|φm−1) are equivalent. Hence they +are of no use in (22). +4 +Empirical results +4.1 +Dataset +We use the data for the US starting from the early days of the pandemic until the end of +March 2022, which captures all major waves of infections related to the Covid-19 pandemic, +including the recent Omicron wave. +The data is originally published by the Center for +Disease Control and Prevention (CDS) and can be tracked on https://covid.cdc.gov/ +covid-data-tracker/#datatracker-home. The data on the number of recovered cases +ceased to be reported following December 2020. We, therefore, treat this data as missing +for the periods when the data is not reported. In these cases, the score function is set to +zero, similar to the estimation of the MF-TVP-SIRD model. For the out-of-sample analysis +that involves a real-time recursive prediction exercise, we use the US data vintages that +are available as of the day of the prediction, which is available on the Covid-19 Data- +Hub. The data-hub includes the daily vintages of Covid-19 pandemic related datasets; see +https://covid19datahub.io/articles/data.html and Guidotti and Ardia (2020) and +Guidotti (2022) for implementation details and the latest version of the dataset. +4.1.1 +A brief account of the pandemic’s course in the US +We display the evolution of the daily confirmed cases and deaths throughout the sample in +Figure 1. +[Insert Figure 1 about here] +The US exhibits extensive heterogeneity throughout the sample regarding their experience +related to the pandemic, as shown in Figure 1. +At the onset of the pandemic, the US +opted for imposing mixed strategies involving full and partial lockdowns and voluntary +quarantine in different regions. As of March 2020, the US reported the highest number of +daily confirmed cases worldwide and therefore was the center of the pandemic when the +country was struggling with the first wave. In late 2020, the Alpha variant was the virus’s +dominant strain, which can be detected as the second wave in Figure 1. Nevertheless, with +20 + +the start of 2021, the US began an intensive inoculation campaign to fight the pandemic. +In 2021 two major pandemic waves were experienced with the emergence of the Delta +and Omicron variants, among other strains. +While the Omicron variant notably led to +record values for the number of daily confirmed cases, the daily number of deaths was still +comparable to that of the Alpha variant owing to the success of vaccination efforts. Hence, +this relatively rich and heterogeneous dataset involving all sorts of pandemic experiences +enables us to examine the econometric model’s success in tracking the parameter changes +in response to policy implementations and changes in the virus’s characteristics. +4.2 +Full sample results +This section discusses the full sample estimates of the main parameters for the models with +fixed and time-varying parameters, as described in (2) and (6), respectively. We further +evaluate the model’s parameter estimates with time-varying parameters when asymptomatic +cases are explicitly considered, as shown in (17). We start our analysis with the full sample +estimates of the model parameters for the SIRD model with fixed parameters described in +(2). These are displayed in Panel A of Table 1. +[Insert Table 1 about here] +The model with fixed parameters reflects the stance of the pandemic over the last two years, +on average, as the parameters are kept fixed. The basic reproduction rate, R0, as the main +summary statistics on the course of the pandemic when the whole population is considered +susceptible, is displayed in the last column of Table 1. The median estimate of the R0 for +the US shows that over the sample period of more than two years since the start of the +pandemic in March 2020, the estimate is 1.64 with little uncertainty. The fact that the R0 +well exceeds 1 reflects that the pandemic is not contained yet on average with the repeated +waves. Notice that this value might be inaccurate because the number of susceptible people +has reduced to some extent with the progress of the pandemic. We, therefore, display the +evolution of the effective reproduction rate, eRt in Figure 2. +[Insert Figure 2 about here] +Figure 2 indicates that the effective reproduction rate declines over time with the reduced +number of susceptible people. +As the main reason for the reduction in the number of +susceptible people is the infected cases, the reduction in eRt closely follows the waves of the +21 + +pandemic. The final value as of the end of March 2022 is still 1.3, indicating that a large +part of the population is not infected yet. +The effect of pharmaceutical or nonpharmaceutical interventions is reflected in the re- +production rate through the ’implied’ model parameters as discussed in Section 2.3. These +effects are all reflected as implied time variation in these parameters captured by the TVP- +SIRD model, which we discuss in the remainder of this section13. Before discussing the +evolution of the model parameters, we first start with displaying the fitted values of the +daily number of confirmed and death cases in Figure 3 to consider the overall performance +of the TVP-SIRD model in fitting the pandemic dataset. +[Insert Figure 3 about here] +The left panel of the Figure 3 shows the satisfactory fitting performance of the TVP-SIRD +model for the daily confirmed cases. Both level and seasonal patterns could be matched +using the model framework that can capture daily seasonal behavior in addition to level. We +display the fitted values of the daily number of death cases on the right panel of Figure 3. +This panel confirms the model’s ability to capture daily death cases’ level and seasonal +patterns. Next, we display the evolution of the (level of the) model parameters and the +estimated effective reproduction rate, eRt, using the TVP-SIRD model in Figure 4. +[Insert Figure 4 about here] +In the first two graphs, we display the variation in the infection and death rates, and in +the bottom set, we display the effective reproduction rate, eRt.14 For clarity of demonstra- +tion, we display the evolution of the parameters and the effective reproduction rate in two +subfigures representing two subperiods. On the left, we only display the periods until June +2020; on the right, we display the remaining periods. In these two subperiods, the scale +of the variation of the parameters differs considerably, and providing two graphs for two +13We also perform a statistical evaluation for the presence of time variation. In all the tests, the null +hypothesis boils down to the hypothesis that the coefficients of the score functions are zero. As indicated +by Calvori et al. (2017), the test against the time-varying parameter alternative checks whether there is +any autocorrelation in the scores of the fixed parameter model. If that is the case, such autocorrelations +can be exploited to improve the model’s fit by using the likelihood scores as drivers for the time-varying +parameter as in the TVP-SIRD model. Our test results indicate decisively favor time variation in the model +parameters. We thank an anonymous referee for pointing out this. +14The data on recovery ceased to be published after December 2020. We treat these periods of 2021 and +2022, where the recovery data is unavailable, as missing data. The estimate of the recovery rate remains +constant for these periods when the data is missing since the score functions for these periods are set to 0; +see Creal et al. (2014) and Lucas et al. (2016) for a similar approach. Accordingly, we do not display this +parameter’s evolution, estimated as 0.0091, in large part of our sample period. +22 + +different subperiods enhances the display’s visual quality. Finally, in the bottom graph of +the effective reproduction rate, we split the sample into seven enumerated subperiods with +distinct characteristics to motivate the variation in the eRt. +The first period labeled as (1), starting from early 2020 until April 2020, is the emergence +period of the pandemic. +During these periods, the World Health Organisation (WHO) +officially declared the pandemic on March 11, and the national public health agency in the +US imposed various measures to contain the pandemic, including the ban of large gatherings +and travel restrictions. Until the end of April, ’stay-at-home’ quarantines were effective in +several locations, and many testing facilities for effectively isolating the infected cases were +established. Our estimates suggest that the effective reproduction rate reduced from values +as high as 35 to around 6 in mid-April. This reduction is in line with early studies reporting a +basic reproduction rate (which is very close to the effective rate at the onset of the pandemic) +of around 5.7 using the early dataset from Wuhan, China, the point of emergence of the +pandemic, see Sanche et al. (2020) and around 5.3 for the US, see Peirlinck et al. (2020). +In the second period (2), comprising the period from mid-April until June, the eRt steadily +fluctuates around the value of 2.5 with the accumulation of the bulk datasets, similar to +the studies reported elsewhere, see Petersen et al. (2020) for example for a comparison +of SARS-COV-2 parameters to that of the SARS-COV and influenza viruses. The third +period, (3), captures the summer period of 2020. This period experienced the economy- +wide reopening and relaxation of various measures. Many large-scale events resulted in large +gatherings15 that lead to an increase in the implied infection rate as can be seen in the first +subfigure. The increase in the infection rate had overcome the increase in the death rate, +which led the effective reproduction rate to increase to values around five again. The fourth +period, (4), captures the winter period that includes the holiday season of Thanksgiving +followed by the Christmas period, with an estimated number of more than 2 million people +flying on airlines during the Thanksgiving period.16 In addition to the changing mobility +of the susceptible people, there was also a new variant of the virus, denoted as Alpha, +first detected in December 2020. Davies et al. (2021) report that this variant is 43%-90% +more transmissible than the predecessor lineage. Therefore, following the demonstration +15See for example New York Times article on 80th +Motorcycle rally in South Dakota, +where +there were more than 400,000 audiences in the gathering, https://www.nytimes.com/2020/11/06/us/ +sturgis-coronavirus-cases.html. +16See the article https://www.masslive.com/coronavirus/2020/11/thanksgiving-travel-many-americans/ +-flying-for-holiday-despite-cdcs-pleas-not-to-due-to-covid-19-transmission-risk.html for ex- +ample. +23 + +in subsection 2.3, the time-varying mixture of these two variants might be one underlying +source of the increasing infection rate. +The year 2021 started with a massive inoculation campaign with the availability of the +Covid-19 vaccines with an efficacy rate as high as 90%, see Polack et al. (2020). This period +(5) experienced a significant and rapid drop in the effective reproduction rate, which fell +below the critical value of 1 for the first time since the start of the pandemic. Therefore, +in the first six months of 2021, the US successfully contained the pandemic, thanks to the +vaccination campaign, which led to 67% of the overall adult population receiving at least +one dose. In the second half of 2021, the proportion of vaccinated people in the population +has remained relatively steady, above 60%. Furthermore, containment measures have also +remained stable to a large extent. Therefore, the changes in the implied parameters in +periods (6) and (7) mainly stem from the emergence of new variants with new structural +parameter values. Indeed, in period (6), which spans the summer and early fall of 2021, the +Delta variant was the dominant strain, according to the CDC estimates. The Delta variant +seems to be around 60% more transmissible than the already highly infectious Alpha variant, +see Callaway et al. (2021), which can also be traced in the course of the infection rate and, +thereby, the effective reproduction rate. Finally, in the last period, (7), which captures the +remaining period until the end of March 2022, the Omicron variant has been the dominant +variant which is much more contagious than the previous strains but less severe compared +to those, see Karim and Karim (2021). This rapid infection rate surge due to the Omicron +variant can be captured nicely using our model framework. Moreover, the increase in the +death rate remained relatively moderate and lower than that of the Delta variant, confirming +the findings on the Omicron variant. As a result, the effective reproduction rate soared to +as high as five. As of March 2022, the rate again fell below the critical value of 1. The +full-sample findings demonstrate that the model with ’implied’ time-varying parameters can +capture various factors, such as changes in the number of susceptible people either due to +pharmaceutical or nonpharmaceutical interventions or the emergence of new variants of the +virus accurately and promptly, confirming stylized facts. +4.3 +Accounting for unreported cases +The results discussed in previous sections are computed using official statistics, including +only the reported cases. In this section, we present our findings when we account for this +24 + +selection bias using the information on the number of excess deaths and the number of tests +together with these tests’ positivity rates. We display the evolution of the estimated rate +of total infections to the number of reported infected cases, 1/(1 − δt), in Figure 5. +[Insert Figure 5 about here] +Figure 5 indicates that the actual number of infected cases, including the asymptomatic +cases, might be three times more than the reported cases, especially during the peaks +of the first two pandemic waves. However, our estimation results suggest a considerable +uncertainty around this ratio with a 95% credible interval between almost two and five on +average. This finding is consistent with the CDC estimates reported as around 4 for the +period until the end of 2020; see Reese et al. (2020) and the related web source17 for details. +Similar findings are also reported by Angulo et al. (2021) based on the data from four +regional and one nationwide seroprevalence surveys.18 These serosurveys serve as a crucial +data source for measuring the number of infected cases because the survey participants are +selected randomly, thereby overcoming selection bias. Our results align with the reported +results in Angulo et al. (2021), where they estimate this rate as four using the nationwide +serosurvey conducted during July-August 2020 in 47 states. While this is the case for most +waves, including Delta and Omicron waves, throughout 2021, an important finding is in the +late summer of 2021. Our results show that the total number of infected cases might be as +high as seven times (with a wide 95% credibility interval between [4-10]) of those reported +cases. This finding is because, during this period, we observe a rapid surge in the number of +excess deaths and relatively greater test positivity ratios. This implies that the low number +of confirmed cases in the summer of 2021 in the US might be mainly due to these unreported +cases. According to the CDS estimates based on recurring serosurveys, the seroprevalence +estimates, that is, the percentage of people with antibodies against the virus, soars from +20% in July to around 30% in September, which is in line with our findings.19 We display +the evolution of the death rates (based on the total number of deaths, including the excess +deaths) and the effective reproduction rate in Figure 6. +[Insert Figure 6 about here] +17https://www.cdc.gov/coronavirus/2019-ncov/cases-updates/burden.html +18A seroprevalence survey uses antibody tests to estimate the percentage of people in a population who +have antibodies against the virus. The number of people in a specific population who have been previously +infected with the virus is estimated using these test outputs. +19Notice that the CDS report involves seroprevalence of infection-induced antibodies (nucleocapsid anti- +body) which is distinct from the vaccination-induced antibody (spike antibody). Therefore, these estimates +genuinely represent the total infections; see Jones et al. (2022) for example for details. +25 + +Considering the death rate, the discrete evolution of this parameter is due to the weekly +frequency of the excess death dataset. +This parameter is updated only at the weekly +frequency when the data is observed and remains fixed for the other periods. While the +death rate exhibits a similar pattern, we can track the surge in the rate in late summer of +2021, in accord with the previous discussion on the total number of infected cases, thanks +to the excess death data. This finding is also confirmed in the evolution of the effective +reproduction rate level, eRl,t, where the eRl,t is computed as high as nine during these +periods. +5 +Real-time performance of the models +The results in the previous section display our findings based on the estimates using the +full sample dataset. +These results indicate that our flexible modeling structure can ac- +commodate various forms of parameter changes reflecting the pandemic’s course. However, +exploring the model’s real-time performance would uncover whether this additional flexibil- +ity brought by the time-varying parameters could provide timely and accurate information +on the pandemic’s real-time stance. Therefore, in this section, we discuss the model param- +eters’ estimation results in real-time using the model with fixed parameters and the model +with time-varying parameters. We aim to provide a thorough real-time analysis in the sense +that we make use of the complete vintage data publicly available at the time of the predic- +tion. Given that the pandemic data were revised substantially at times, using vintage data +provides the actual predictive performance of the complete models. The vintage dataset +is obtained from the Covid-19 Data Hub20, see Guidotti and Ardia (2020) and Guidotti +(2022) for details on the dataset. +5.1 +Predictive performance at the daily frequency +We use a rolling window for performing the SIRD model’s estimations with fixed parameters +rather than expanding window21 following the evidence of time variation in parameters in +the previous section. Specifically, using the dataset from t−M, t−M +1, . . . , t, we estimate +the SIRD model, and the resulting parameter estimates are those for the period t. We +20https://covid19datahub.io/ +21We include the forecasting performance using an expanding window in the earlier versions of this paper. +The results are decisively inferior compared to all moving window approaches. Therefore, we do not display +those results here, but results are available upon request. +26 + +repeat this process by recursively adding one observation (and dropping one observation at +the beginning of the sample for the rolling window). We consider three cases by setting +M = 30, 45, and 60, i.e., starting from one month of data up to two months of data. For +capturing seasonality in these rolling window regressions, we consider daily dummy variables +representing the days of the week. These models are denoted as RW-30, RW-45, and RW-60, +respectively. For the TVP-SIRD model, we use the data up to period t using an expanding +window rather than a rolling window, as the parameters, in this case, are time-varying. We +also include a restricted version of the TVP-SIRD model, where we allow for time variation +only in the infection rate, β, denoted as TVP-SIRD-β, following similar approaches, see +Fern´andez-Villaverde and Jones (2022) for example. Finally, we also include a time-varying +parameter model that falls into the parameter-driven model category. +In that case, we +consider the computationally least costly alternative by imposing Normal distributions for +observation and parameter evolution. For capturing the seasonality, we use the same model +framework that we employ in the TVP-SIRD model with the critical distinction of including +the error term in the state equations capturing seasonal patterns. The resulting specification +leads to a standard inference using the Kalman filter and simulation smoother, which is still +tractable in cases when the data is scarce; see Durbin and Koopman (2012) for details. This +model is denoted as KF. For a given model, the predictive distribution of the observation +at t0 + 1 conditional on the information available at t0, Ωt0, is given by +p(yt0+1|Ωt0) = +� +f(yt0+1|Φ)f(Φ|yt0)dΦ, +(23) +where f(Φ|yt0) is the posterior distribution of the model parameters, estimated using the +data until t0, gathered in the parameter set, Φ, given the observations until t0. p(yt0+1|Φ) +is the density of the observation yt0+1, which can be written as +f(yt0+1|Φ) = +� +θt0+1 +f(yt0+1|θt0+1, Φ)f(θt0+1|Φ, Ωt0). +(24) +We can use the posterior simulator to obtain the distribution of the model parameters and +estimate the predictive distribution using the draws from the simulator as (y(m) +t0+1|Ωt0, Φ(m)), +where m represents the mth draw from the posterior simulator. +We display the results +involving Root Mean Squared Forecast Errors (RMSFEs) of the competing models relative +to the TVP-SIRD model considering the prediction of the daily confirmed cases in Table 2. +27 + +[Insert Table 2 about here] +We perform equal predictive accuracy tests for the out-of-sample comparisons to evaluate +the relative model performance. Specifically, for all the comparisons, we perform Diebold- +Mariano (DM) type of pairwise comparison tests of equal predictive accuracy between the +competing models with HAC standard errors and small sample correction suggested by +Harvey et al. (1997) using squared error contributions as loss functions. The cells with +white backgrounds contain statistically insignificant values at the conventional significance +level of 5%. Table 2 indicates a clear-cut result. The TVP-SIRD model outperforms all +competing models up to 15 days horizon. This indicates the superior performance of our +flexible modeling structure in the short and medium-term forecasting of the confirmed +cases up to two weeks. This outperformance deteriorates for the horizons exceeding two +weeks. In this case, although some models provide relative RMSFEs lower than unity, this +relative performance is statistically insignificant at conventional significance levels. This +is due to increasing uncertainty surrounding these point predictions leading to statistical +insignificance. +Focusing on pairwise evaluations, comparing the TVP-SIRD model with the TVP-SIRD- +β model reveals the importance of modeling time variation not only in the infection rate but +also in the remaining parameters, at least for short and medium-term predictions. For the +horizons longer than two weeks, the two models perform alike with relative RMSFEs very +close to unity. Comparison of the TVP-SIRD model with the parameter-driven model with +Normal distributions for the observables and the parameters, denoted as KF, indicates the +merits of deterministic updating with a proper specification of the data structure over more +flexibility of the parameter-driven models. In this case, the TVP-SIRD model performs +significantly better than the KF model for up to 10 days, and the two perform statistically +indifferent for longer horizons. Finally, the trade-off between the flexible and less flexible +models is apparent when we compare short and long-horizon performances of the regressions +with a 30-day moving window versus a 60-day moving window. While the regressions with +a 30-day moving window outperform the counterpart with a 60-day moving window for the +predictions up to two weeks due to flexibility, the latter model outperforms the former due +to the reduction in the variance despite the increasing bias. +We display the results involving RMSFEs of the competing models relative to the TVP- +SIRD model when we consider the prediction of the daily death cases in Table 3. +28 + +[Insert Table 3 about here] +Table 3 indicates mixing results. First, comparing the TVP-SIRD model with the TVP- +SIRD-β model indicates the importance of the time variation in the death rates. In this +case, the TVP-SIRD model with the time-varying death rate outperforms the TVP-SIRD-β +with the fixed death rate at all horizons, and this outperformance is statistically signifi- +cant. Comparison of the TVP-SIRD model with the KF model indicates that the former +outperforms the latter significantly at all horizons except the 1-day ahead forecast, indi- +cating the importance of proper modeling of pandemic count data. Finally, we compare +the TVP-SIRD model performance with the regression models. +The TVP-SIRD model +performs better than the regression model with a 60-day moving window at all horizons +except the 30 days horizon. When the window size is shortened to 45 and 30-day leading +to more flexibly changing parameters, the superior performance of the TVP-SIRD model +remains significant at longer horizons exceeding ten days. However, the predictions become +statistically indifferent for short horizons thanks to the flexibility of these regression models +with shorter windows. Overall, it seems that the flexibility of the TVP-SIRD model pays +off even more at longer horizons for predicting the daily death cases compared to confirmed +cases.22 +5.2 +Predictive performance at weekly frequency +In the previous section, we display a horse race for the predictive performances of a set of +competing models at a daily frequency. However, since the outburst of the pandemic, many +models, including various forms of epidemiological models, curve fitting frameworks, or ma- +chine learning setups, have predicted the pandemic’s key variables, including confirmed and +death cases. Luckily, the CDC-funded Influenza Forecasting Centers of Excellence worked +closely with global, federal, state, and local public health officials to integrate infectious +disease forecasting in a so-called forecast hub providing predictions of the outstanding fore- +casting sources.23. In this section, we compare the TVP-SIRD model with these prominent +competitors. Since these forecasts are provided at weekly frequency, we estimate the TVP- +SIRD model at the weekly frequency in real time using vintage data as in the previous +22The predictive performance of the daily death cases is closely related to the number of ICU patients +due to Covid-19, which is a critical factor for the decision-makers, with a correlation exceeding 0.8. The +predictive results of forecasting the number of ICU patients are very similar to those of death cases, and +these are presented in Section H of the supplementary material. +23See https://covid19forecasthub.org/doc/.Wethankananonymousrefereeforpointingoutthissource. +for further details on this initiative. +29 + +case.24 We discard the forecasts that have less than 30 forecast readings, leaving us with 19 +forecast sources for the prediction of the weekly confirmed cases and 28 for the prediction +of the weekly death cases. We display the forecast sources that range from Microsoft to +MIT-based models in the supplementary material in Table G.3. +We display the number of models that our TVP-SIRD model outperforms in Table 4 for +horizons including h = 1, 2, 3, 4, i.e., from the 1-week horizon up to the 1-month horizon. +[Insert Table 4 about here] +Table 4 reveals that in the short horizons, the TVP-SIRD model can beat the majority +of these outstanding forecast sources with 11 outperformance out of 19 sources for the +prediction of the confirmed cases and 19 out of 28 sources for the prediction of the death +cases considering 1-week horizon. However, this superior predictive ability monotonically +erodes with the increasing forecast horizon. In line with the prediction results using daily +frequency, the TVP-SIRD model is more successful in predicting the death cases at long +horizons compared to the confirmed cases with better performance than 30% (20%) of the +forecast sources at 3-week (4-week) horizon for death cases versus 15% (10%) for confirmed +cases prediction. Therefore, we conclude that the TVP-SIRD model successfully predicts the +critical Covid-19 pandemic-related variables at the short and medium horizon. It performs +comparably to the leading pandemic forecasting tools at long horizons. +We provide a more detailed picture of the dynamic performance of the TVP-SIRD model +over time compared to the forecasting tools in Figure 7 for the 1-week horizon.25 +[Insert Figure 7 about here] +Rather than providing pairwise comparisons with every single model, Figure 7 displays +the evolution of relative RMSFEs of the TVP-SIRD model with an ensemble model (EM) +over time where relative RMSFEs (rRMSFE) are computed recursively in real time using +vintage datasets. The EM is computed using a forecast combination scheme generated using +the space of individual models. In line with the advantages of forecast combinations over +individual models in many settings, the EM provides better predictions than the individual +forecast sources. Thus, it is a gold standard in predicting confirmed and death cases; see +https://covid19forecasthub.org/doc/reports/ for details. In Figure 7, we also include +24We also evaluate the weekly forecasts by aggregating our daily forecasts. However, this yields worse +results compared to using weekly data for estimation. Therefore, we provide the forecasting results regarding +weekly data setup. +25We display the results for longer horizons in Section G of the supplementary material. +30 + +the actual number of cases to compare the relative predictive performance taking the timing +of the various phases of the pandemic into account. In the left and right panels of Figure 7, +we display the RMSFE of the TVP-SIRD model relative to the EM for the prediction of the +confirmed cases and death cases, respectively. A value lower than unity indicates the better +performance of the TVP-SIRD model relative to EM. Considering the confirmed cases, on +average, the TVP-SIRD model performs closely to the EM as the rRMSFE is very close +to unity in most periods. A striking finding is that the TVP-SIRD model performs better +than the EM, specifically at the onset of the pandemic waves. This result indicates that the +TVP-SIRD model provides timely predictions at the onset of the pandemic waves reflecting +the flexible model structure that can immediately accommodate the changing conditions. +However, this picture reverses when the pandemic wave is at its peak. Once the data on +the new pandemic wave is accumulated, the EM provides better predictions, especially on +the timing of the turning point down the hill. Focusing on predicting weekly death cases +at 1-week horizon, we observe that EM performs much better than the individual forecast +sources. Unlike the previous comparison of the TVP-SIRD model to individual forecast +sources, the EM model performs better than the TVP-SIRD model over time as relative +RMSFEs exceed unity persistently. Still, the pattern discussed in the prediction of the +confirmed cases is also apparent here. Again, the performance of the TVP-SIRD model +improves at the onset of the pandemic waves and deteriorates once the pandemic’s peak +is passed. Overall, our results indicate that the TVP-SIRD model performs favorably well +at daily and weekly frequency against very compelling competitors in forecasting of key +Covid-19 pandemic aggregates. +6 +Potential extensions +In the previous sections, we display the potential of the TVP-SIRD model both in in- +sample fit and out-of-sample forecasting. This section provides a potential extension to the +TVP-SIRD model. Since the pandemic is a global phenomenon, multiple countries have +had common experiences, with some countries having relatively larger part of idiosyncratic +variations. Departing from this observation, we extend the model to a multi-country setting +31 + +using a factor model structure. Consider the following model for country i = 1, . . . , K, +∆Ci,t|Ωt−1 +∼ +Poisson(βi,t +Si,t−1 +Ni Ii,t−1) +∆Rci,t|Ωt−1 +∼ +Poisson(γi,tIi,t−1) +∆Di,t|Ωt−1 +∼ +Poisson(νi,tIi,t−1) +∆Ii,t += +∆Ci,t − ∆Rci,t − ∆Di,t. +(25) +We assume that country-specific parameters of the TVP-SIRD model admit a factor struc- +ture for their level, while the seasonal patterns are idiosyncratic.26 Specifically, consider +the decomposition as in (7), where the level component evolves according to the following +factor structure +θi,l,t += +τi,lθl,t + ˆθi,l,t +θl,t += +θl,t−1 + αθlsθl,t−1 +ˆθi,l,t += +ˆθi,l,t−1 + αˆθi,lsˆθi,l,t−1 +(26) +for parameter θt = ˜βt, ˜γt and ˜νt, and θl,t is the common level component, respectively. +Here, the key difference between the factor model and a country-specific model is that +common level factor θl,t is loaded by all country-specific information with the coefficient τi,l +for country i, 27 and thus, the corresponding score function becomes +s˜β,t += +∇1, ˜β,t+···+∇i, ˜β,t+···+∇K, ˜β,t +S ˜β,t +s˜γ,t += +∇1,˜γ,t+···+∇i,˜γ,t+···+∇K,˜γ,t +S˜γ,t +s˜ν,t += +∇1,˜ν,t+···+∇i,˜ν,t+···+∇K,˜ν,t +S˜ν,t +(27) +where +∇i,˜β,t += +(∆Ci,t − λi,β,t) τ1,i +∇i,˜γ,t += +(∆Rci,t − λi,γ,t) (1 − γi,t)τ2,i +∇i,˜ν,t += +(∆Di,t − λi,ν,t) (1 − νi,t)τ3,i +(28) +and +S˜β,t += +λ1,β,tτ 2 +1,1 + · · · + λK,β,tτ 2 +1,K +S˜γ,t += +λ1,γ,t(1 − γ1,t)2τ 2 +2,1 + · · · + λK,γ,t(1 − γK,t)2τ 2 +2,K +S˜ν,t += +λ1,ν,t(1 − ν1,t)2τ 2 +3,1 + · · · + λK,ν,t(1 − νK,t)2τ 2 +3,K. +(29) +26Imposing a factor structure to the seasonal component is a straightforward extension of the model. +However, our experience with this model shows that identifying a common seasonal factor poses more +challenges than a level factor. Since the level factor is the key component of the parameters, we consider a +factor structure only in the level component. +27In this extension, we opt for a factor structure in the parameters similar to seasonality modeling. +Alternatively, we could also proceed with a factor representation of the data, using principal components, +for example, and a SIRD model corresponding to each component, similar to factor GARCH models; see +Zhang and Chan (2009) for such a strategy. +32 + +The score functions of the idiosyncratic parts are the same as in the TVP-SIRD model. +We provide details on these derivations in Section D.2 of the supplementary material. We +denote this model as the factor TVP-SIRD model. In the following application, we only +consider a factor structure in the infection rate, βt, keeping the remaining parameters as +wholly idiosyncratic as before. However, the extension of the factor structure to the re- +maining parameters is similar. Still, the parameters in their current form are not identified, +as none of the components are observed. To identify the factors and idiosyncratic compo- +nents separately, we set τ1 as one for the first country and fix the initial condition for the +common factor, which enables the identification of the location of the factor and idiosyn- +cratic components separately. We consider four countries in the application: US, Germany, +Italy, and Brazil. We display the evolution of the number of daily active infected cases for +Germany, Italy, and Brazil in Figure 8, while we provide a comprehensive analysis of the +US in previous sections. +[Insert Figure 8 about here] +Figure 8 indicates that the pandemic’s evolution in Europe, represented by Germany and +Italy follows a similar trajectory. On the other hand, Brazil’s pandemic trajectory exhibits +a unique pattern, counter to Germany and Italy at times. Still, the countries’ patterns +converge with the last wave, i.e., the Omicron wave. +We display the estimates of the +fixed parameters related to the common factor in Table 5. We display the evolution of +the common factor infection rate and country-specific effective reproduction rates, eRts, in +Figure 9. +[Insert Table 5 and Figure 9 about here] +Table 5 indicates that the common factor is affected by the past score function, derived in +(27), considerably with a coefficient close to 0.6 leading to a time-varying pattern. However, +the 95% HPDI covers a wide range of values between 0.5 and 0.8. +Factor loadings for +Germany and Italy are very similar, as expected, with values around 0.8 and bear little +uncertainty. The lowest loading is for Brazil with a value of 0.3 with almost 0 and 0.5 for +the bounds of 95% HPDI. +The upper left panel of Figure 9 displays the evolution of the common factor of infection +rate. Following the estimates of factor loadings, this factor is mainly influenced by the +pandemic trajectory of the US, Germany, and Italy. We can observe that the common factor +33 + +can nicely capture all significant waves with the relatively higher values corresponding to +the initial wave at the onset of the pandemic and the recent two waves, i.e., Delta and +Omicron waves. The common impact of these two variants can also be observed by the +increased eRt of the US, Germany, and Italy, particularly for the Delta variant. Finally, +the relatively more idiosyncratic behavior of the pandemic in Brazil can be traced by the +corresponding eRt in the lower right panel of Figure 9. For Brazil, the eRt fluctuates around +one for a large part of the sample period leading to the unique pattern of active infected +cases as shown in the most right panel of Figure 8. These results show the efficacy of the +factor TVP-SIRD model in capturing both the common and idiosyncratic patterns of the +Covid-19 pandemic. +7 +Conclusion +This paper puts forward the time-varying parameters SIRD model for timely and accurate +measurement of the pandemic’s current stance and accurate predictions of its future tra- +jectory. Our modeling framework falls into the class of ’generalized autoregressive score +models’. These models involve parameters evolving deterministically according to an au- +toregressive process in the direction implied by the score function. Therefore the resulting +approach permits a flexible yet parsimonious and statistically coherent framework to oper- +ate efficiently in scarce data environments. We demonstrate the proposed model’s potential +using daily and weekly US data using full sample estimation and out-of-sample forecasting +using a recursive real-time prediction exercise. +Our results show that the proposed framework can nicely track the stance of the pan- +demic. Our findings suggest that there is considerable fluctuation in the rate of infection +and death rates. We further extend the model to include the infected individuals who do not +show symptoms and are therefore not diagnosed. We show that this sample selection might +have a sizable impact on the estimated reproduction rate. We extend the model framework +in various directions, including a mixed-frequency setup blending daily and weekly Covid-19 +pandemic-related critical data and a factor model setup by blending datasets from various +countries. Results indicate the potential of our flexible model structure. +34 + +References +Acemoglu D, Chernozhukov V, Werning I, Whinston MD. 2021. 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Efficient factor garch models and factor-dcc models. Quantitative +Finance 9: 71–91. +Zhang Y, You C, Cai Z, Sun J, Hu W, Zhou XH. 2020. Prediction of the covid-19 outbreak +based on a realistic stochastic model. medRxiv . +38 + +Tables and Figures +Table 1: Estimation results of the SIRD model with fixed parameters and the TVP-SIRD model +Panel A: Fixed parameters SIRD model +Panel B: TVP-SIRD model +βl +γl (×10−1) +νl (×10−2) +R0 +αβt +αγt +ανt +Median +0.0122 +0.0746 +0.0133 +1.6392 +0.4822 +0.6334 +0.3514 +2.5% per. +0.0120 +0.0744 +0.0131 +1.6155 +0.4553 +0.6253 +0.3375 +97.5% per. +0.0124 +0.0747 +0.0134 +1.6596 +0.5104 +0.6421 +0.3682 +Note: The table displays the estimation results of the model in (2). We display the posterior median and the +2.5% and 97.5% percentiles of the posterior distributions of the corresponding parameter shown in the first +row. +Table 2: Relative RMSFEs of the competing models relative to the TVP- +SIRD model - Daily confirmed cases +RW−30 +RW−45 +RW−60 +KF +TVP-SIRD-β +h = 1 +2.111 +2.414 +2.826 +1.443 +1.251 +h = 5 +1.747 +1.959 +2.162 +1.206 +1.183 +h = 10 +1.183 +1.550 +1.562 +1.197 +1.088 +h = 15 +1.325 +1.317 +1.142 +0.971 +1.027 +h = 20 +1.054 +1.024 +0.798 +0.865 +0.967 +h = 25 +1.013 +0.960 +0.857 +0.829 +1.002 +h = 30 +0.949 +0.862 +0.662 +0.774 +0.973 +Note: The table displays the Root Mean Squared Forecast Errors (RMSFE) of the +competing models in predicting the daily confirmed cases relative to the TVP-SIRD +model introduced in (6). RW-M stands for Rolling Window with M observations +as the sample size for M = 30, 45, 60. KF stands for the time-varying parameter +version of the SIRD model using a state space model framework with Normal er- +ror terms. TVP-SIRD-β denotes the restricted version of the TVP-SIRD model, +where we allow for time variation only in the infection rate, β. Statistical signifi- +cance of relative Root Mean Squared Forecast Errors (RMSFE) is tested using the +Diebold-Mariano (DM) test using the measures of squared forecast error contri- +butions together with the HAC covariance matrix and a finite sample correction, +Harvey et al. (1997). The bold values are statistically INsignificant at the con- +ventional significance level of 5%. +39 + +Table 3: Relative RMSFEs of the competing models relative to the TVP- +SIRD model - Daily death cases +RW−30 +RW−45 +RW−60 +KF +TVP-SIRD-β +h = 1 +0.974 +1.051 +1.310 +0.776 +3.055 +h = 5 +0.949 +0.995 +1.288 +1.285 +2.852 +h = 10 +1.183 +1.060 +1.413 +1.332 +2.799 +h = 15 +1.147 +1.105 +1.336 +1.175 +2.642 +h = 20 +1.130 +1.170 +1.326 +1.186 +2.566 +h = 25 +1.096 +1.110 +1.214 +1.301 +2.266 +h = 30 +1.074 +1.068 +1.097 +1.294 +2.099 +Note: The table displays the Root Mean Squared Forecast Errors (RMSFE) of the +competing models in predicting the daily death cases relative to the TVP-SIRD +model introduced in (6). RW-M stands for Rolling Window with M observations +as the sample size for M = 30, 45, 60. KF stands for the time-varying parameter +version of the SIRD model using a state space model framework with Normal er- +ror terms. TVP-SIRD-β denotes the restricted version of the TVP-SIRD model, +where we allow for time variation only in the infection rate, β. Statistical signifi- +cance of relative Root Mean Squared Forecast Errors (RMSFE) is tested using the +Diebold-Mariano (DM) test using the measures of squared forecast error contri- +butions together with the HAC covariance matrix and a finite sample correction, +Harvey et al. (1997). The bold values are statistically INsignificant at the con- +ventional significance level of 5%. +Table 4: Number of models in the Forecast Hub that the TVP-SIRD +model outperforms +1-week +2-week +3-week +4-week +Confirmed cases (19) +11 +5 +3 +2 +Death cases (28) +19 +14 +9 +5 +Note: The table displays the number of the models in the Forecast Hub with +more than 30 predictions that have greater RMSFE than the TVP-SIRD +model. The total number of the models considered in the comparison is +indicated in the left column in the parenthesis. +Table 5: Estimation results of the factor TVP-SIRD +model +αβl,t +τGer +τIt +τBr +Median +0.6347 +0.7840 +0.8228 +0.2982 +2.5% per. +0.4944 +0.7409 +0.7522 +0.0722 +97.5% per. +0.7750 +0.8271 +0.8934 +0.5242 +Note: The table displays the estimation results of the model in +(25). We display the posterior median and the 2.5% and 97.5% +percentiles of the posterior distributions of the corresponding +parameter shown in the first row. +40 + +Figure 1: The evolution of the daily number of confirmed and death cases in the US +Confirmed +Death +0 +200 +400 +600 +800 +1,000 +1,200 +1,400 +0 +200 +400 +600 +800 +1,000 +1,200 +1,400 +4/23/20 +7/22/20 +10/20/20 +1/18/21 +4/18/21 +7/17/21 +10/15/21 +1/13/22 +0 +1,000 +2,000 +3,000 +4,000 +5,000 +0 +1,000 +2,000 +3,000 +4,000 +5,000 +4/23/20 +7/22/20 +10/20/20 +1/18/21 +4/18/21 +7/17/21 +10/15/21 +1/13/22 +Note: The graphs show the evolution of the daily confirmed and death cases in the US over the sample from +March 2020 until the end of March 2022. +Figure 2: The evolution of the effective reproduction rate, eRt, estimated using the FP- +SIRD model +Jul 2020 +Jan 2021 +Jul 2021 +Jan 2022 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +Note: The graphs show the evolution of the effective reproduction rate, eRt, estimated using the SIRD +model with fixed parameters displayed in (2) in the US over the sample from March 2020 until the end of +March 2022. The 95% (HPDI) Highest Posterior Density Intervals are computed using the posterior output. +Figure 3: The fitted values of the daily number of confirmed and death cases using the +TVP-SIRD model +Confirmed +Death +0 +200 +400 +600 +800 +1,000 +1,200 +1,400 +0 +200 +400 +600 +800 +1,000 +1,200 +1,400 +4/23/20 +7/22/20 +10/20/20 +1/18/21 +4/18/21 +7/17/21 +10/15/21 +1/13/22 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +4/23/20 +7/22/20 +10/20/20 +1/18/21 +4/18/21 +7/17/21 +10/15/21 +1/13/22 +Note: The graphs show the evolution of the daily confirmed and death cases in the US and the fitted values +using the TVP-SIRD model over the sample from March 2020 until the end of March 2022. +41 + +Figure 4: The evolution of the level values for infection and death rates and effective +reproduction rate, βl,t, νl,t, and eRt, over the sample from March 2020 until March 2022 +βl,t +Mar 31 +Apr 14 +Apr 28 +May 12 +May 26 +2020 +0 +0.05 +0.1 +0.15 +0.2 +0.25 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +νl,t +Mar 31 +Apr 14 +Apr 28 +May 12 +May 26 +2020 +0 +1 +2 +3 +4 +5 +6 +7 +10-3 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10-4 +eRl,t +Mar 31 +Apr 14 +Apr 28 +May 12 +May 26 +2020 +01 +2.5 +5 +10 +15 +20 +25 +30 +35 +(2) +(1) +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +1 +2 +3 +4 +5 +6 +7 +(3) +(4) +(5) +(6) +(7) +Note: The graphs show the evolution of the time-varying parameters, βl,t, the rate of infection νl,t, the death +rate, and the effective reproduction rate, eRt, estimated using the TVP-SIRD model introduced in (6) for +the US. The 95% (HPDI) Highest Posterior Density Intervals are computed using the posterior output. +Figure 5: The evolution of the ratio of total infections to the number of reported infected +cases +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +Note: The graph shows the evolution of the ratio of total infections to the number of reported infected +cases estimated using the MF-TVP-SIRD model introduced in (17) for the US. The 95% (HPDI) Highest +Posterior Density Intervals are computed using the posterior output. +42 + +Figure 6: The evolution of the level values for the death rate and effective reproduction +rate, νl,t and eRl,t starting from March 2020 until March 2022 +νl,t +Apr 14 +Apr 21 +Apr 28 +May 05 +May 12 +May 19 +May 26 +Jun 02 +2020 +0 +1 +2 +3 +4 +5 +6 +7 +8 +10-3 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.5 +1 +1.5 +2 +2.5 +3 +10-3 +eRl,t +Apr 14 +Apr 21 +Apr 28 +May 05 +May 12 +May 19 +May 26 +Jun 02 +2020 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +5 +10 +15 +20 +25 +Note: The graphs show the evolution of the time-varying parameters, νl,t, the death rate, and eRl,t, the +effective reproduction rate, estimated using the MF-TVP-SIRD model introduced in (17) for the US. The +95% (HPDI) Highest Posterior Density Intervals are computed using the posterior output. +Figure 7: The evolution of the relative RMSFE of the ensemble model’s 1-week ahead +predictions relative to those of the TVP-SIRD model +Weekly confirmed cases +Weekly death cases +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +1.2 +1.3 +1.4 +0 +2000 +4000 +6000 +8000 +10000 +12000 +14000 +16000 +18000 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +1.8 +0 +0.5 +1 +1.5 +2 +2.5 +104 +Note: The graphs show the evolution of the relative Root Mean Squared Forecast Error (rRMSFE) for the +weekly 1-week ahead predictions of the ensemble model from Forecast-Hub relative to the TVP-SIRD model +estimated using weekly data for the period starting from July 2020 until March 2022. The solid line shows +the rRMSFEs computed using expanding window. The dashed line indicates actual realizations of weekly +confirmed and death cases. +43 + +Figure 8: The evolution of the number of active infected cases in countries used for factor +TVP-SIRD model estimation +Germany +Italy +Brazil +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +106 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.5 +1 +1.5 +2 +2.5 +106 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.5 +1 +1.5 +2 +2.5 +106 +Note: The graphs show the evolution of the active infected cases in Germany, Italy, and Brazil over the +sample from March 2020 until the end of March 2022. +The number of recovered cases is absent in all +countries in half of the sample. Therefore, we use γ = 0.07, corresponding to a recovery duration of 14 days +when computing the active infected cases. +Figure 9: The evolution of common infection rate and country-specific eRts +Infection rate factor +eRt of Germany +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +0.6 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.5 +1 +1.5 +2 +2.5 +eRt of Italy +eRt of Brazil +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.5 +1 +1.5 +2 +2.5 +3 +Apr 2020 +Jul 2020 +Oct 2020 +Jan 2021 +Apr 2021 +Jul 2021 +Oct 2021 +Jan 2022 +Apr 2022 +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +Note: The graphs show the evolution of the common infection rate and country-specific eRts for Germany, +Italy, and Brazil over the sample from March 2020 until the end of March 2022 using the factor TVP-SIRD +model as introduced in (25). +44 + diff --git a/XtFRT4oBgHgl3EQf-zgt/content/tmp_files/load_file.txt b/XtFRT4oBgHgl3EQf-zgt/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..69d1b4bd4f5b2a8aec1d922ae99dc63590fb0107 --- /dev/null +++ b/XtFRT4oBgHgl3EQf-zgt/content/tmp_files/load_file.txt @@ -0,0 +1,1322 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf,len=1321 +page_content='Bridging the Covid-19 Data and the Epidemiological Model using Time-Varying Parameter SIRD Model Cem C¸akmaklı ∗ ,1 and Yasin S¸im¸sek †,2 1Ko¸c University 2Duke University February 1, 2023 Abstract This paper extends the canonical model of epidemiology, the SIRD model, to allow for time-varying parameters for real-time measurement and prediction of the trajectory of the Covid-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Time variation in model parameters is captured using the generalized autoregressive score modeling structure designed for the typical daily count data related to the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The resulting specification permits a flexible yet parsi- monious model with a low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The model is extended to allow for un- reported cases using a mixed-frequency setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Results suggest that these cases’ effects on the parameter estimates might be sizeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Full sample results show that the flex- ible framework accurately captures the successive waves of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' A real-time exercise indicates that the proposed structure delivers timely and precise information on the pandemic’s current stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This superior performance, in turn, transforms into accurate predictions of the confirmed and death cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Keywords: Covid-19, SIRD, Observation driven models, Score models, Count data, Time- varying parameters JEL Classification: C13, C32, C51, I19 ∗Correspondence to: Cem C¸akmaklı, Ko¸c University, Rumelifeneri Yolu 34450 Sarıyer Istanbul Turkey, e–mail: ccakmakli@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' †e–mail: yasin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='simsek@duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='13692v1 [econ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='EM] 31 Jan 2023 1 Introduction The outbreak of the new coronavirus, the Covid-19 pandemic, is one of the most severe health crises the world has encountered in the last decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Since the onset of the pan- demic in early January 2020, it has exhibited a varying pattern for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' First, countries have repeatedly taken various measures to reduce the transmission of the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These measures involve complete lockdowns, such as full closure of business and curfew, and partial lockdown that implies a partial closure of daily routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These interventions seem to mitigate the spread of the virus at various stages of the pandemic to the extent that people comply with the ’shelter-in-place’ orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Second, mutations of the virus might lead to changes in its main characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Furthermore, the death rate seems to be par- tially lowered thanks to vaccination campaigns, increasing medical knowledge, and ongoing research on the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the pandemic evolves rapidly with successive waves of infections, efficient and timely monitoring is crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Making prompt and effective decisions of imposing or relaxing lockdown measures for policymakers and taking timely precautions for individuals critically relies on knowledge about the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, epidemiological models for estimat- ing and, perhaps even more crucially, for predicting the pandemic’s trajectory come to the forefront.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Conventional statistical epidemiology models mainly involve structural parame- ters that remain constant throughout the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, suppose these interventions are effective and cause changes in the pandemic’s natural course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Essentially, even if the underlying structural parameters related to the pandemic remain unaltered, there might be various reasons for the resulting estimated parameter to be time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These include changes in the people’s attitude towards the disease1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Besides, the virus might undergo some mutations which alter the contagiousness and fatality of the virus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see, for example, Karim and Karim (2021) for the recent Omicron variant of the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Hence, these muta- tions translate into changes in key structural model parameters that alter smoothly with the changing weight of infections of each virus variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These observations are the depar- ture point of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Specifically, we develop a computationally simple and statistically 1Arias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (Forthcoming) uses the terminology of behavioral parameters to refer to the behavioral aspects that lead to time variation in the structural parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Similarly, Avery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) refers to the changes in the parameters as potential endogeneity of these parameters as the precautions taken could be a function of current cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We thoroughly discuss these issues and the inability to measure the observations of units/compartments, such as the number of susceptible people required for estimating the models in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 1 coherent model that allows for time variation in the epidemiological model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We start our analysis by confronting a simple version of the workhorse epidemiological model with the existing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' From the perspective of econometrics, we specify a counting process for modeling the course of the Covid-19 pandemic for the US based on the SIRD model, which is an abbreviation representing the four identified states of the pandemic as Susceptible, Infected, Recovered, and Death.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The SIRD model depicts these states’ evolution depending on the number of infected individuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Kermack and McKendrick (1927);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Allen (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Provided that these daily counts of susceptible, infected, recovered, and death cases are available, the model is well-identified conditional on the infected cases’ initial value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The pandemic’s course is determined by the contestation of these forces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', the parameters governing infection and resolution (either recovery or death) rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As a result, if the rate of infection (multiplied by the share of susceptible people in the population) is larger than the resolution rate, the number of infections evolves according to a nonstationary process representing the virus’s increasing spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In contrast, the opposite case results in a stationary process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we opt for a Bayesian estimation strategy of the parameters for computing reliable credible intervals for inference conditional on the available data rather than utilizing asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Equipped with these tools, we extend the econometric model by allowing for time vari- ation in the structural parameters by resorting to the Generalized Autoregressive Score (henceforth GAS) modeling framework, which is a class of observation-driven models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The proposed model permits a flexible yet feasible framework to track structural parameters’ evolution timely and accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' A relevant aspect of our specification is its relatively low computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The computational cost might be crucial, especially at the beginning of the pandemic, when the data is scarce, plaguing the inference of flexible models, and the uncertainty is overwhelming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Furthermore, since the typical Covid-19 dataset exhibits a sizeable seasonal pattern, the model is extended to capture these stark seasonal variations using a seasonal component for each parameter by resorting to the frequency domain and GAS framework model structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We extend the model further, taking undocumented cases (as these infected individuals do not show symptoms) into consideration by exploit- ing the information on testing for infection following Grewelle and De Leo (2020) and on the number of excess deaths, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', the total number of deaths caused by the pandemic that is in excess of the reported death cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Since the typical excess death data is at the weekly frequency, unlike the remaining daily counts, the resulting extension permits a mixed 2 frequency time-varying parameter epidemiological model blending datasets with mixed fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, we also consider the model in a multi-country setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In particular, we put forward the factor TVP-SIRD model, where we consider four countries jointly, focusing on the common and idiosyncratic patterns of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The resulting model can efficiently capture the common behavior in the diffusion of the pandemic throughout the world for monitoring the stance of the pandemic from a global perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We consider the US dataset related to the pandemic to demonstrate the proposed frame- work’s efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The US has ample experience in containing the virus with differential mo- mentum in fighting the disease at various pandemic stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the initial US response to the pandemic in terms of imposing restrictions and establishing a containment infras- tructure such as mass testing was not fast, the US was among the first countries to start a massive inoculation campaign at the start of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These distinct experiences provide a testing ground with various patterns to examine the proposed model’s efficacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our results indicate that the model parameters exhibit substantial changes over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The virus’ trans- mission had reduced considerably with the start of 2021 thanks to the massive vaccination campaign in the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, this pattern is interrupted by the two successive waves of the Delta variant and then the Omicron variant, with a rapid increase in virus transmission and a relatively low death rate captured by our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The mixed frequency extension of the model that also captures unreported cases suggests that the actual number of infected cases is potentially as high as three times more than the reported cases for some periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Provided by the wide 95% credibility set ranging from two to five, this finding is consistent with the estimates around four provided by the Center for Diseases Control and Prevention (CDC) for the period until the end of 2020, see Reese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The multi-country analysis involving Germany, Italy, and Brazil on top of the US using the factor TVP-SIRD model indicates that the pandemic diffusion shares a sizable common pattern with Euro- pean countries, including Germany and Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, the diffusion of the pandemic is relatively more idiosyncratic in Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We examine the model’s performance in real-time by conducting a recursive estima- tion and forecasting exercise with real-time datasets predicting 1- to 30-day ahead number of confirmed and death cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Results indicate that the proposed model with time-varying parameters provides timely information on the pandemic’s current stance ahead of the com- peting models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the results are relatively mixed for long horizons from 2-week up to a 1-month ahead, our model yields superior forecasting performance up to 2-week horizon 3 against many competitors, including a linear Gaussian state-space model and a subclass of our model framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, the resulting model is instrumental in providing cru- cial information on the stance of the weeks ahead of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We further confront the weekly predictions using our model with those of the models that are included in the Forecast Hub2, which is a forecast data repository with the predictions created by dozens of leading infectious disease modeling teams from around the globe, in coordination with the CDC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our comparison indicates that the proposed TVP-SIRD model outperforms more than half of those leading models when 1-week ahead forecasts are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, this outperformance reduces gradually with the increasing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The Forecast Hub addition- ally provides an ensemble model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' a forecast combination scheme generated using individual models built on different assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The results show that our model provides superior forecasts, specifically at the onset of the pandemic when the data is scarce, reflecting our proposed framework’s flexible yet parsimonious model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The literature on estimating the SIRD model (with fixed parameters) and variants to evaluate the current stance of the Covid-19 pandemic has exploded since its outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Rela- tively earlier analyses include Read et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) and Lourenco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020), which estimate a SIRD-based model with the data from China for the former and from the UK and Italy for the latter using a likelihood-based inference strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) blend data related to Covid-19 for China with mobility data and estimate the epidemiological model using Bayesian inference to predict the spread of the infection domestically and internationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) conduct a similar analysis employing a modified SIRD model together with a network structure and mobility data to uncover the size of the undocumented cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see also Horta¸csu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) extend the standard SIRD model with many additional compartments and estimate some parameters using Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Identification of the model parameters in these models hinges upon the data availability for each compartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Otherwise, parameter values are set based on the pandemic’s stylized facts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Manski and Molinari (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Atkeson (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Korolev (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Several factors might lead to the time variation in the parameters of the epidemiological model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the one hand, lockdown measures implemented by the policymakers isolate the infected from the susceptible individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, the parameter governing the infection rate, that is, the average number of contacts of an individual, is likely to alter with lockdown conduct, see Hale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the other hand, advancements in the fight 2https://covid19forecasthub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='org/ 4 against Covid-19, including the recovery of drugs and vaccination, could effectively mitigate the course of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In addition, the installment or the lack of medical equipment such as ventilators might alter the rate of recovery or, in other words, the duration of the state of being infected, see for example Greenhalgh and Day (2017) on time variation in recovery rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Accordingly, Anastassopoulou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) use a least-squares-based approach on a rolling window of daily observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' They document the time variation of parameters in the SIRD-based model using Chinese data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Tan and Chen (2020) also employ a similar but more articulated rolling window strategy to capture the time variation in the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Other frameworks with time-varying model parameters almost exclusively allow for the time variation only in the infection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' An application before the Covid-19 outbreak includes, for example, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2016) among others, who utilize a Gaussian process prior to the incidence rate involving the infection rate using a Bayesian nonparametric structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In the context of the Covid-19 pandemic, Kucharski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) estimate a modified SIRD model using a parameter-driven model framework allowing the infection rate to follow a geometric random walk with the remaining parameters kept as constant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Arroyo-Marioli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2021) for a similar approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Similarly, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) and Fern´andez-Villaverde and Jones (2022) allow for time variation in the infection rate, keeping the remaining parameters constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Arias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (Forthcoming), on the other hand, extends the model with time variation in the remaining parameters and provides a Bayesian inference methodology of the resulting model using a particle filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) proposes an econometric specification where the growth rate of infections follows an autoregressive process around a deterministic trend with a structural break.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this paper, we propose an alternative modeling strategy to capture the time vari- ation in the structural parameters of the SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the one hand, our modeling framework is statistically consistent with the typical count data structure related to the pandemic, unlike the models that either employ least-squares or likelihood-based inference using Gaussian distribution, that is, the Kalman filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the other hand, our framework is computationally inexpensive, unlike the models that are statistically consistent but compu- tationally costly such as the particle filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This computational efficiency might be crucial, most notably when the data is scarce, and uncertainty about the pandemic is abounding at the start of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our framework belongs to the observation-driven models class, specifically the GAS models proposed by Creal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' GAS models involve many celebrated econometric models like the Generalized Autoregressive Heteroskedasticity 5 (GARCH) model and various variants as a specific case, and thus, they proved to be useful in both fitting and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Koopman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2016) provide a comprehensive analysis of these models’ predictive power compared to parameter-driven models in many settings, including models with count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Observation-driven models for count data are considered, in many cases, independent of the analysis of the Covid-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Davis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2003) provide a comprehensive analysis of observation-driven models with a particular focus on data with (conditional) Poisson distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2006) derive an integer-valued analog of the GARCH model (IN-GARCH) using Poisson distribution instead of Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Fokianos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2009) consider the Poisson autoregression of linear and nonlinear forms like the IN- GARCH model as a specific case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Chen and Lee (2016) extend the Poisson autoregression to allow for smooth regime switches in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our framework naturally extends these approaches to the epidemiological model framework for each of the core compartments of the SIRD model using a multivariate structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The remainder of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Section 2 describes the canonical SIRD model and introduces the SIRD model with time-varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Section 3 dis- cusses econometric issues, including identifying model parameters and how to account for sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This section further elaborates on our estimation strategy and the result- ing simulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Section 4 presents estimation results using full sample data from the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In Section 5, we evaluate our model framework’s real-time performance in capturing the pandemic’s current stance and forecasting compared to frequently used competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Section 6 discusses potential extensions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, we conclude in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 2 Model specification 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 The canonical model of the pandemic, the SIRD model We start our analysis by discussing the epidemiological model denoted as the SIRD model of Kermack and McKendrick (1927).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Specifically, the SIRD model categorizes a population into four classes of individuals representing four distinct states of the pandemic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Susceptible (S(t)), Infected (I(t)), Recovered (Rc(t)) and Death (D(t)) in period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The susceptible group does not yet have immunity to disease, and individuals in this group have the pos- sibility of getting infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the other hand, the recovered group consists of individuals who are immune to the disease, and finally, D(t) represents individuals who have suc- 6 cumbed to the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The Susceptible-Infected-Recovered-Death (SIRD) model builds on the principle that a fraction of the infected individuals in the population, I(t) N , can transmit the disease to susceptible ones, S(t), with a (structural) infection rate of β by assuming a quadratic matching in the spirit of gravity law, see Acemoglu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2021) for details on alternative matching structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, the number of newly infected individuals in the current period is βS(t) I(t) N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The newly infected individuals, that is, confirmed cases, C(t), should be deducted from the susceptible individuals in the current period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Meanwhile, in each period, a fraction γ of the infected people recover from the disease, which reduces the number of actively infected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Similarly, a fraction ν of the infected people have succumbed to the disease, further reducing the number of actively infected individuals3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Hence, a fraction γ + ν of the infections are ‘resolved’ in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This structure leads to the following sets of equations: ˙S(t) = −βS(t) I(t) N ˙Rc(t) = γI(t) ˙D(t) = νI(t) ˙I(t) = ˙S(t) + ˙Rc(t) + ˙D(t) (1) where ˙x corresponds to dx/dt, and we assume that the population remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 Econometric analysis of the SIRD model with fixed parameters The parameters of interest are the structural parameters β, γ, and ν that provide informa- tion on the transmission and resolution rates of the Covid-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' A central metric that characterizes the course of the pandemic is the effective reproduction number, eR(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The effective reproduction number refers to the speed of the diffusion, which can be com- puted by the ratio of newly confirmed cases, denoted as ˙C(t), to the resolved cases, that is, ˙C(t)/( ˙Rc(t) + ˙D(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, it serves as a threshold parameter of many epidemiological models for examining whether the disease will be extinct or spread further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Accordingly, using (1) eR(t) is identical to β S(t) N /(γ + ν) and when t = 0, it is identical to β/(γ + ν), 3We note the difference between the term death rate and the terms case fatality ratio or mortality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the case fatality ratio refers to the ratio of the (cumulative) number of deaths to the (cumulative) number of the infected individuals, the mortality rate measures the proportion of deaths due to a specific disease among the entire population for a given period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the other hand, the death rate, νt, measures the portion of the actively infected population who succumbed to Covid-19 for a given period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 4In fact, the number of deaths reduces the total population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We assume that the total number of deaths is negligible compared to the population for the tractability of the resulting SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 7 in which case it is denoted as the basic reproduction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this sense, a value of eR(t) being less than unity indicates that the pandemic is contained, and if it exceeds unity, this implies that the spread of the pandemic continues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our primary motivation for employing the model from the econometrics perspective is to conform to this canonical epidemiological model with the existing datasets and pinpoint the pandemic’s stance timely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For that pur- pose, we first discretize (1) as the typical Covid-19 dataset involves daily observations on the counts of individuals belonging to these states of health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Motivated by this, we specify a counting process for the states using the Poisson distribution conditional on past cases of active infections implying a nonhomogenous Poisson process for all the counts see, for example, Allen (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Yan (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Rizoiu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2018) for earlier examples, and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) in the Covid-19 context for a similar approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We specify the following for the stochastic evolution of the counts of these states;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ∆Ct|Ωt−1 ∼ Poisson(β St−1 N It−1) ∆Rct|Ωt−1 ∼ Poisson(γIt−1) ∆Dt|Ωt−1 ∼ Poisson(νIt−1) ∆It = ∆Ct − ∆Rct − ∆Dt, (2) where Ωt stands for information set that is available up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We assume that ∆Ct, ∆Rct, and ∆Dt, representing the daily counts of the pandemic states, are independent conditional on Ωt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The final identity leads to an autoregressive process for the number of active infections, It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The resulting distribution for the number of active infections is a Skellam distribution (conditional on Ωt−1) with the mean πt−1It−1, where πt−1 = (1+β(1− eR−1 t−1)) and the variance as β(1+eRt−1)It−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Here, we use the identity in the last equation of (2) together with the definition of eRt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, the stationarity of the resulting process depends on whether eRt−1 < 1 or eRt−1 ≥ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', whether the pandemic is taken under control or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In addition, in case St−1 N ≈ 1 or in other words, eRt−1 ≈ R0, the first and second unconditional moments are as follows, E[It] = πtI0 V ar(It) = β(1 + R−1 0 ) πt−1(1−πt) 1−π I0, (3) where we assume that the initial condition, I0, is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' If the initial condition is considered a parameter to be estimated, then the variance is further amplified with a factor in the 8 initial condition’s variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Accordingly, the unconditional moments of the pandemic states are linear functions of these unconditional moments of It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We refer to Section A of the supplementary material for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 Motivation for time variation in parameters The canonical model’s structural parameters represent the characteristics of the Covid-19 virus in terms of infectiousness and effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, unless there is a change in these characteristics, such as the emergence of a virus variant, the pandemic’s underlying structural parameters remain unaffected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, when confronting the epidemiological model with the real datasets of the pandemic, often, it is not feasible to observe/measure all the categories/compartments of the pandemic precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These measurement problems might arise from various reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For example, non-pharmaceutical interventions might be one of the underlying reasons, as identifying the number of perfectly isolated people who comply with stay-at-home orders is not easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Even in a full lockdown scenario, some citizens might either break the rules or work in essential sectors that are always kept open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In addition, some behavioral shifts might be reflected in these parameters because, confronted with a high number of infections or alerted by the strict public measures, people would self-isolate even more with the fear of getting infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These swings in attitudes are also reflected as time variations in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, some authors refer to this distinction by rephrasing these as ’behavioral’ parameters, see for example Arias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (Forthcoming) or ’potential endogeneity of parameters’, see for example Avery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Here we refer to a broader stance and refer to these parameters as ’implied’ parameters in the sense that throughout the paper, the term ’parameter’ refers to ’implied parameters’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Another motivation for the time-variation is the emergence of the new variant(s) of the virus with altered characteristics, such as the Delta and Omicron variants, for example, see Karim and Karim (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' With the increasing number of infections by the new variant, the number of confirmed cases would correspond to a mixture of the prevailing variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The weights of this mixture gradually evolve depending on the prevalence of more infectious variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' To elaborate further, we reconsider the first equation of the SIRD model in (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', the equation concerning the number of confirmed cases, this time considering the measurement error and the virus variants explicitly as ∆Ct ∝ S∗ t−1 N (β1I1,t−1 + β2I2,t−1) (4) 9 where v = 1, 2 denotes the vth variant of the virus and Iv,t−1 are the number of people infected with this variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For ease of demonstration, we assume that in a given period t, only two variants can be spread in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' S∗ t−1 indicates the number of susceptible people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, we observe the number of susceptible people only with a measurement error that we denote with Ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, the observed number of susceptible people is St−1 = S∗ t−1 + Ξt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Moreover, the total number of active infections is the sum of the number of variant-specific infections over the type of infections, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', It = I1,t + I2,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Considering the following decomposition, we can define the time-varying parameter as βt St−1 N It−1 ≈ S∗ t−1 N (β1I1,t−1 + β2I2,t−1) = S∗ t−1 N ((β2 − β1)I2,t−1 + β1It−1) = St−1−Ξt−1 N ((β2 − β1)I2,t−1 + β1It−1) = St−1 N β1It−1 + St−1 N (β2 − β1)I2,t−1 − Ξt−1 N β1It−1 − Ξt−1 N (β2 − β1)I2,t−1 (5) The last three terms on the right-hand side represent various sources of factors that can lead to time variation in β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' First, if in society only the first variant of the virus with the infection rate β1 prevails and susceptible people are counted perfectly, then these terms drop from the expression and βt = β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, if a second variant of the virus emerges with the infection rate β2 then we would observe a smooth change in the parameter with the increasing number of infections I2,t−1 relative to I1,t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' If the dominance of the second variant does not materialize immediately, then the changes will be smooth until βt = β2, where the prevailing variant will be the second variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the other hand, if the number of susceptible could not be efficiently measured, then we would have a nonzero Ξt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This measurement error would magnify these changes further as, in this case, the third and fourth terms would contribute to the changes in the implied parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We provide a detailed analysis of these underlying causes of time variation related to measurement errors especially using the vaccination dataset and additionally in case of the probability of reinfection in Section B of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We refer to that section for a more detailed analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this analytical demonstration, we only focus on the rate of infection for the discussion’s compactness but modeling the underlying drivers of the rate of recovery and death could be conducted similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Essentially, for the remaining parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', the rates of recovery and death, these arguments related to the difficulties in measurement and time-varying weight of the mixture of virus variants in the society might play an integral 10 role in the time variation in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In addition, advancements in the fight against Covid-19, including recovery of drugs and vaccination, could effectively mitigate the course of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4 SIRD model with time-varying parameters - the TVP-SIRD model In this section, we put forward the SIRD model with time-varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We use the framework of the Generalized Autoregressive Score model for modeling the time variation in parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This framework encompasses a wide range of celebrated models in econo- metrics, including the GARCH model and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Briefly, the GAS model relies on the intuitive principle of modeling the time variation in key parameters in an autoregressive manner which evolves in the direction implied by the score function and thereby improving the (local) likelihood;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Creal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2013) for a detailed analysis of the GAS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As in the case of the GARCH model, it effectively captures the time dependence in long lags in a parsimonious yet quite flexible structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Consider the SIRD model with time-varying parameters as βt, γt, and νt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the parameter for the rate of infection βt is positive, the parameters γt and νt are on the unit line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we transform βt using logarithmic transformation and γt and νt using logit transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 Let the parameter with a ˜(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=') denote the corresponding transformations as ˜βt = ln(βt), ˜γt = logit(γt) and ˜νt = logit(νt) where logit refers to the inverse of the logistic transformation as logit(x) = log( x 1−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The resulting Time-Varying Parameters - SIRD (TVP-SIRD) model is as follows ∆Ct|Ωt−1 ∼ Poisson(βt St−1 N It−1) ∆Rct|Ωt−1 ∼ Poisson(γtIt−1) ∆Dt|Ωt−1 ∼ Poisson(νtIt−1) ∆It = ∆Ct − ∆Rct − ∆Dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (6) We decompose the transformed parameters further into a smooth level component and a high-frequency seasonal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Because the typical daily Covid-19 dataset exhibits an immense daily seasonal pattern potentially due to frictions in reporting, we also put a 5Essentially, we require an additional requirement that γt + νt ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We go through a detailed specification for the convenient transformation of the parameters and consider three cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In the first case, all three parameters are subject to only logarithmic transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In the second case, the parameters γt, νt ∈ [0, 1] using logistic transformation, while in the third case, we impose an additional restriction of γt + νt ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Results show that the final restriction γt + νt ∈ [0, 1] is not binding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, it imposes important challenges on the feasibility of the estimation, especially when we enhance the model to allow for seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we proceed with the second case throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The specification search and the findings are displayed in Section C of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 11 particular emphasis on the seasonal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Specifically, consider the decomposition as θt = θl,t + θs,t (7) where for parameter θt = ˜βt, ˜γt and ˜νt, respectively, θl,t and θs,t are the level and seasonal components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6 For the level parameter, the evolution of the parameters is specified as θl,t = ωθ + βθθl,t−1 + αθsθ,t−1 (8) where sθ,t for θt = ˜β, ˜γt and ˜νt are the (scaled) score functions of the joint likelihood which is identical to that for the level parameter due to the additive structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Since the SIRD model’s likelihood function is constituted by the (conditionally) independent Poisson processes, each score function is derived using the corresponding compartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Specifically, let ∇˜β,t = ∂L(∆Ct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜βt) ∂ ˜βt , ∇˜γ,t = ∂L(∆Rct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ˜γt) ∂ ˜γt and ∇˜ν,t = ∂L(∆Dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ˜νt) ∂ ˜νt denote the score functions for period t observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We specify sθ,t such that the score functions are scaled by their variance as sθ,t = ∇θ,t Var(∇θ,t) for θt = ˜β, ˜γt and ˜νt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7 In the specific case of the SIRD model, this modeling strategy leads to the following specification for the (scaled score functions) in terms of the corresponding link function s˜β,t = ∆Ct−1−λ1,t−1 λ1,t−1 s˜γ,t = ∆Rt−1−λ2,t−1 λ2,t−1 1 (1−γt) s˜ν,t = ∆Dt−1−λ3,t−1 λ3,t−1 1 (1−νt) (9) where λ1,t = βt It−1St−1 N , λ2,t = γtIt−1 and λ3,t = νtIt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The resulting specification implies an intuitive updating rule because the parameters (in the logarithmic form) are updated using a combination of the previous parameter value and a function of the previous per- centage deviation from the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We refer to Section D of the supplementary material for the details on the derivation of (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' One drawback of the specification in (8) is that when βθ is close to unity, identification of ωθ is cumbersome, see for example Kastner and Fr¨uhwirth-Schnatter (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This is also our experience when estimating the model using 6Such additive structure in the transformed parameters leads to a multiplicative seasonal structure in exponential form as a function of original parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see, for example, Hansen and Schmidtblaicher (2021) for a similar approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 7Alternative approaches for scaling the score function include the standard deviation rather than the variance and the score function without scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our findings suggest that using the variance as the scaling function leads to smoother and more robust evolution of parameters over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 12 real datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we restrict the βθ parameter to be unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this case, we remove the intercept parameter ωθ, but the level of the time-varying parameters is estimated as the initial condition θl,0 along with other model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8 For the seasonal component, we specify a structure using frequency domain for capturing the daily seasonality in a given week adequately departing from Hansen and Schmidtblaicher (2021)9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Consider the model θs,t = �s/2 j=1 θjs,t (10) with θjs,t = cos(Λj)θjs,t−1 + sin(Λj)θ∗ js,t−1 + ψjsθ,t−1 θ∗ js,t = − sin(Λj)θjs,t−1 + cos(Λj)θ∗ js,t−1 + ψ∗ Jsθ,t−1 (11) where Λj = 2πj s for j = 1, 2, 3 and θt = ˜β, ˜γt and ˜νt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The structure in (10) and (11) provides a quite flexible yet parsimonious model structure for capturing seasonal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Essentially, it can be shown that in case the score functions are zero, it reduces to θjs,t = − sin(Λjt)θjs,t−1 + cos(Λjt)θ∗ js,t−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (12) This structure implies that the cycle is captured by the three periodic series with frequencies Λ1 = 2π 7 , Λ2 = 4π 7 and Λ3 = 6π 7 each with period 7, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the first series has the fundamental frequency, the remaining parts could be obtained by integrating the fundamental frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Proietti and Pedregal (2022) for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The sine and cosine terms together function as two orthogonal bases generalizing the model in Hansen and Schmidtblaicher (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As in the level case, the score function is identical to the general score function owing to the linear decomposition of the parameters into the level and seasonal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This structure facilitates the estimation substantially and enables us to capture the potential link between level and seasonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The specification in (6)-(11) leads to quite rich dynamics both in terms of mean and the variance of the resulting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These rich dynamics enable us to accurately capture the pandemic’s evolution reflected in the timely and prompt response of the parameters to the 8We also compute the Bayes factor of this model relative to the unrestricted model for formal model comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We find that the Bayes factor is to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='99, indicating that the restriction is also supported by the data, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 9We also consider models for seasonality that exploits the time domain using daily components for modeling seasonality where the corresponding coefficients are time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our findings suggest that models in the frequency domain facilitate the estimation and performs better than their time domain counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display these results in Section E of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 13 pandemic states’ data changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' To elaborate further, we also consider the implied moments of the resulting process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display these findings in Section F of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 3 Econometric inference Two major issues plague the inference of the SIRD model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' First, from the epidemiology point of view, the SIRD model could be extended in various directions by incorporating other phases, or in other words, compartments of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As this implies additional parameters to be estimated, identifying these parameters is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Second, detection of the infected individuals might be burdensome as some of them do not show symptoms, yet they are infectious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this section, we first discuss the challenges of the econometric inference, and second, we introduce the details of the simulation-based Bayesian estimation strategy for the econometric inference of the TVP-SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 Identification of model parameters The pandemic’s course in the evolution of active cases depends on the structural parameters, β, γ and ν that are used to construct π in (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The initial condition I0 is also required because the process might be nonstationary if the pandemic is not contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We estimate the models starting from the period when the number of cumulative confirmed cases exceeds 1000, and we use this first observation in our sample as the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='10 The structural parameters represent the compartments of the SIRD model where the compartments refer to the specific phases of the disease as ’susceptible’, ’infected’, ’recovered’, or ’death’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Still, it is possible to extend the model with additional compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For example, it is known that the virus has an incubation period in which the susceptible person is ’exposed’ to the virus but not yet affected by it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Nevertheless, she can transmit the virus to other people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Departing from this point Korolev (2020), for example, discusses identification problems of the structural parameters regarding the SIRD model with an additional compartment of ’exposed’ case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, these compartments require additional parameters to be estimated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Lourenco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' If the specific compartments’ data, such as the number of infected cases where the virus is in the incubation period, is available, these structural 10Different starting points (such as the periods when the number of cumulative confirmed cases exceeds 10000) yield very similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Results are available upon request by the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 14 parameters are well identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, while these additional compartments provide further refinements to the SIRD model, these refinements plague the identification of the structural parameters if additional data is missing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see, for example, Atkeson (2020) who discusses the identification of the structural parameters regarding the SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' He demonstrates how different parameter setups might result in very similar initial phases of the pandemic but result in divergent patterns in the long-run absence of the data on the model’s compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 Accounting for sample selection A fundamental underlying assumption of the model specification in previous sections is that the variables of the infected, recovered, and succumbed individuals represent the aggregate numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, one of the stylized facts related to the Covid-19 pandemic is the presence of infected individuals who do not have any symptoms, denoted as asymptomatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These hard-to-detect cases complicate the analysis as it leads to a selection bias in econometric inference, among other factors, see Manski and Molinari (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These unreported infection cases prohibit the tests from being randomly assigned, plaguing econometric inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This section provides a model extension based on some assumptions on the model structure to capture asymptomatic infected individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We use two sources of additional datasets to extract the total number of active cases, including the reported and unreported ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The first additional data we use is the number of excess deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These include the estimates of additional deaths directly or indirectly attributed to Covid-19 in excess of the published number of deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This projection provides weekly estimates of excess deaths, and these weekly counts of deaths are compared with historical trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Accordingly, it provides an essential source of identification for the unreported cases, as potentially a significant part of the excess deaths might be attributed to this sample selection11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The second variable we exploit is the number of positives in the tested individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' To motivate the idea, let Pt denote an indicator function that takes the value one if an individual is infected in period t and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Further, let Tt denote another indicator function, which takes the value one if an individual is tested for the infection in period t and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Using the Bayes 11Please see, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='cdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='gov/nchs/nvss/vsrr/covid19/excess_deaths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='htm and https: //ourworldindata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='org/excess-mortality-covid for additional discussion and methodology on the computation of excess death 15 rule, we can show that, P(Pt = 1) = P(Pt = 1|Tt = 1)P(Tt = 1) P(Tt = 1|Pt = 1) , (13) see also Stock (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In case the assignment for testing of an individual is carried out randomly, then P(Tt = 1) = P(Tt = 1|Pt = 1) and there is no identification problem due to sample selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Neither, P(Pt = 1) nor P(Tt = 1|Pt = 1) are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Nevertheless, P(Tt = 1) could be computed as the fraction of tested individuals in the population in period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Furthermore, P(Pt = 1|Tt = 1) could be considered as the daily positive test rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Equipped with these, identification of P(Tt = 1|Pt = 1) boils down to the identification of P(Pt = 1), the true prevalence of the infection, including asymptomatic cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Departing from Grewelle and De Leo (2020), we make use of a parametric identification strategy for approximation of P(Tt = 1|Pt = 1), P(Tt = 1|Pt = 1) = exp(−kρt) (14) where ρt is the fraction of positives in the tested individuals in period t, and k is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Briefly, the underlying idea stems from the fact that detecting infections, including asymptomatic individuals, would improve with the increasing number of testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In that sense, the fraction of tested individuals in the population should be related to the ratio of reported infections to the total number of infections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' With the increasing number of testing on the population, this fraction approaches one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the other hand, if testing is concentrated only on symptomatic individuals, this fraction approaches a lower bound, captured by the parameter exp(−k), where the functional form admits exponential decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Unlike the daily dataset we employ for estimating the TVP-SIRD model, the excess death data is at the weekly frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we extend our model framework to allow for a mixed-frequency dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Let I∗ t be the number of infected individuals involving asymptomatic and symptomatic cases, and let S∗ t and Rc∗ t denote the total number of susceptible and recovered individuals, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Let δt = 1 − exp(−kρt) denote the fraction of the unreported infection cases among all infection cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Using (14), these could be computed as ∆C∗ t = ∆Ct 1−δt ∆Rc∗ t = ∆Rct 1−δt for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (15) 16 Further denote the total number of weekly deaths as ∆ ¯Dw t which is computed as, ∆ ¯Dw t = ∆Dw t + ∆EDt for t = 7k and k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , (16) where Dw t stands for the reported deaths at the weekly frequency and ∆EDt for the Excess Death numbers estimated in period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, the mixed frequency TVP-SIRD (MF-TVP- SIRD) model in terms of the total numbers can be written as ∆C∗ t |Ωt−1 ∼ Poisson(βt S∗ t−1 N I∗ t−1) ∆Rc∗ t |Ωt−1 ∼ Poisson(γtI∗ t−1) for t = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , T ∆ ¯Dw t |Ωt−1 ∼ Poisson( t� s=t−6 νsI∗ s−1) for t = 7k and k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ∆C∗ t = −∆S∗ t = ∆I∗ t + ∆Rc∗ t + ∆ ¯Dd t (17) Here ∆ ¯Dd t is computed as νtI∗ t−1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The evolution of the model parameters decomposed as in (7) follow the recursions in (8) and (10) using the (scaled) score functions displayed in (9) for the parameters βt and γt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the score function of the νt, we have the following expression due to the change in the frequency of the Poisson process s˜ν,t = ∆ ¯Dt−¯λ3,t−1 ¯λ3,t−1 1 (1−νt) (18) with ¯λ3,t−1 = νt t� s=t−6 I∗ s−1 for t = 7k and k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , that is for the periods where the weekly data is released, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The score function takes the value 0 when the weekly excess death variable is not observed, which completes the specification of the MF-TVP- SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 Estimation strategy and the simulation algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 Bayesian inference We use simulation-based Bayesian estimation techniques for inference on model param- eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Bayesian inference involves updating the prior distributions of model parameters with the data likelihood to form the parameters’ posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Considering the SIRD model, Bayesian inference is especially appealing since the inference is conditional on the data at hand and does not require asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we can compute 12Notice that the sum of the independent Poisson processes is a Poisson process, which enables us to switch between daily and weekly frequency when necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 17 the credible intervals when the underlying process of the number of infected cases, It, is nonstationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This property is especially reassuring in our case since, obviously, nonsta- tionarity, or in other words, effective reproduction rate being greater than one, eRt > 1, is an inevitable feature of the pandemic, at least locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Here we demonstrate the likelihood function and prior specifications for the TVP- SIRD model because the SIRD model with fixed parameters boils down to a special case of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The likelihood function is based on the specification in (6) where we specify conditionally independent Poisson distributions for each of the components of the SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Let yt = (∆Ct, ∆Rct, ∆Dt)′ be the vector of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Notice that It = It−1 + ∆Ct − ∆Rct − ∆Dt, and thus the number of active infections can be computed using the information set Ωt = (y′ t, It−1)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Accordingly, we have the following likelihood function f(yt|Ωt−1) = λ∆Ct 1,t exp(−λ1,t) Γ(∆Ct+1) λ∆Rct 2,t exp(−λ2,t) Γ(∆Rct+1) λ∆Dt 3,t exp(−λ3,t) Γ(∆Dt+1) , (19) where λ1,t = βt St−1It−1 N , λ2,t = γtIt−1 and λ3,t = νtIt−1 and Γ(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=') is the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Note that the time-varying parameters decomposed as in (7) follow the recursions in (8) and (10) using the (scaled) score functions displayed in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We want to obtain posterior results driven by the data rather than the prior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we impose rather diffuse prior specifications for the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This strategy implies that for the representative model parameters φ, we specify the following improper prior specifications f(φ) ∝ 1 (20) for φ ∈ Φ = (Θ′ l,0, α′, ψ ′ ˜β, ψ∗′ ˜β , ψ ′ ˜γ, ψ∗′ ˜γ , ψ ′ ˜ν, ψ∗′ ˜ν )′ where Θl,0 = (βl,0, γl,0, νl,0)′, α = (αβ, αγ, αν)′, ψθ = (ψ1,θ, ψ2,θ, ψ3,θ)′ and ψ∗ θ = (ψ∗ 1,θ, ψ∗ 2,θ, ψ∗ 3,θ)′ for θt = ˜β, ˜γt and ˜νt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the MF-TVP- SIRD model, we specify the prior distribution for the additional k parameter noninformative in the positive domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 Simulation scheme For the SIRD model with fixed parameters, the likelihood with Poisson distributions as in (19) with fixed parameters and noninformative or conjugate priors in the form of Gamma distribution lead to a Gamma distribution for the posterior distributions of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, these can be sampled using the plain Gibbs sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the TVP- 18 SIRD model, the fact that we have time-varying parameters with deterministic recursions leads to nonstandard posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we cannot use standard distribu- tions we can easily simulate for the inference, as is the case for the Gibbs sampler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Instead, we resort to the (adaptive) random walk Metropolis-Hastings (MH) algorithm within the Gibbs sampler;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Robert and Casella (2013) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The algorithm is as follows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Sample α from f(α|ST , IT , Φ−α) using MH step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Sample Θl,0 from f(Θl,0|RcT , IT , Φ−Θl,0) using MH step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Sample ψθ from f(ψθ|Y T , Φ−ψθ) using MH step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Sample ψ∗ θ from f(ψ∗ θ|Y T , Φ−ψ∗ θ) using MH step Here Y T = (Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , Yt, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , YT )′ for Yt = St, It, Rct, Dt indicating the full sample of the count data regarding the states of the pandemic, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Φ−X indicates the vector of parameters Φ excluding the parameters X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the MH steps, the candidate generating density is constructed using the random walk specification as φm = φm−1 + Σ1/2 φ εm (21) where φm is the parameter draw depending on the step at the iteration m, and εm follows a standard (multivariate) t−distribution with degrees of freedom 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the starting values of the parameters to initialize the sampler, Φ0, and for the covariance matrix ΣΦ0, we use the maximum likelihood estimate of the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we use the mode and the inverse Hessian of the likelihood function at the mode in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' To improve the sampler’s performance, we follow the adaptive scheme described in Haario et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This scheme involves replacing ΣΦ0 with χSM +ϵI, once we obtain a sufficient number of draws to replace the inverse Hessian of the likelihood function with the simulated curvature of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Here Sm corresponds to the empirical covariance matrix computed using the draws up to step M, I indicates the identity matrix, and ϵ is a small number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ϵI ensures a nonsingular empirical covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In addition, we use χ for optimizing the sampler’s performance for the candidate generating density to be efficient enough to cover the tails of the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Let φcand ∼ q(φcand|φm−1) be a draw from the candidate generating density in iteration m of the sampler, the candidate is accepted with probability π = min � 1, q(φm−1|φcand)p(φcand|Y T ) q(φcand|φm−1)p(φm−1|Y T ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (22) 19 Here p(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='|Y T ) refers to the posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Note that, due to the symmetry of the random walk specification, q(φm−1|φcand) and q(φcand|φm−1) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Hence they are of no use in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 4 Empirical results 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 Dataset We use the data for the US starting from the early days of the pandemic until the end of March 2022, which captures all major waves of infections related to the Covid-19 pandemic, including the recent Omicron wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The data is originally published by the Center for Disease Control and Prevention (CDS) and can be tracked on https://covid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='cdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='gov/ covid-data-tracker/#datatracker-home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The data on the number of recovered cases ceased to be reported following December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We, therefore, treat this data as missing for the periods when the data is not reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In these cases, the score function is set to zero, similar to the estimation of the MF-TVP-SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the out-of-sample analysis that involves a real-time recursive prediction exercise, we use the US data vintages that are available as of the day of the prediction, which is available on the Covid-19 Data- Hub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The data-hub includes the daily vintages of Covid-19 pandemic related datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see https://covid19datahub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='io/articles/data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='html and Guidotti and Ardia (2020) and Guidotti (2022) for implementation details and the latest version of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 A brief account of the pandemic’s course in the US We display the evolution of the daily confirmed cases and deaths throughout the sample in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Figure 1 about here] The US exhibits extensive heterogeneity throughout the sample regarding their experience related to the pandemic, as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' At the onset of the pandemic, the US opted for imposing mixed strategies involving full and partial lockdowns and voluntary quarantine in different regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As of March 2020, the US reported the highest number of daily confirmed cases worldwide and therefore was the center of the pandemic when the country was struggling with the first wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In late 2020, the Alpha variant was the virus’s dominant strain, which can be detected as the second wave in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Nevertheless, with 20 the start of 2021, the US began an intensive inoculation campaign to fight the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In 2021 two major pandemic waves were experienced with the emergence of the Delta and Omicron variants, among other strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the Omicron variant notably led to record values for the number of daily confirmed cases, the daily number of deaths was still comparable to that of the Alpha variant owing to the success of vaccination efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Hence, this relatively rich and heterogeneous dataset involving all sorts of pandemic experiences enables us to examine the econometric model’s success in tracking the parameter changes in response to policy implementations and changes in the virus’s characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 Full sample results This section discusses the full sample estimates of the main parameters for the models with fixed and time-varying parameters, as described in (2) and (6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We further evaluate the model’s parameter estimates with time-varying parameters when asymptomatic cases are explicitly considered, as shown in (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We start our analysis with the full sample estimates of the model parameters for the SIRD model with fixed parameters described in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These are displayed in Panel A of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Table 1 about here] The model with fixed parameters reflects the stance of the pandemic over the last two years, on average, as the parameters are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The basic reproduction rate, R0, as the main summary statistics on the course of the pandemic when the whole population is considered susceptible, is displayed in the last column of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The median estimate of the R0 for the US shows that over the sample period of more than two years since the start of the pandemic in March 2020, the estimate is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='64 with little uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The fact that the R0 well exceeds 1 reflects that the pandemic is not contained yet on average with the repeated waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Notice that this value might be inaccurate because the number of susceptible people has reduced to some extent with the progress of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We, therefore, display the evolution of the effective reproduction rate, eRt in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Figure 2 about here] Figure 2 indicates that the effective reproduction rate declines over time with the reduced number of susceptible people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As the main reason for the reduction in the number of susceptible people is the infected cases, the reduction in eRt closely follows the waves of the 21 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The final value as of the end of March 2022 is still 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3, indicating that a large part of the population is not infected yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The effect of pharmaceutical or nonpharmaceutical interventions is reflected in the re- production rate through the ’implied’ model parameters as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These effects are all reflected as implied time variation in these parameters captured by the TVP- SIRD model, which we discuss in the remainder of this section13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Before discussing the evolution of the model parameters, we first start with displaying the fitted values of the daily number of confirmed and death cases in Figure 3 to consider the overall performance of the TVP-SIRD model in fitting the pandemic dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Figure 3 about here] The left panel of the Figure 3 shows the satisfactory fitting performance of the TVP-SIRD model for the daily confirmed cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Both level and seasonal patterns could be matched using the model framework that can capture daily seasonal behavior in addition to level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the fitted values of the daily number of death cases on the right panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This panel confirms the model’s ability to capture daily death cases’ level and seasonal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Next, we display the evolution of the (level of the) model parameters and the estimated effective reproduction rate, eRt, using the TVP-SIRD model in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Figure 4 about here] In the first two graphs, we display the variation in the infection and death rates, and in the bottom set, we display the effective reproduction rate, eRt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='14 For clarity of demonstra- tion, we display the evolution of the parameters and the effective reproduction rate in two subfigures representing two subperiods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the left, we only display the periods until June 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' on the right, we display the remaining periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In these two subperiods, the scale of the variation of the parameters differs considerably, and providing two graphs for two 13We also perform a statistical evaluation for the presence of time variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In all the tests, the null hypothesis boils down to the hypothesis that the coefficients of the score functions are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As indicated by Calvori et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2017), the test against the time-varying parameter alternative checks whether there is any autocorrelation in the scores of the fixed parameter model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' If that is the case, such autocorrelations can be exploited to improve the model’s fit by using the likelihood scores as drivers for the time-varying parameter as in the TVP-SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our test results indicate decisively favor time variation in the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We thank an anonymous referee for pointing out this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 14The data on recovery ceased to be published after December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We treat these periods of 2021 and 2022, where the recovery data is unavailable, as missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The estimate of the recovery rate remains constant for these periods when the data is missing since the score functions for these periods are set to 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Creal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2014) and Lucas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2016) for a similar approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Accordingly, we do not display this parameter’s evolution, estimated as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0091, in large part of our sample period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 22 different subperiods enhances the display’s visual quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, in the bottom graph of the effective reproduction rate, we split the sample into seven enumerated subperiods with distinct characteristics to motivate the variation in the eRt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The first period labeled as (1), starting from early 2020 until April 2020, is the emergence period of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' During these periods, the World Health Organisation (WHO) officially declared the pandemic on March 11, and the national public health agency in the US imposed various measures to contain the pandemic, including the ban of large gatherings and travel restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Until the end of April, ’stay-at-home’ quarantines were effective in several locations, and many testing facilities for effectively isolating the infected cases were established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our estimates suggest that the effective reproduction rate reduced from values as high as 35 to around 6 in mid-April.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This reduction is in line with early studies reporting a basic reproduction rate (which is very close to the effective rate at the onset of the pandemic) of around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7 using the early dataset from Wuhan, China, the point of emergence of the pandemic, see Sanche et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) and around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 for the US, see Peirlinck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In the second period (2), comprising the period from mid-April until June, the eRt steadily fluctuates around the value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 with the accumulation of the bulk datasets, similar to the studies reported elsewhere, see Petersen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) for example for a comparison of SARS-COV-2 parameters to that of the SARS-COV and influenza viruses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The third period, (3), captures the summer period of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This period experienced the economy- wide reopening and relaxation of various measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Many large-scale events resulted in large gatherings15 that lead to an increase in the implied infection rate as can be seen in the first subfigure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The increase in the infection rate had overcome the increase in the death rate, which led the effective reproduction rate to increase to values around five again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The fourth period, (4), captures the winter period that includes the holiday season of Thanksgiving followed by the Christmas period, with an estimated number of more than 2 million people flying on airlines during the Thanksgiving period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='16 In addition to the changing mobility of the susceptible people, there was also a new variant of the virus, denoted as Alpha, first detected in December 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Davies et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2021) report that this variant is 43%-90% more transmissible than the predecessor lineage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, following the demonstration 15See for example New York Times article on 80th Motorcycle rally in South Dakota, where there were more than 400,000 audiences in the gathering, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='nytimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='com/2020/11/06/us/ sturgis-coronavirus-cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='html.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 16See the article https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='masslive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='com/coronavirus/2020/11/thanksgiving-travel-many-americans/ flying-for-holiday-despite-cdcs-pleas-not-to-due-to-covid-19-transmission-risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='html for ex- ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 23 in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3, the time-varying mixture of these two variants might be one underlying source of the increasing infection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The year 2021 started with a massive inoculation campaign with the availability of the Covid-19 vaccines with an efficacy rate as high as 90%, see Polack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This period (5) experienced a significant and rapid drop in the effective reproduction rate, which fell below the critical value of 1 for the first time since the start of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, in the first six months of 2021, the US successfully contained the pandemic, thanks to the vaccination campaign, which led to 67% of the overall adult population receiving at least one dose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In the second half of 2021, the proportion of vaccinated people in the population has remained relatively steady, above 60%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Furthermore, containment measures have also remained stable to a large extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, the changes in the implied parameters in periods (6) and (7) mainly stem from the emergence of new variants with new structural parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Indeed, in period (6), which spans the summer and early fall of 2021, the Delta variant was the dominant strain, according to the CDC estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The Delta variant seems to be around 60% more transmissible than the already highly infectious Alpha variant, see Callaway et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2021), which can also be traced in the course of the infection rate and, thereby, the effective reproduction rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, in the last period, (7), which captures the remaining period until the end of March 2022, the Omicron variant has been the dominant variant which is much more contagious than the previous strains but less severe compared to those, see Karim and Karim (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This rapid infection rate surge due to the Omicron variant can be captured nicely using our model framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Moreover, the increase in the death rate remained relatively moderate and lower than that of the Delta variant, confirming the findings on the Omicron variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As a result, the effective reproduction rate soared to as high as five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' As of March 2022, the rate again fell below the critical value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The full-sample findings demonstrate that the model with ’implied’ time-varying parameters can capture various factors, such as changes in the number of susceptible people either due to pharmaceutical or nonpharmaceutical interventions or the emergence of new variants of the virus accurately and promptly, confirming stylized facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 Accounting for unreported cases The results discussed in previous sections are computed using official statistics, including only the reported cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this section, we present our findings when we account for this 24 selection bias using the information on the number of excess deaths and the number of tests together with these tests’ positivity rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the evolution of the estimated rate of total infections to the number of reported infected cases, 1/(1 − δt), in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Figure 5 about here] Figure 5 indicates that the actual number of infected cases, including the asymptomatic cases, might be three times more than the reported cases, especially during the peaks of the first two pandemic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, our estimation results suggest a considerable uncertainty around this ratio with a 95% credible interval between almost two and five on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This finding is consistent with the CDC estimates reported as around 4 for the period until the end of 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Reese et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2020) and the related web source17 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Similar findings are also reported by Angulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2021) based on the data from four regional and one nationwide seroprevalence surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='18 These serosurveys serve as a crucial data source for measuring the number of infected cases because the survey participants are selected randomly, thereby overcoming selection bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our results align with the reported results in Angulo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2021), where they estimate this rate as four using the nationwide serosurvey conducted during July-August 2020 in 47 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While this is the case for most waves, including Delta and Omicron waves, throughout 2021, an important finding is in the late summer of 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our results show that the total number of infected cases might be as high as seven times (with a wide 95% credibility interval between [4-10]) of those reported cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This finding is because, during this period, we observe a rapid surge in the number of excess deaths and relatively greater test positivity ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This implies that the low number of confirmed cases in the summer of 2021 in the US might be mainly due to these unreported cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' According to the CDS estimates based on recurring serosurveys, the seroprevalence estimates, that is, the percentage of people with antibodies against the virus, soars from 20% in July to around 30% in September, which is in line with our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='19 We display the evolution of the death rates (based on the total number of deaths, including the excess deaths) and the effective reproduction rate in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Figure 6 about here] 17https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='cdc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='gov/coronavirus/2019-ncov/cases-updates/burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='html 18A seroprevalence survey uses antibody tests to estimate the percentage of people in a population who have antibodies against the virus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The number of people in a specific population who have been previously infected with the virus is estimated using these test outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 19Notice that the CDS report involves seroprevalence of infection-induced antibodies (nucleocapsid anti- body) which is distinct from the vaccination-induced antibody (spike antibody).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, these estimates genuinely represent the total infections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (2022) for example for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 25 Considering the death rate, the discrete evolution of this parameter is due to the weekly frequency of the excess death dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This parameter is updated only at the weekly frequency when the data is observed and remains fixed for the other periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the death rate exhibits a similar pattern, we can track the surge in the rate in late summer of 2021, in accord with the previous discussion on the total number of infected cases, thanks to the excess death data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This finding is also confirmed in the evolution of the effective reproduction rate level, eRl,t, where the eRl,t is computed as high as nine during these periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 5 Real-time performance of the models The results in the previous section display our findings based on the estimates using the full sample dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These results indicate that our flexible modeling structure can ac- commodate various forms of parameter changes reflecting the pandemic’s course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, exploring the model’s real-time performance would uncover whether this additional flexibil- ity brought by the time-varying parameters could provide timely and accurate information on the pandemic’s real-time stance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, in this section, we discuss the model param- eters’ estimation results in real-time using the model with fixed parameters and the model with time-varying parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We aim to provide a thorough real-time analysis in the sense that we make use of the complete vintage data publicly available at the time of the predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Given that the pandemic data were revised substantially at times, using vintage data provides the actual predictive performance of the complete models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The vintage dataset is obtained from the Covid-19 Data Hub20, see Guidotti and Ardia (2020) and Guidotti (2022) for details on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 Predictive performance at the daily frequency We use a rolling window for performing the SIRD model’s estimations with fixed parameters rather than expanding window21 following the evidence of time variation in parameters in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Specifically, using the dataset from t−M, t−M +1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , t, we estimate the SIRD model, and the resulting parameter estimates are those for the period t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We 20https://covid19datahub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='io/ 21We include the forecasting performance using an expanding window in the earlier versions of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The results are decisively inferior compared to all moving window approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we do not display those results here, but results are available upon request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 26 repeat this process by recursively adding one observation (and dropping one observation at the beginning of the sample for the rolling window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We consider three cases by setting M = 30, 45, and 60, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', starting from one month of data up to two months of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For capturing seasonality in these rolling window regressions, we consider daily dummy variables representing the days of the week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These models are denoted as RW-30, RW-45, and RW-60, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the TVP-SIRD model, we use the data up to period t using an expanding window rather than a rolling window, as the parameters, in this case, are time-varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We also include a restricted version of the TVP-SIRD model, where we allow for time variation only in the infection rate, β, denoted as TVP-SIRD-β, following similar approaches, see Fern´andez-Villaverde and Jones (2022) for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, we also include a time-varying parameter model that falls into the parameter-driven model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In that case, we consider the computationally least costly alternative by imposing Normal distributions for observation and parameter evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For capturing the seasonality, we use the same model framework that we employ in the TVP-SIRD model with the critical distinction of including the error term in the state equations capturing seasonal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The resulting specification leads to a standard inference using the Kalman filter and simulation smoother, which is still tractable in cases when the data is scarce;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Durbin and Koopman (2012) for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This model is denoted as KF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For a given model, the predictive distribution of the observation at t0 + 1 conditional on the information available at t0, Ωt0, is given by p(yt0+1|Ωt0) = � f(yt0+1|Φ)f(Φ|yt0)dΦ, (23) where f(Φ|yt0) is the posterior distribution of the model parameters, estimated using the data until t0, gathered in the parameter set, Φ, given the observations until t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' p(yt0+1|Φ) is the density of the observation yt0+1, which can be written as f(yt0+1|Φ) = � θt0+1 f(yt0+1|θt0+1, Φ)f(θt0+1|Φ, Ωt0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (24) We can use the posterior simulator to obtain the distribution of the model parameters and estimate the predictive distribution using the draws from the simulator as (y(m) t0+1|Ωt0, Φ(m)), where m represents the mth draw from the posterior simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the results involving Root Mean Squared Forecast Errors (RMSFEs) of the competing models relative to the TVP-SIRD model considering the prediction of the daily confirmed cases in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 27 [Insert Table 2 about here] We perform equal predictive accuracy tests for the out-of-sample comparisons to evaluate the relative model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Specifically, for all the comparisons, we perform Diebold- Mariano (DM) type of pairwise comparison tests of equal predictive accuracy between the competing models with HAC standard errors and small sample correction suggested by Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (1997) using squared error contributions as loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The cells with white backgrounds contain statistically insignificant values at the conventional significance level of 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Table 2 indicates a clear-cut result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The TVP-SIRD model outperforms all competing models up to 15 days horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This indicates the superior performance of our flexible modeling structure in the short and medium-term forecasting of the confirmed cases up to two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This outperformance deteriorates for the horizons exceeding two weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this case, although some models provide relative RMSFEs lower than unity, this relative performance is statistically insignificant at conventional significance levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This is due to increasing uncertainty surrounding these point predictions leading to statistical insignificance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Focusing on pairwise evaluations, comparing the TVP-SIRD model with the TVP-SIRD- β model reveals the importance of modeling time variation not only in the infection rate but also in the remaining parameters, at least for short and medium-term predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For the horizons longer than two weeks, the two models perform alike with relative RMSFEs very close to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Comparison of the TVP-SIRD model with the parameter-driven model with Normal distributions for the observables and the parameters, denoted as KF, indicates the merits of deterministic updating with a proper specification of the data structure over more flexibility of the parameter-driven models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this case, the TVP-SIRD model performs significantly better than the KF model for up to 10 days, and the two perform statistically indifferent for longer horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, the trade-off between the flexible and less flexible models is apparent when we compare short and long-horizon performances of the regressions with a 30-day moving window versus a 60-day moving window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' While the regressions with a 30-day moving window outperform the counterpart with a 60-day moving window for the predictions up to two weeks due to flexibility, the latter model outperforms the former due to the reduction in the variance despite the increasing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the results involving RMSFEs of the competing models relative to the TVP- SIRD model when we consider the prediction of the daily death cases in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 28 [Insert Table 3 about here] Table 3 indicates mixing results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' First, comparing the TVP-SIRD model with the TVP- SIRD-β model indicates the importance of the time variation in the death rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this case, the TVP-SIRD model with the time-varying death rate outperforms the TVP-SIRD-β with the fixed death rate at all horizons, and this outperformance is statistically signifi- cant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Comparison of the TVP-SIRD model with the KF model indicates that the former outperforms the latter significantly at all horizons except the 1-day ahead forecast, indi- cating the importance of proper modeling of pandemic count data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, we compare the TVP-SIRD model performance with the regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The TVP-SIRD model performs better than the regression model with a 60-day moving window at all horizons except the 30 days horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' When the window size is shortened to 45 and 30-day leading to more flexibly changing parameters, the superior performance of the TVP-SIRD model remains significant at longer horizons exceeding ten days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, the predictions become statistically indifferent for short horizons thanks to the flexibility of these regression models with shorter windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Overall, it seems that the flexibility of the TVP-SIRD model pays off even more at longer horizons for predicting the daily death cases compared to confirmed cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 Predictive performance at weekly frequency In the previous section, we display a horse race for the predictive performances of a set of competing models at a daily frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, since the outburst of the pandemic, many models, including various forms of epidemiological models, curve fitting frameworks, or ma- chine learning setups, have predicted the pandemic’s key variables, including confirmed and death cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Luckily, the CDC-funded Influenza Forecasting Centers of Excellence worked closely with global, federal, state, and local public health officials to integrate infectious disease forecasting in a so-called forecast hub providing predictions of the outstanding fore- casting sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In this section, we compare the TVP-SIRD model with these prominent competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Since these forecasts are provided at weekly frequency, we estimate the TVP- SIRD model at the weekly frequency in real time using vintage data as in the previous 22The predictive performance of the daily death cases is closely related to the number of ICU patients due to Covid-19, which is a critical factor for the decision-makers, with a correlation exceeding 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The predictive results of forecasting the number of ICU patients are very similar to those of death cases, and these are presented in Section H of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 23See https://covid19forecasthub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='org/doc/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='Wethankananonymousrefereeforpointingoutthissource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' for further details on this initiative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 29 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='24 We discard the forecasts that have less than 30 forecast readings, leaving us with 19 forecast sources for the prediction of the weekly confirmed cases and 28 for the prediction of the weekly death cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the forecast sources that range from Microsoft to MIT-based models in the supplementary material in Table G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the number of models that our TVP-SIRD model outperforms in Table 4 for horizons including h = 1, 2, 3, 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', from the 1-week horizon up to the 1-month horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Table 4 about here] Table 4 reveals that in the short horizons, the TVP-SIRD model can beat the majority of these outstanding forecast sources with 11 outperformance out of 19 sources for the prediction of the confirmed cases and 19 out of 28 sources for the prediction of the death cases considering 1-week horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, this superior predictive ability monotonically erodes with the increasing forecast horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In line with the prediction results using daily frequency, the TVP-SIRD model is more successful in predicting the death cases at long horizons compared to the confirmed cases with better performance than 30% (20%) of the forecast sources at 3-week (4-week) horizon for death cases versus 15% (10%) for confirmed cases prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we conclude that the TVP-SIRD model successfully predicts the critical Covid-19 pandemic-related variables at the short and medium horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' It performs comparably to the leading pandemic forecasting tools at long horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We provide a more detailed picture of the dynamic performance of the TVP-SIRD model over time compared to the forecasting tools in Figure 7 for the 1-week horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='25 [Insert Figure 7 about here] Rather than providing pairwise comparisons with every single model, Figure 7 displays the evolution of relative RMSFEs of the TVP-SIRD model with an ensemble model (EM) over time where relative RMSFEs (rRMSFE) are computed recursively in real time using vintage datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The EM is computed using a forecast combination scheme generated using the space of individual models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In line with the advantages of forecast combinations over individual models in many settings, the EM provides better predictions than the individual forecast sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Thus, it is a gold standard in predicting confirmed and death cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see https://covid19forecasthub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='org/doc/reports/ for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In Figure 7, we also include 24We also evaluate the weekly forecasts by aggregating our daily forecasts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, this yields worse results compared to using weekly data for estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we provide the forecasting results regarding weekly data setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 25We display the results for longer horizons in Section G of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 30 the actual number of cases to compare the relative predictive performance taking the timing of the various phases of the pandemic into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In the left and right panels of Figure 7, we display the RMSFE of the TVP-SIRD model relative to the EM for the prediction of the confirmed cases and death cases, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' A value lower than unity indicates the better performance of the TVP-SIRD model relative to EM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Considering the confirmed cases, on average, the TVP-SIRD model performs closely to the EM as the rRMSFE is very close to unity in most periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' A striking finding is that the TVP-SIRD model performs better than the EM, specifically at the onset of the pandemic waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This result indicates that the TVP-SIRD model provides timely predictions at the onset of the pandemic waves reflecting the flexible model structure that can immediately accommodate the changing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, this picture reverses when the pandemic wave is at its peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Once the data on the new pandemic wave is accumulated, the EM provides better predictions, especially on the timing of the turning point down the hill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Focusing on predicting weekly death cases at 1-week horizon, we observe that EM performs much better than the individual forecast sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Unlike the previous comparison of the TVP-SIRD model to individual forecast sources, the EM model performs better than the TVP-SIRD model over time as relative RMSFEs exceed unity persistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Still, the pattern discussed in the prediction of the confirmed cases is also apparent here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Again, the performance of the TVP-SIRD model improves at the onset of the pandemic waves and deteriorates once the pandemic’s peak is passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Overall, our results indicate that the TVP-SIRD model performs favorably well at daily and weekly frequency against very compelling competitors in forecasting of key Covid-19 pandemic aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 6 Potential extensions In the previous sections, we display the potential of the TVP-SIRD model both in in- sample fit and out-of-sample forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' This section provides a potential extension to the TVP-SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Since the pandemic is a global phenomenon, multiple countries have had common experiences, with some countries having relatively larger part of idiosyncratic variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Departing from this observation, we extend the model to a multi-country setting 31 using a factor model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Consider the following model for country i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' , K, ∆Ci,t|Ωt−1 ∼ Poisson(βi,t Si,t−1 Ni Ii,t−1) ∆Rci,t|Ωt−1 ∼ Poisson(γi,tIi,t−1) ∆Di,t|Ωt−1 ∼ Poisson(νi,tIi,t−1) ∆Ii,t = ∆Ci,t − ∆Rci,t − ∆Di,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (25) We assume that country-specific parameters of the TVP-SIRD model admit a factor struc- ture for their level, while the seasonal patterns are idiosyncratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='26 Specifically, consider the decomposition as in (7), where the level component evolves according to the following factor structure θi,l,t = τi,lθl,t + ˆθi,l,t θl,t = θl,t−1 + αθlsθl,t−1 ˆθi,l,t = ˆθi,l,t−1 + αˆθi,lsˆθi,l,t−1 (26) for parameter θt = ˜βt, ˜γt and ˜νt, and θl,t is the common level component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' the key difference between the factor model and a country-specific model is that common level factor θl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t is loaded by all country-specific information with the coefficient τi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='l for country i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 27 and thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' the corresponding score function becomes s˜β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = ∇1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ˜β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t+···+∇i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ˜β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t+···+∇K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' ˜β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t S ˜β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t s˜γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = ∇1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t+···+∇i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t+···+∇K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t S˜γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t s˜ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = ∇1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t+···+∇i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t+···+∇K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t S˜ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t (27) where ∇i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = (∆Ci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t − λi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t) τ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='i ∇i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = (∆Rci,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t − λi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t) (1 − γi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t)τ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='i ∇i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='˜ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = (∆Di,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t − λi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t) (1 − νi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t)τ3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='i (28) and S˜β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='tτ 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 + · · · + λK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='tτ 2 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='K S˜γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t(1 − γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t)2τ 2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 + · · · + λK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t(1 − γK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t)2τ 2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='K S˜ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t = λ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t(1 − ν1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t)2τ 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 + · · · + λK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t(1 − νK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='t)2τ 2 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (29) 26Imposing a factor structure to the seasonal component is a straightforward extension of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, our experience with this model shows that identifying a common seasonal factor poses more challenges than a level factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Since the level factor is the key component of the parameters, we consider a factor structure only in the level component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 27In this extension, we opt for a factor structure in the parameters similar to seasonality modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Alternatively, we could also proceed with a factor representation of the data, using principal components, for example, and a SIRD model corresponding to each component, similar to factor GARCH models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' see Zhang and Chan (2009) for such a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 32 The score functions of the idiosyncratic parts are the same as in the TVP-SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We provide details on these derivations in Section D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 of the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We denote this model as the factor TVP-SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' In the following application, we only consider a factor structure in the infection rate, βt, keeping the remaining parameters as wholly idiosyncratic as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, the extension of the factor structure to the re- maining parameters is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Still, the parameters in their current form are not identified, as none of the components are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' To identify the factors and idiosyncratic compo- nents separately, we set τ1 as one for the first country and fix the initial condition for the common factor, which enables the identification of the location of the factor and idiosyn- cratic components separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We consider four countries in the application: US, Germany, Italy, and Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the evolution of the number of daily active infected cases for Germany, Italy, and Brazil in Figure 8, while we provide a comprehensive analysis of the US in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Figure 8 about here] Figure 8 indicates that the pandemic’s evolution in Europe, represented by Germany and Italy follows a similar trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' On the other hand, Brazil’s pandemic trajectory exhibits a unique pattern, counter to Germany and Italy at times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Still, the countries’ patterns converge with the last wave, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', the Omicron wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the estimates of the fixed parameters related to the common factor in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the evolution of the common factor infection rate and country-specific effective reproduction rates, eRts, in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' [Insert Table 5 and Figure 9 about here] Table 5 indicates that the common factor is affected by the past score function, derived in (27), considerably with a coefficient close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6 leading to a time-varying pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' However, the 95% HPDI covers a wide range of values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Factor loadings for Germany and Italy are very similar, as expected, with values around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8 and bear little uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The lowest loading is for Brazil with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 with almost 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 for the bounds of 95% HPDI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The upper left panel of Figure 9 displays the evolution of the common factor of infection rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Following the estimates of factor loadings, this factor is mainly influenced by the pandemic trajectory of the US, Germany, and Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We can observe that the common factor 33 can nicely capture all significant waves with the relatively higher values corresponding to the initial wave at the onset of the pandemic and the recent two waves, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=', Delta and Omicron waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The common impact of these two variants can also be observed by the increased eRt of the US, Germany, and Italy, particularly for the Delta variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Finally, the relatively more idiosyncratic behavior of the pandemic in Brazil can be traced by the corresponding eRt in the lower right panel of Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' For Brazil, the eRt fluctuates around one for a large part of the sample period leading to the unique pattern of active infected cases as shown in the most right panel of Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These results show the efficacy of the factor TVP-SIRD model in capturing both the common and idiosyncratic patterns of the Covid-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 7 Conclusion This paper puts forward the time-varying parameters SIRD model for timely and accurate measurement of the pandemic’s current stance and accurate predictions of its future tra- jectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our modeling framework falls into the class of ’generalized autoregressive score models’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' These models involve parameters evolving deterministically according to an au- toregressive process in the direction implied by the score function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore the resulting approach permits a flexible yet parsimonious and statistically coherent framework to oper- ate efficiently in scarce data environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We demonstrate the proposed model’s potential using daily and weekly US data using full sample estimation and out-of-sample forecasting using a recursive real-time prediction exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our results show that the proposed framework can nicely track the stance of the pan- demic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Our findings suggest that there is considerable fluctuation in the rate of infection and death rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We further extend the model to include the infected individuals who do not show symptoms and are therefore not diagnosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We show that this sample 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Infectious Disease Modelling 5: 563–574.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Zhang K, Chan L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Efficient factor garch models and factor-dcc models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Quantitative Finance 9: 71–91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Zhang Y, You C, Cai Z, Sun J, Hu W, Zhou XH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Prediction of the covid-19 outbreak based on a realistic stochastic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' medRxiv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 38 Tables and Figures Table 1: Estimation results of the SIRD model with fixed parameters and the TVP-SIRD model Panel A: Fixed parameters SIRD model Panel B: TVP-SIRD model βl γl (×10−1) νl (×10−2) R0 αβt αγt ανt Median 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0133 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3514 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0131 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3375 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0134 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3682 Note: The table displays the estimation results of the model in (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the posterior median and the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% percentiles of the posterior distributions of the corresponding parameter shown in the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Table 2: Relative RMSFEs of the competing models relative to the TVP- SIRD model - Daily confirmed cases RW−30 RW−45 RW−60 KF TVP-SIRD-β h = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='111 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='414 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='826 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='443 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='251 h = 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='747 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='959 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='162 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='206 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='183 h = 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='183 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='550 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='562 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='197 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='088 h = 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='317 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='971 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='027 h = 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='054 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='967 h = 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='829 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='002 h = 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='973 Note: The table displays the Root Mean Squared Forecast Errors (RMSFE) of the competing models in predicting the daily confirmed cases relative to the TVP-SIRD model introduced in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' RW-M stands for Rolling Window with M observations as the sample size for M = 30, 45, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' KF stands for the time-varying parameter version of the SIRD model using a state space model framework with Normal er- ror terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' TVP-SIRD-β denotes the restricted version of the TVP-SIRD model, where we allow for time variation only in the infection rate, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Statistical signifi- cance of relative Root Mean Squared Forecast Errors (RMSFE) is tested using the Diebold-Mariano (DM) test using the measures of squared forecast error contri- butions together with the HAC covariance matrix and a finite sample correction, Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The bold values are statistically INsignificant at the con- ventional significance level of 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 39 Table 3: Relative RMSFEs of the competing models relative to the TVP- SIRD model - Daily death cases RW−30 RW−45 RW−60 KF TVP-SIRD-β h = 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='974 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='051 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='310 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='776 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='055 h = 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='288 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='285 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='852 h = 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='183 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='060 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='413 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='332 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='799 h = 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='147 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='105 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='336 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='175 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='642 h = 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='130 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='170 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='326 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='186 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='566 h = 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='096 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='110 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='214 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='301 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='266 h = 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='074 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='068 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='097 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='294 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='099 Note: The table displays the Root Mean Squared Forecast Errors (RMSFE) of the competing models in predicting the daily death cases relative to the TVP-SIRD model introduced in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' RW-M stands for Rolling Window with M observations as the sample size for M = 30, 45, 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' KF stands for the time-varying parameter version of the SIRD model using a state space model framework with Normal er- ror terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' TVP-SIRD-β denotes the restricted version of the TVP-SIRD model, where we allow for time variation only in the infection rate, β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Statistical signifi- cance of relative Root Mean Squared Forecast Errors (RMSFE) is tested using the Diebold-Mariano (DM) test using the measures of squared forecast error contri- butions together with the HAC covariance matrix and a finite sample correction, Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The bold values are statistically INsignificant at the con- ventional significance level of 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Table 4: Number of models in the Forecast Hub that the TVP-SIRD model outperforms 1-week 2-week 3-week 4-week Confirmed cases (19) 11 5 3 2 Death cases (28) 19 14 9 5 Note: The table displays the number of the models in the Forecast Hub with more than 30 predictions that have greater RMSFE than the TVP-SIRD model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The total number of the models considered in the comparison is indicated in the left column in the parenthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Table 5: Estimation results of the factor TVP-SIRD model αβl,t τGer τIt τBr Median 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7840 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2982 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7409 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='0722 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5242 Note: The table displays the estimation results of the model in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' We display the posterior median and the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% and 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5% percentiles of the posterior distributions of the corresponding parameter shown in the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 40 Figure 1: The evolution of the daily number of confirmed and death cases in the US Confirmed Death 0 200 400 600 800 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='200 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='400 0 200 400 600 800 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='200 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='400 4/23/20 7/22/20 10/20/20 1/18/21 4/18/21 7/17/21 10/15/21 1/13/22 0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 0 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 4/23/20 7/22/20 10/20/20 1/18/21 4/18/21 7/17/21 10/15/21 1/13/22 Note: The graphs show the evolution of the daily confirmed and death cases in the US over the sample from March 2020 until the end of March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Figure 2: The evolution of the effective reproduction rate, eRt, estimated using the FP- SIRD model Jul 2020 Jan 2021 Jul 2021 Jan 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6 Note: The graphs show the evolution of the effective reproduction rate, eRt, estimated using the SIRD model with fixed parameters displayed in (2) in the US over the sample from March 2020 until the end of March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The 95% (HPDI) Highest Posterior Density Intervals are computed using the posterior output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Figure 3: The fitted values of the daily number of confirmed and death cases using the TVP-SIRD model Confirmed Death 0 200 400 600 800 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='200 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='400 0 200 400 600 800 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='000 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='200 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='400 4/23/20 7/22/20 10/20/20 1/18/21 4/18/21 7/17/21 10/15/21 1/13/22 0 1 2 3 4 5 6 0 1 2 3 4 5 6 4/23/20 7/22/20 10/20/20 1/18/21 4/18/21 7/17/21 10/15/21 1/13/22 Note: The graphs show the evolution of the daily confirmed and death cases in the US and the fitted values using the TVP-SIRD model over the sample from March 2020 until the end of March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 41 Figure 4: The evolution of the level values for infection and death rates and effective reproduction rate, βl,t, νl,t, and eRt, over the sample from March 2020 until March 2022 βl,t Mar 31 Apr 14 Apr 28 May 12 May 26 2020 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='25 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='06 νl,t Mar 31 Apr 14 Apr 28 May 12 May 26 2020 0 1 2 3 4 5 6 7 10-3 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 1 2 3 4 5 6 7 8 9 10-4 eRl,t Mar 31 Apr 14 Apr 28 May 12 May 26 2020 01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 5 10 15 20 25 30 35 (2) (1) Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 1 2 3 4 5 6 7 (3) (4) (5) (6) (7) Note: The graphs show the evolution of the time-varying parameters, βl,t, the rate of infection νl,t, the death rate, and the effective reproduction rate, eRt, estimated using the TVP-SIRD model introduced in (6) for the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The 95% (HPDI) Highest Posterior Density Intervals are computed using the posterior output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Figure 5: The evolution of the ratio of total infections to the number of reported infected cases Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 1 2 3 4 5 6 7 8 9 10 11 Note: The graph shows the evolution of the ratio of total infections to the number of reported infected cases estimated using the MF-TVP-SIRD model introduced in (17) for the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The 95% (HPDI) Highest Posterior Density Intervals are computed using the posterior output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 42 Figure 6: The evolution of the level values for the death rate and effective reproduction rate, νl,t and eRl,t starting from March 2020 until March 2022 νl,t Apr 14 Apr 21 Apr 28 May 05 May 12 May 19 May 26 Jun 02 2020 0 1 2 3 4 5 6 7 8 10-3 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 3 10-3 eRl,t Apr 14 Apr 21 Apr 28 May 05 May 12 May 19 May 26 Jun 02 2020 0 1 2 3 4 5 6 7 8 9 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 5 10 15 20 25 Note: The graphs show the evolution of the time-varying parameters, νl,t, the death rate, and eRl,t, the effective reproduction rate, estimated using the MF-TVP-SIRD model introduced in (17) for the US.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The 95% (HPDI) Highest Posterior Density Intervals are computed using the posterior output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Figure 7: The evolution of the relative RMSFE of the ensemble model’s 1-week ahead predictions relative to those of the TVP-SIRD model Weekly confirmed cases Weekly death cases Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 104 Note: The graphs show the evolution of the relative Root Mean Squared Forecast Error (rRMSFE) for the weekly 1-week ahead predictions of the ensemble model from Forecast-Hub relative to the TVP-SIRD model estimated using weekly data for the period starting from July 2020 until March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The solid line shows the rRMSFEs computed using expanding window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The dashed line indicates actual realizations of weekly confirmed and death cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 43 Figure 8: The evolution of the number of active infected cases in countries used for factor TVP-SIRD model estimation Germany Italy Brazil Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 106 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 106 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 106 Note: The graphs show the evolution of the active infected cases in Germany, Italy, and Brazil over the sample from March 2020 until the end of March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' The number of recovered cases is absent in all countries in half of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Therefore, we use γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='07, corresponding to a recovery duration of 14 days when computing the active infected cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' Figure 9: The evolution of common infection rate and country-specific eRts Infection rate factor eRt of Germany Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='6 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 eRt of Italy eRt of Brazil Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 3 Apr 2020 Jul 2020 Oct 2020 Jan 2021 Apr 2021 Jul 2021 Oct 2021 Jan 2022 Apr 2022 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content='5 Note: The graphs show the evolution of the common infection rate and country-specific eRts for Germany, Italy, and Brazil over the sample from March 2020 until the end of March 2022 using the factor TVP-SIRD model as introduced in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} +page_content=' 44' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XtFRT4oBgHgl3EQf-zgt/content/2301.13692v1.pdf'} diff --git a/Z9E5T4oBgHgl3EQfDg58/vector_store/index.pkl b/Z9E5T4oBgHgl3EQfDg58/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d4bb21bd53b846577c90f1e365d866e711521304 --- /dev/null +++ b/Z9E5T4oBgHgl3EQfDg58/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:12bbdac23f1ab1ff73d7aec96a5dbaf3e0549f861e42e0cf74c7cd2163a04522 +size 152304 diff --git a/ZNAyT4oBgHgl3EQfifg0/vector_store/index.pkl b/ZNAyT4oBgHgl3EQfifg0/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..c1234668ef28b45c6856ee8edadb0828215a897a --- /dev/null +++ b/ZNAyT4oBgHgl3EQfifg0/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b09996eefdc90b6044199c9144aa2ca791fb5a266cacf9dc1697a8cc3a4eae14 +size 249063 diff --git a/_NFLT4oBgHgl3EQfDi7Y/content/tmp_files/2301.11980v1.pdf.txt b/_NFLT4oBgHgl3EQfDi7Y/content/tmp_files/2301.11980v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2aa91c290595a19f017172a4c51b37cd1cc4bfbb --- /dev/null +++ b/_NFLT4oBgHgl3EQfDi7Y/content/tmp_files/2301.11980v1.pdf.txt @@ -0,0 +1,218 @@ +A semi-analytical model to simulate the spin-diode effect and +accelerate its use in neuromorphic computing +Chlo´e Chopin1, Leandro Martins2, Luana Benetti2, Simon de Wergifosse1, Alex Jenkins2, +Ricardo Ferreira2, and Flavio Abreu Araujo1 +1Institute of Condensed Matter and Nanosciences, UCLouvain, Louvain-la-Neuve, Belgium, chloe.chopin@uclouvain.be +2International Iberian Nanotechnology Laboratory, Braga, Portugal +The spin-diode effect is studied both experimentally and with our original semi-analytical method. The latter is based on an +improved version of the Thiele equation approach (TEA) that we combine to micromagnetic simulation data to accurately model +the non-linear dynamics of spin-torque vortex oscillator (STVO). This original method, called data-driven Thiele equation approach +(DD-TEA), absorbs the difference between the analytical model and micromagnetic simulations to provide a both ultra-fast and +quantitative model. The DD-TEA model predictions also agree very well with the experimental data. The reversal of the spin-diode +effect with the chirality of the vortex, the impact of the input current and the origin of a variation at half of the STVO frequency are +presented as well as the ability of the model to reproduce the experimental behavior. Finally, the spin-diode effect and its simulation +using the DD-TEA model are discussed as a promising perspective in the framework of neuromorphic computing. +Index Terms—Neuromorphic, Spin-diode effect, Spintronics, Vortex. +I. INTRODUCTION +S +PIN-TORQUE +vortex-based +oscillators +(STVOs) +are +nanoscale devices with high potential for applications like +radio-frequency (rf)-generation [1], rf detection [2], or neuro- +morphic computing [3]. They are based on the magnetic tunnel +junction (MTJ) composed of two magnetic layers decoupled +by an insulating spacer (see Fig. 1). One of the magnetic +layer is called the polarizer and has a fixed magnetization +while the other magnetic layer is called the free layer as its +magnetization can be freely controlled using a dc current for +example. By carefully choosing the geometry of the free layer +of the MTJ, a magnetic vortex can be nucleated as the nano- +dot magnetization ground state [4]. It has an in-plane curling +magnetization except at the vortex core where it points out-of- +plane. Depending on the circulation direction of the magneti- +zation which is either anti-clockwise or clockwise, a vortex has +either a positive or negative chirality. In addition, a vortex has +a positive or negative polarity depending on the magnetization +of the vortex core (either pointing up or down, respectively). +Its dynamics is impacted by the two topological parameters +mentioned above, by the polarizer orientation and by the input +current. Furthermore, the latter generates an Amp`ere-Oersted +field (AOF) which can not be neglected as it modifies the vor- +tex core dynamics depending on its chirality [5], [6]. A spin- +diode effect [7] arises in a STVO when a rf current is injected +into the device. The combination of the input current and the +oscillation of magnetization of the MTJ due to the vortex core +motion gives rise to an output oscillating voltage measured by +tunnel magnetoresistance. The dc component, which depends +on the input current frequency, is then extracted (see Fig. 2). +This phenomenon is suitable for applications like non-volatile +memory [8] or neuromorphic computing [3]. The spin-diode +effect can be easily measured experimentally and understood +at some extend using micromagnetic simulations [8]. However, +micromagnetic simulations are very time consuming and the +h +z +X +R +Y +X +Fig. 1. +A spin-torque vortex oscillator is represented with the polarizer in +green (its magnetization is along the y-axis), the insulating layer in gray and +the free layer in blue. The chirality of the vortex core is represented by white +arrows. The AOF field generated by the input current is symbolized by gray +arrows. Adapted from [9] +. +results offer low resolution in terms of rf frequency in contrast +to what is needed to explain the experimental features. So, +an original semi-analytical model called data-driven Thiele +equation approach (DD-TEA) [9] is used to fill this gap. This +DD-TEA model is a promising tool to study the spin-diode +effect. Here, its predictions are compared to experimental data +and its potential use in neuromorphic computing is presented. +II. METHODS +Experimental data are obtained for a STVO with a radius of +R = 500 nm and a free layer with a thickness of h = 9 nm. +The polarizer is along the y-axis (see Fig. 1). The DD-TEA +model [9] is based on the TEA [10] where the vortex is seen +as a quasi-particle and the evolution of the vortex core position +X is defined by +G(ez × ˙X) + D ˙X = ∂(W ex + W ms + W Oe) +∂X ++ FST, +(1) +arXiv:2301.11980v1 [cond-mat.mes-hall] 27 Jan 2023 + +where G and D are the gyro-vector and the damping terms, +W ex, W ms, W Oe are respectively the exchange, magnetostatic, +and the AOF contribution to the potential energy, and FST is +the spin-torque force. Due to the assumptions used in TEA [6], +only qualitative results are obtained [5]. Thanks to a limited +amount of micromagnetic simulations and our data-driven +approach, the DD-TEA model allows to achieve both fast and +quantitative results by absorbing the difference between the +pure analytical model and micromagnetic simulations data [9]. +The spin-diode effect is computed as follow +∆V ≃ ∆RSTVO +2 +Iac +T2 − T1 +� T2 +T1 +∆my sin(2πfrft)dt, +(2) +with ∆V the rectified voltage, ∆RSTVO the device resistance +variation, Iac and frf the input current amplitude and frequency, +T1 and T2 two moments in time and ∆my the variation of +the normalized magnetization computed from the vortex core +position. The experimental results and the DD-TEA model +predictions are shown in Fig. 2. +III. RESULTS +Figure 2 shows a very nice agreement between experimental +data and DD-TEA predictions. The experimental measure- +ments show a reversal of the spin-diode effect depending on +the vortex chirality. This reversal comes from the impact of the +AOF [8] and our ultra-fast DD-TEA model is able to capture +this fine-grained feature. Also, a bump can be seen around +half of the STVO frequency in the experimental data and the +origin of this variation is revealed by the model. Thanks to +its speed, the rf input frequency interval is small enough so +the model offers a high resolution. This allows to capture this +phenomenon with, in addition, a complete knowledge of the +vortex core dynamics leading to this behavior. It can then be +shown that this behavior is due to a fractional synchronization +pattern [11]. The combination of a STVO with the spin-diode +effect can be used either to build synapses with non-volatile +memory [8], as the information is stored in the vortex chirality, +or as neuron with its non-linear dynamics. Furthermore, DD- +TEA can simulate a neuron with two different activation +functions depending on the vortex chirality. +IV. CONCLUSION +Our DD-TEA model is used to predict the vortex core +dynamics. It successfully shows and explains the spin-diode +reversal effect, the impact of the input current and even +explains experimental features, namely the fractional syn- +chronization phenomenon. These features can be used for +neuromorphic computing either as neurons or synapses. As +this model is both ultra-fast and quantitative, the next step is +to apply it for the simulation of full neuromorphic circuits. +ACKNOWLEDGEMENT +Computational resources have been provided by the Con- +sortium des Equipements de Calcul Intensif (CECI), funded +by the Fonds de la Recherche Scientique de Belgique (F.R.S.- +FNRS) under Grant No. 2.5020.11 and by the Walloon Region. +F.A.A. is a Research Associate and S.d.W. is a FRIA grantee, +both of the F.R.S.-FNRS. +6 +4 +2 +0 +2 +4 +6 +C + +20 +40 +60 +80 +100 +120 +RF source frequency fRF (MHz) +6 +4 +2 +0 +2 +4 +6 +C +Spin-diode Voltage (mV) +Model prediction +Experimental data +Fig. 2. Spin-diode voltage as a function of the input current frequency from +(top) experimental results and (bottom) model predictions. Curves in red (resp. +blue) correspond to positive (resp. negative) chirality. The input power ranges +between 0 dBm (darkest curve) and -5 dBm (lightest curve) with a step of +-1 dBm. +REFERENCES +[1] A. Dussaux, B. Georges, J. Grollier, V. Cros, A. Khvalkovskiy, +A. Fukushima, M. Konoto, H. Kubota, K. Yakushiji, S. Yuasa et al., +“Large microwave generation from current-driven magnetic vortex os- +cillators in magnetic tunnel junctions,” Nature Communications, vol. 1, +no. 1, pp. 1–6, 2010. +[2] D. Markovi´c, N. Leroux, A. Mizrahi, J. Trastoy, V. Cros, P. Bortolotti, +L. Martins, A. Jenkins, R. Ferreira, and J. Grollier, “Detection of the +microwave emission from a spin-torque oscillator by a spin diode,” +Physical Review Applied, vol. 13, no. 4, p. 044050, 2020. +[3] J. Grollier, D. Querlioz, K. Camsari, K. Everschor-Sitte, S. Fukami, and +M. D. Stiles, “Neuromorphic spintronics,” Nature Electronics, vol. 3, +no. 7, pp. 360–370, 2020. +[4] K. Y. Guslienko, “Magnetic vortex state stability, reversal and dynamics +in restricted geometries,” Journal of Nanoscience and Nanotechnology, +vol. 8, no. 6, pp. 2745–2760, 2008. +[5] F. Abreu Araujo, C. Chopin, and S. de Wergifosse, “Ampere–oersted +field splitting of the nonlinear spin-torque vortex oscillator dynamics,” +Scientific Reports, vol. 12, no. 1, pp. 1–8, 2022. +[6] S. de Wergifosse, C. Chopin, and F. Abreu Araujo, “Quantitative and +realistic description of the magnetic potential energy of spin-torque +vortex oscillators,” arXiv preprint arXiv:2206.13438, 2022. +[7] A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, +K. Tsunekawa, D. Djayaprawira, N. Watanabe, and S. Yuasa, “Spin- +torque diode effect in magnetic tunnel junctions,” Nature, vol. 438, no. +7066, pp. 339–342, 2005. +[8] L. Martins, A. S. Jenkins, L. S. E. Alvarez, J. Borme, T. B¨ohnert, +J. Ventura, P. P. Freitas, and R. Ferreira, “Non-volatile artificial synapse +based on a vortex nano-oscillator,” Scientific Reports, vol. 11, no. 1, pp. +1–7, 2021. +[9] F. Abreu Araujo, C. Chopin, and S. de Wergifosse, “Data-driven thiele +equation approach for solving the full nonlinear spin-torque vortex +oscillator dynamics,” arXiv preprint arXiv:2206.13596, 2022. +[10] A. Thiele, “Steady-state motion of magnetic domains,” Physical Review +Letters, vol. 30, no. 6, p. 230, 1973. +[11] S. Urazhdin, P. Tabor, V. Tiberkevich, and A. Slavin, “Fractional syn- +chronization of spin-torque nano-oscillators,” Physical Review Letters, +vol. 105, no. 10, p. 104101, 2010. + diff --git a/_NFLT4oBgHgl3EQfDi7Y/content/tmp_files/load_file.txt b/_NFLT4oBgHgl3EQfDi7Y/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8638db0cbe8e341a4baca83a86a5b0b4a1b2c740 --- /dev/null +++ b/_NFLT4oBgHgl3EQfDi7Y/content/tmp_files/load_file.txt @@ -0,0 +1,193 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf,len=192 +page_content='A semi-analytical model to simulate the spin-diode effect and accelerate its use in neuromorphic computing Chlo´e Chopin1, Leandro Martins2, Luana Benetti2, Simon de Wergifosse1, Alex Jenkins2, Ricardo Ferreira2, and Flavio Abreu Araujo1 1Institute of Condensed Matter and Nanosciences, UCLouvain, Louvain-la-Neuve, Belgium, chloe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='chopin@uclouvain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='be 2International Iberian Nanotechnology Laboratory, Braga, Portugal The spin-diode effect is studied both experimentally and with our original semi-analytical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The latter is based on an improved version of the Thiele equation approach (TEA) that we combine to micromagnetic simulation data to accurately model the non-linear dynamics of spin-torque vortex oscillator (STVO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' This original method, called data-driven Thiele equation approach (DD-TEA), absorbs the difference between the analytical model and micromagnetic simulations to provide a both ultra-fast and quantitative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The DD-TEA model predictions also agree very well with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The reversal of the spin-diode effect with the chirality of the vortex, the impact of the input current and the origin of a variation at half of the STVO frequency are presented as well as the ability of the model to reproduce the experimental behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Finally, the spin-diode effect and its simulation using the DD-TEA model are discussed as a promising perspective in the framework of neuromorphic computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Index Terms—Neuromorphic, Spin-diode effect, Spintronics, Vortex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' INTRODUCTION S PIN-TORQUE vortex-based oscillators (STVOs) are nanoscale devices with high potential for applications like radio-frequency (rf)-generation [1], rf detection [2], or neuro- morphic computing [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' They are based on the magnetic tunnel junction (MTJ) composed of two magnetic layers decoupled by an insulating spacer (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' One of the magnetic layer is called the polarizer and has a fixed magnetization while the other magnetic layer is called the free layer as its magnetization can be freely controlled using a dc current for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' By carefully choosing the geometry of the free layer of the MTJ, a magnetic vortex can be nucleated as the nano- dot magnetization ground state [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' It has an in-plane curling magnetization except at the vortex core where it points out-of- plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Depending on the circulation direction of the magneti- zation which is either anti-clockwise or clockwise, a vortex has either a positive or negative chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' In addition, a vortex has a positive or negative polarity depending on the magnetization of the vortex core (either pointing up or down, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Its dynamics is impacted by the two topological parameters mentioned above, by the polarizer orientation and by the input current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Furthermore, the latter generates an Amp`ere-Oersted field (AOF) which can not be neglected as it modifies the vor- tex core dynamics depending on its chirality [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' A spin- diode effect [7] arises in a STVO when a rf current is injected into the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The combination of the input current and the oscillation of magnetization of the MTJ due to the vortex core motion gives rise to an output oscillating voltage measured by tunnel magnetoresistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The dc component, which depends on the input current frequency, is then extracted (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' This phenomenon is suitable for applications like non-volatile memory [8] or neuromorphic computing [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The spin-diode effect can be easily measured experimentally and understood at some extend using micromagnetic simulations [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' However, micromagnetic simulations are very time consuming and the h z X R Y X Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' A spin-torque vortex oscillator is represented with the polarizer in green (its magnetization is along the y-axis), the insulating layer in gray and the free layer in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The chirality of the vortex core is represented by white arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The AOF field generated by the input current is symbolized by gray arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Adapted from [9] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' results offer low resolution in terms of rf frequency in contrast to what is needed to explain the experimental features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' So, an original semi-analytical model called data-driven Thiele equation approach (DD-TEA) [9] is used to fill this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' This DD-TEA model is a promising tool to study the spin-diode effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Here, its predictions are compared to experimental data and its potential use in neuromorphic computing is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' METHODS Experimental data are obtained for a STVO with a radius of R = 500 nm and a free layer with a thickness of h = 9 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The polarizer is along the y-axis (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The DD-TEA model [9] is based on the TEA [10] where the vortex is seen as a quasi-particle and the evolution of the vortex core position X is defined by G(ez × ˙X) + D ˙X = ∂(W ex + W ms + W Oe) ∂X + FST, (1) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='11980v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='mes-hall] 27 Jan 2023 where G and D are the gyro-vector and the damping terms, W ex, W ms, W Oe are respectively the exchange, magnetostatic, and the AOF contribution to the potential energy, and FST is the spin-torque force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Due to the assumptions used in TEA [6], only qualitative results are obtained [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Thanks to a limited amount of micromagnetic simulations and our data-driven approach, the DD-TEA model allows to achieve both fast and quantitative results by absorbing the difference between the pure analytical model and micromagnetic simulations data [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The spin-diode effect is computed as follow ∆V ≃ ∆RSTVO 2 Iac T2 − T1 � T2 T1 ∆my sin(2πfrft)dt, (2) with ∆V the rectified voltage, ∆RSTVO the device resistance variation, Iac and frf the input current amplitude and frequency, T1 and T2 two moments in time and ∆my the variation of the normalized magnetization computed from the vortex core position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The experimental results and the DD-TEA model predictions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' RESULTS Figure 2 shows a very nice agreement between experimental data and DD-TEA predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The experimental measure- ments show a reversal of the spin-diode effect depending on the vortex chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' This reversal comes from the impact of the AOF [8] and our ultra-fast DD-TEA model is able to capture this fine-grained feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Also, a bump can be seen around half of the STVO frequency in the experimental data and the origin of this variation is revealed by the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Thanks to its speed, the rf input frequency interval is small enough so the model offers a high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' This allows to capture this phenomenon with, in addition, a complete knowledge of the vortex core dynamics leading to this behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' It can then be shown that this behavior is due to a fractional synchronization pattern [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' The combination of a STVO with the spin-diode effect can be used either to build synapses with non-volatile memory [8], as the information is stored in the vortex chirality, or as neuron with its non-linear dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Furthermore, DD- TEA can simulate a neuron with two different activation functions depending on the vortex chirality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' CONCLUSION Our DD-TEA model is used to predict the vortex core dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' It successfully shows and explains the spin-diode reversal effect, the impact of the input current and even explains experimental features, namely the fractional syn- chronization phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' These features can be used for neuromorphic computing either as neurons or synapses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' As this model is both ultra-fast and quantitative, the next step is to apply it for the simulation of full neuromorphic circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' ACKNOWLEDGEMENT Computational resources have been provided by the Con- sortium des Equipements de Calcul Intensif (CECI), funded by the Fonds de la Recherche Scientique de Belgique (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='- FNRS) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='5020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='11 and by the Walloon Region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' is a Research Associate and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' is a FRIA grantee, both of the F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content='-FNRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 6 4 2 0 2 4 6 C + 20 40 60 80 100 120 RF source frequency fRF (MHz) 6 4 2 0 2 4 6 C Spin-diode Voltage (mV) Model prediction Experimental data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFLT4oBgHgl3EQfDi7Y/content/2301.11980v1.pdf'} +page_content=' Spin-diode voltage as a function of the input current frequency from (top) experimental results and (bottom) model predictions.' metadata={'source': 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Abreu∗ +Instituto Tecnológico de Aeronáutica, 12228–900, São José dos Campos, SP, Brazil +Carlos Junqueira-Junior† +Arts et Métiers Institute of Technology, DynFluid, CNAM, HESAM University, 151 Boulevard de l’Hôpital, 75013, Paris, +France +Eron T. V. Dauricio‡ +Instituto Tecnológico de Aeronáutica, 12228–900, São José dos Campos, SP, Brazil +João Luiz F. Azevedo§ +Instituto de Aeronáutica e Espaço, 12228–904, São José dos Campos, SP, Brazil +The present study is concerned with large-eddy simulations (LES) of supersonic jet flows. +The work addresses, in particular, the simulation of a perfectly expanded free jet flow with an +exit Mach number of 1.4 and an exit temperature equal to the ambient temperature. Calcula- +tions are performed using a nodal discontinuous Galerkin method. The present effort studies +the effects of mesh and polynomial refinement on the solution. The present calculations con- +sider computational meshes and plynomial orders such that the number of degrees of freedom +(DOFs) in the solution ranges from 50 to 410 million. Mean velocity results and root mean +square (RMS) values of velocity fluctuations indicate a better agreement with experimental +data as the resolution is increased. The generated data provide a good understanding of the +effects of increasing the discretization refinement for LES calculations of jet flows. The present +results can guide future simulations of similar flow configurations. +I. Introduction +With the progress of computing power in the last years, the large-eddy simulation (LES) formulation appears as +an alternative to Reynolds-averaged Navier-Stokes (RANS) methods due to its reasonable cost when compared to the +direct numerical simulation (DNS) of the Navier-Stokes equations or even physical experiments. LES can provide +valuable information on complex configurations such as shear layers [1, 2] and detached flows [3, 4] due to its capability +to generate unsteady data for flow and temperature fields with high-frequency fluctuations, which are necessary for +aerodynamics, acoustics, loads, and heat transfer analyses. +The authors are interested in the LES of jet flows from aircraft and rockets engines [5–7, 9, 10]. More specifically, +on the perfectly expanded configuration, when the jet exit pressure matches the ambient pressure, at 1.4 Mach +number. Recent work highlights [8] the effects of structured second-order finite-difference and unstructured nodal +discontinuous-Galerkin spatial discretizations [11, 12] on the flow of interest at a fixed number of degrees of freedom +(DOF). The results indicate good agreement with experimental and numerical data, where the spatial resolution is +sufficient and with the same order of error in the coarser mesh regions. Therefore, the current study addresses the effects +of refinement on the LES of a supersonic jet flow configuration using the FLEXI framework [13]. The solver applies +an unstructured nodal discontinuous Galerkin spatial discretization that allows evaluating the influence of mesh and +polynomial (hp) refinement. +∗Ph.D. Candidate, Graduate Program in Space Sciences and Technologies, Departamento de Ciência e Tecnologia Aeroespacial, DCTA/ITA; +E-mail: mecabreu@yahoo.com.br. +†Research Engineer, Arts et Métiers Institute of Technology, DynFluid laboratory; E-mail: junior.junqueira@ensam.eu. +‡Ph.D. Candidate, Graduate Program in Space Sciences and Technologies, Departamento de Ciência e Tecnologia Aeroespacial, DCTA/ITA; +E-mail: eron.tiago90@gmail.com. +§Senior Research Engineer, Aerodynamics Division, Departamento de Ciência e Tecnologia Aeroespacial, DCTA/IAE/ALA; E-mail: +joaoluiz.azevedo@gmail.com. Fellow AIAA. +1 +arXiv:2301.01582v1 [physics.flu-dyn] 4 Jan 2023 + +The literature [14–18] does not agree on the mesh requirements for adequately solving the LES formulation due +to employing different numerical methods for solving jet flows. The present paper studies the effects of mesh and +polynomial refinement along with mesh topology to identify the minimum mesh requirements for adequately solving the +problem of interest. The research group improved the baseline grid from Ref. [8] with local mesh refinement in the +vicinity of the lipline and with an increase in the number of elements, ranging from 6.2 × 106 to 15.4 × 106 elements. +The jet flow calculations use second-order and third-order polynomials. The simulations present 50 to 410 million +DOFs when combining grid and polynomial refinement. +The generated data for mean velocity and RMS of velocity fluctuations are investigated and compared with +experimental data [19] at different regions of the domain where the jet is developing. The paper is organized to introduce +the reader to the description of physical and numerical formulation in the second section. Then, one can find details of +the experimental configuration and the numerical setup in sections three and four. Finally, the results and the concluding +remarks close the work in sections five and six, respectively. +II. Numerical Formulation +A. Governing Equations +The work has the interest in the solution of the filtered Navier-Stokes equations. The filtering strategy is based on a +spatial filtering process that separates the flow into a resolved part and a non resolved part. Usually the filter size is +obtained from the mesh size. The filtered Navier-Stokes equations in conservative form can be written by +𝜕 ¯Q +𝜕𝑡 + ∇ · F( ¯Q, ∇ ¯Q) = 0, +(1) +where ¯Q = [ ¯𝜌, ¯𝜌 ˜𝑢, ¯𝜌˜𝑣, ¯𝜌 ˜𝑤, ¯𝜌 ˇ𝐸]𝑇 is the vector of filtered conserved variables and F is the flux vector. The flux vector +can be divided into the Euler fluxes and the viscous flux, F = F𝑒 − F𝑣. The fluxes with the filtered variables may be +written as +F𝑒 +𝑖 = +������������ +¯𝜌 ˜𝑢𝑖 +¯𝜌 ˜𝑢 ˜𝑢𝑖 + 𝛿1𝑖 ¯𝑝 +¯𝜌˜𝑣 ˜𝑢𝑖 + 𝛿2𝑖 ¯𝑝 +¯𝜌 ˜𝑤 ˜𝑢𝑖 + 𝛿3𝑖 ¯𝑝 +( ¯𝜌 ˇ𝐸 + ¯𝑝) ˜𝑢𝑖 +������������ +F𝑣 +𝑖 = +������������ +0 +𝜏𝑚𝑜𝑑 +1𝑖 +𝜏𝑚𝑜𝑑 +2𝑖 +𝜏𝑚𝑜𝑑 +3𝑖 +˜𝑢 𝑗𝜏𝑚𝑜𝑑 +𝑖 𝑗 +− 𝑞𝑚𝑜𝑑 +𝑖 +������������ +, for 𝑖 = 1, 2, 3, +(2) +where ˜𝑢𝑖 or ( ˜𝑢, ˜𝑣, ˜𝑤) are the Favre averaged velocity components, ¯𝜌 is the filtered density, ¯𝑝 is the filtered pressure and +¯𝜌 ˇ𝐸 is the filtered total energy per unit volume. The terms 𝜏𝑚𝑜𝑑 +𝑖 𝑗 +and 𝑞𝑚𝑜𝑑 +𝑖 +are the modified viscous stress tensor and +heat flux vector, respectively, and 𝛿𝑖 𝑗 is the Kronecker delta. The filtered total energy per unit volume, according to the +definition proposed by Vreman [20] in its "system I", is given by +¯𝜌 ˇ𝐸 = +¯𝑝 +𝛾 − 1 + 1 +2 ¯𝜌 ˜𝑢𝑖 ˜𝑢𝑖. +(3) +The filtered pressure, Favre averaged temperature and filtered density are correlated using the ideal gas equation of +state ¯𝑝 = ¯𝜌𝑅 ˜𝑇, and 𝑅 is the gas constant, written as 𝑅 = 𝑐 𝑝 − 𝑐𝑣. The properties 𝑐 𝑝 and 𝑐𝑣 are the specific heat at +constant pressure and volume, respectively. The modified viscous stress tensor may be written as +𝜏𝑚𝑜𝑑 +𝑖 𝑗 += (𝜇 + 𝜇𝑆𝐺𝑆) +� 𝜕 ˜𝑢𝑖 +𝜕𝑥 𝑗 ++ 𝜕 ˜𝑢 𝑗 +𝜕𝑥𝑖 +� +− 2 +3 (𝜇 + 𝜇𝑆𝐺𝑆) +� 𝜕 ˜𝑢𝑘 +𝜕𝑥𝑘 +� +𝛿𝑖 𝑗 +(4) +where 𝜇 is the dynamic viscosity coefficient, calculated by Sutherland’s Law, and 𝜇𝑆𝐺𝑆 is the SGS dynamic viscosity +coefficient, which is provided by the subgrid-scale model. The strategy of modeling the subgrid-scale contribution as an +additional dynamic viscosity coefficient is based on the Boussinesq hyphotesis. The modified heat flux vector, using the +same modeling strategy, is given by +𝑞𝑚𝑜𝑑 +𝑖 += −(𝑘 + 𝑘𝑆𝐺𝑆) 𝜕 ˜𝑇 +𝜕𝑥𝑖 +(5) +where 𝑘 is the thermal conductivity coefficient of the fluid and 𝑘𝑆𝐺𝑆 is the SGS thermal conductivity coefficient given by +𝑘𝑆𝐺𝑆 = 𝜇𝑆𝐺𝑆𝑐 𝑝 +𝑃𝑟𝑆𝐺𝑆 +(6) +2 + +and 𝑃𝑟𝑆𝐺𝑆 is the SGS Prandtl number. The present work employs only the static Smagorinsky model [21] to calculate +the subgrid-scale contribution. +B. Nodal Discontinuous Galerkin Method +The nodal discontinuous Galerkin method used in this work is based on the modeling proposed by Kopriva and +Gassner [11], and Hindenlang et al. [12]. In this discretization, the domain is divided into multiples hexahedral elements. +This choice of elements permits that the interpolating polynomial be defined as a tensor product basis with degree 𝑁 in +each space direction. The implementation is simpler and improve the computational efficiency of the code. +In this method, the elements from the physical domain are mapped onto a reference unit cube elements 𝐸 = [−1, 1]3. +The equations, presented in Eq. (1) need also to be mapped to this new reference domain, leading to +𝐽 𝜕 ¯Q +𝜕𝑡 + ∇𝜉 · ¯F = 0, +(7) +where ∇𝜉 is the divergence operator with respect to the reference element coordinates, 𝜉 = (𝜉1, 𝜉2, 𝜉3)𝑇 , 𝐽 = |𝜕x/𝜕𝜉| is +the Jacobian of the coordinate transformation and ¯F is the contravariant flux vector. +The discontinuous Galerkin formulation is obtained multiplying Eq. (7) by the test function 𝜓 = 𝜓(𝜉) and integrating +over the reference element 𝐸 +∫ +𝐸 +𝐽 𝜕 ¯Q +𝜕𝑡 𝜓𝑑𝜉 + +∫ +𝐸 +∇𝜉 · ¯F 𝜓𝑑𝜉 = 0. +(8) +It is possible to obtain the weak form of the scheme by integrating by parts the second term in Eq. (8) +𝜕 +𝜕𝑡 +∫ +𝐸 +𝐽 ¯Q𝜓𝑑𝜉 + +∫ +𝜕𝐸 +( ¯F · �𝑁)∗𝜓𝑑𝑆 − +∫ +𝐸 +¯F · (∇𝜉𝜓)𝑑𝜉 = 0, +(9) +where �𝑁 is the unit normal vector of the reference element faces. Because the discontinuous Galerkin scheme allows +discontinuities in the interfaces, the surface integral above is ill-defined. In this case, a numerical flux, ¯F ∗, is defined, +and a Riemann solver is used to compute the value of this flux based on the discontinuous solutions given by the +elements sharing the interface. +For the nodal form of the discontinuous Galerkin formulation, the solution in each element is approximated by a +polynomial interpolation of the form +¯Q(𝜉) ≈ +𝑁 +∑︁ +𝑝,𝑞,𝑟=0 +¯Qℎ(𝜉1 +𝑝, 𝜉2 +𝑞, 𝜉3 +𝑟, 𝑡)𝜙𝑝𝑞𝑟 (𝜉), +(10) +where ¯Qℎ(𝜉1 +𝑝, 𝜉2 +𝑞, 𝜉3 +𝑟, 𝑡) is the value of the vector of conserved variables at each interpolation node in the reference +element and 𝜙𝑝𝑞𝑟 (𝜉) is the interpolating polynomial. For hexahedral elements, the interpolating polynomial is a tensor +product basis with degree N in each space direction +𝜙𝑝𝑞𝑟 (𝜉) = 𝑙𝑝(𝜉1)𝑙𝑞(𝜉2)𝑙𝑟 (𝜉3), +𝑙𝑝(𝜉1) = +𝑁𝑝 +� +𝑖=0 +𝑖≠𝑝 +𝜉1 − 𝜉1 +𝑖 +𝜉1𝑝 − 𝜉1 +𝑖 +. +(11) +The definitions presented are applicable to other two directions. +The numerical scheme used in the simulation additionally presents the split formulation presented by Pirozzoli [22], +with the discrete form given by Gassner et al. [23], to enhance the stability of the simulation. The split formulation is +employed to Euler fluxes only. The solution and the fluxes are interpolated and integrated at the nodes of a Gauss-Lobatto +Legende quadrature, which presents the summation-by parts property, that is necessary to employ the split formulation. +The Riemann solver used in the simulations is a Roe scheme with entropy fix [24] to ensure that second law of +thermodynamics is respected, even with the split formulation. To be able to adequately handle the viscous flux in the +boundaries of the elements, the lifting scheme of Bassi and Rebay [25] is used, which is also known as BR2. The time +marching method chosen is the five-stage, fourth-order explicit Runge-Kutta scheme of Carpenter and Kennedy [26]. +The shock waves that appear in the simulation are stabilized using the finite-volume sub-cell shock capturing method +of Sonntag and Munz [27]. Even though the methodology used in the simulation solves the discontinuous Galerkin +approach, it only handle discontinuities in the interface of the elements. The shock capturing method permits to stabilize +the simulation with shock waves inside the elements. +3 + +III. Experimental Configuration +The focus of this work is to investigate the influence of mesh and polynomial refinement on the perfectly expanded +jet flow, which is present in many applications, such as supersonic military aircraft and large launch vehicles. The +experimental work of Bridges and Wernet [19] provides data flow properties for different jet flow configurations In +this work, the interest is to simulate the fully expanded free jet flow configuration with a Mach number of 1.4. In this +configuration the jet flow has a static pressure in the nozzle exit plane that equals the ambient static pressure with a +supersonic velocity. For such a flow configuration, the shock waves are weaker when compared to other operating +conditions, which reduces the constraints of mesh refinement and, consequently, the computational cost of the simulation. +The experimental apparatus for the analysed configuration is composed of a convergent-divergent nozzle designed +based on the method of characteristics [19]. The nozzle exit diameter is 50.8 mm. The Reynolds number based on nozzle +exit diameter is 1.58 × 106. The experimental data acquisition applies the Time-Resolved Particle Image Velocimetry +(TRPIV) at a 10 kHZ sample rate. The investigation uses two sets of cameras, one captures the flow along the nozzle +centerline, and the other captures the flow of the mixing layer along the nozzle lipline. +IV. Numerical Setup +A. Geometry and Mesh Configuration +The geometry used for the calculations in this work presents a divergent shape and axis length of 40𝐷, where 𝐷 +is the jet inlet diameter and has external diameters of 16𝐷 and 25𝐷. Figure 1 illustrates a 2-D representation of the +computational domain, indicating the inlet surface in red, the farfield region in blue, the lipline in grey, and the centerline +in black. Two different computational grids are used in the present work. The coarser mesh used here, named M-1 mesh, +is the same grid developed in Ref. [8]. The other computational grid, which is termed M-2 mesh here, was developed +specifically for the present effort and it represents a considerable improvement over the M-1 mesh. The modifications in +the M-2 mesh are both topological and the result of an increase in the number of grid cells. These modifications result +in a much higher refinement level around the jet inlet, encompassing both the original jet as well as the strong mixing +region around the jet. Afterwards, this mesh transitions to a uniform grid point distribution as one moves downstream in +the longitudinal direction. The mesh generation uses a multiblock strategy in order to handle hexahedral cells. +Fig. 1 +2-D schematic representation of the computational domain used on the jet flow simulations. +The grid design attempts to capture the jet flow until a distance of 𝑥/𝐷 = 15 from the inlet surface, indicated in red +in Fig. 1. Then, the size of elements increases as an attempt to dissipate frequencies that could destabilize the simulation. +Previous results [10] support the creation of mesh topology, indicating that a surface with one degree of opening angle +would better represent the middle surface of the jet mixing layer. From this surface, two regions rise to increase the +resolution of the domain. They have a geometrical stretching in the section 𝑥/𝐷 = 0 to enable local refinement in the +shear region of the flow and a uniform distribution in 𝑥/𝐷 = 15. The internal section of the mixing layer connects to +4 + +Sponge region +neone hexahedral block that forms the core of the mesh. That hexahedral block has its dimensions defined to keep the size +of the elements equal to the size of the last cells in the region of the mixing layer. +The grid refinement in the mixing layer is defined based on the literature [5–7, 9, 14]. The grid spacing in the radial +and axial directions along the mixing layer is Δ𝑦0/𝐷 = 0.001 and Δ𝑥0/𝐷 = 0.005, respectively. The centerline presents +651 elements set in geometrical stretching distribution between 𝑥/𝐷 = 0 and 𝑥/𝐷 = 15. Each region of the mixing +layer contains 50 cells, and the Azimuthal direction accommodates 180 elements evenly distributed. Figure 2a presents +the radial mesh refinement in two different longitudinal positions, 𝑥/𝐷 = 0 and 𝑥/𝐷 = 15. Figure 2b illustrates the +axial mesh refinement along the jet centerline and Fig. 3 exhibits a cutplane of the mesh generated for the current paper +and the baseline line mesh used in previous work. The M-1 and M-2 grids have a total of 6.2 and 15.4 million cells, +respectively, and they are created with the GMSH [28] mesh generator. +(a) Radial mesh refinement (Δ𝑦/𝐷) in the longitudinal sections 𝑥/𝐷 = 0 +and 𝑥/𝐷 = 15. +(b) Longitudinal mesh refinement (Δ𝑥/𝐷) along the jet centerline. +Fig. 2 +Distribution of grid spacing indicating radial and longitudinal refinement for the M-2 mesh. +(a) M-1 mesh. +(b) M-2 mesh. +Fig. 3 +Visualization of the half-plane longitudinal cutplanes for the meshes used in the present work. +B. Boundary Conditions +Properties on the jet inflow, (·) 𝑗𝑒𝑡, and farfield, (·) 𝑓 𝑓 , surfaces are indicated in Fig. 1 in red and blue, respectively. +A weakly enforced solution of a Riemann problem with a Dirichlet condition is enforced at the boundaries. The flow is +characterized as perfectly expanded and isothermal, i.e. 𝑝 𝑗𝑒𝑡/𝑝 𝑓 𝑓 = 𝑇𝑗𝑒𝑡/𝑇𝑓 𝑓 = 1, where 𝑝 stands for pressure and 𝑇 +5 + +101 +100 +10-2 +x/D=0 +x/D=15 +10-3 +0 +1 +2 +3 +4 +5 +y/D101 +100 +△x/D +10-1 +10-2 +10-3 +0 +10 +20 +30 +40 +x/DY +7for temperature. The Mach number of the jet at the inlet is 𝑀𝑗𝑒𝑡 = 1.4 and the Reynolds number based on the diameter +of the nozzle is 𝑅𝑒 𝑗𝑒𝑡 = 1.58 × 106. A small velocity component with 𝑀 𝑓 𝑓 = 0.01 in the streamwise direction is +imposed at the farfield to avoid numerical issues. A sponge zone [29] is employed around the farfield boundaries, the +gray area presented in Fig. 1, to damp any oscillations that could be reflected back to the jet. +C. Simulation Settings and DOFs +The current work compares the effects of hp refinement using three different calculations: S1, S2, and S3. The +first simulation uses the M-1 computational grid, while the other two computations apply the M-2 mesh. Both +studies, S1 and S2, employ first-order polynomial reconstructions in order to achieve second-order accuracy in spatial +discretization. Calculation S3 uses second-order polynomial reconstructions in order to achieve a third-order accurate +spatial discretization. The simulations, therefore, consider from 50 to 410 million DOFs. Table 1 indicates the settings +used the three numerical studies performed in the present effort. +Table 1 +Summary of simulations settings. +Simulation +Meshes +Order of +DOF/cell +Cells +Total # of DOF +Accuracy +(106) +(106) +S1 +M-1 +2nd order +8 +6.2 +≈ 50 +S2 +M-2 +2nd order +8 +15.4 +≈ 120 +S3 +M-2 +3rd order +27 +15.4 +≈ 410 +D. Calculation of Statistical Properties +The simulation procedure involves three steps. The first one is to clean off the domain since the computation starts +with a static flow initial condition. The simulations run three flow-through times (FTT) to develop the jet flow. One +FTT is the time required for one particle with the jet velocity to cross the computational domain. In the sequence, the +simulations run an additional three FTT to produce a statistically steady condition. Then, in the last step, data are +collected for another three FTT to obtain the statistical properties of the flow. +The procedure for developing S3 simulation is slightly different. The simulation is a restart from the finished S2 +calculation. The numerical framework FLEXI allows using one solution with different order of accuracy as initialization. +Once the second-order solution was already available, its usage was a short come to initialize the S3 simulation. Hence, +the S3 numerical study runs 0.5 FFT to allow the solution to adapt from second-order accuracy to third-order accuracy. +Then it runs an additional 2 FTT to collect data. Different frequencies of data acquisition were employed in each +simulation. The S1 case applies 160 kHz, while S2 and S3 cases use 205 and 225 kHz, respectively. +The mean and the root mean square (RMS) fluctuations of properties of the flow are calculated along the centerline, +lipline, and different domain surfaces in the streamwise direction. The centerline is defined as the line in the center of +the geometry 𝑦/𝐷 = 0, whereas the lipline is a surface parallel to the centerline and located at the nozzle diameter, +𝑦/𝐷 = 0.5. The results from the lipline are an azimuthal mean from six equally spaced positions. The four surfaces in +the streamwise positions are 𝑥/𝐷 = 2.5, 𝑥/𝐷 = 5.0, 𝑥/𝐷 = 10.0, and 𝑥/𝐷 = 15.0. Surface properties are averaged +using six equally spaced positions in the azimuthal direction. Figure 4 illustrates a Mach number contours snapshot of +the jet flow with the lines and surfaces of data extraction. +V. Results +The results from S1, S2, and S3 simulations are presented in this section and compared to experimental data [19]. +The focus of this work is to assess the resolution requirements for the correct prediction of supersonic jet flows. S1 +and S2 calculations are performed with the same polynomial order of accuracy, while S3 simulation uses third-order +accuracy polynomials. The S1 Numerical study is performed with mesh M-1, with 6.2 × 106 elements, and S2 and S3 +calculations are performed with mesh M-2, with 15.4 × 106 elements. The simulations have approximately 50, 120, and +410 million DOFs. +6 + +Fig. 4 +Snapshot of the jet simulation with the two longitudinal lines and three crossflow lines along which data +is extracted. Mach number contours are shown. +A. Velocity and Density Contours +Initially, the contours of the mean longitudinal velocity component, RMS of longitudinal velocity fluctuation, and +mean density are presented for the three simulations. Figure 5 presents the contours of the mean longitudinal velocity +component on a cut plane at 𝑧/𝐷 = 0. The contours indicate qualitatively improvement in results when increasing the +number of DOFs. One can notice the size of the jet core is bigger in Fig. 5c than in Figs. 5b and 5a. Moreover, the +development of the mixing layer start closer to the jet inlet in S3 simulation than in S2 and S1 calculations. Furthermore, +it is difficult to notice the existence of shock waves in Fig. 5a, which are more clearly visible in Figs. 5b and 5c. +Figure 6 presents the contours of RMS of longitudinal velocity fluctuations on a cut plane at 𝑧/𝐷 = 0. One can +notice the early development of the mixing layer when increasing the number of DOFs in the calculations. In Fig. 6c the +increase in the RMS values for longitudinal velocity fluctuation occurs right after the boundary condition. The same +physical phenomenon occurs farther when decreasing the simulation resolution, which is noticeable when comparing +Fig.s 6b and 6a. The contours of RMS of longitudinal velocity fluctuation also show that the fluctuation levels get +smaller with earlier development of the mixing layer. In Fig. 6c the region of high velocity fluctuation is thinner and +presents smaller values than in Fig. 6b. In Fig. 6a, the region with a high level of velocity fluctuation is the largest, and +it presents the highest values when compared to the other two results. +The last contours presented in this section, Fig. 7, compare the mean density results from the three simulations on a +cut plane at 𝑧/𝐷 = 0. It is possible to observe that each simulation presents different characteristics. In Fig. 7a the +shock waves are weak, being hardly visible with adequate range scales for all simulations. Moreover, one can notice +a few shock waves and expansion waves reflections. The appearance of the first shock waves occurs far from the jet +inlet boundary condition. Figure 7b presents a different behavior of the flow with shock waves and expansion waves +significantly different from those observed in Fig. 7a. The first shock waves appear closer to the jet inlet boundary +conditions, and they are visible, which indicates that they are significantly stronger than those from the S1 simulation. +Another interesting observation is the number of shock waves and expansion waves reflection, which is much larger than +the presented in Fig. 7a. The final density results are presented in Fig. 7c, which is the result from S3 simulation. It is +possible to observe that the shock and expansion waves are better defined, presenting smaller thickness, which is also +evidence of the improvements when increasing the number of DOFs in the simulation. When comparing the results from +Fig. 7c with Fig. 7b, it is possible to observe that the intensity of the shock waves is stronger in S2 simulation than in S3, +even with the larger thickness. This is in agreement with results presented in Figs. 5b and 5c, in which the shock waves +from S2 calculation where more visible. The quantity of shock waves reflections in the S2 and S3 test cases are similar. +B. Velocity Profiles +One can better understand the effects of the hp refinement on the numerical solution when comparing the results +to the experimental data. Figure 8 presents the mean longitudinal velocity component and the RMS values of the +velocity fluctuation distributions along the jet centerline (𝑦/𝐷 = 0) and the jet lipline (𝑦/𝐷 = 0.5). Analyzing the results +presented in Fig. 8a, the improvement in the capacity to capture flow features when increasing the numerical resolution +of the calculations is prominent. One can notice the numerical solution progression of the mean longitudinal velocity +component towards the experimental data in Fig. 8a. The potential core is longer in S3 than in the other cases, which is +presented in detail in Tab. 2. Moreover, the most refined numerical study presents the intensity of the shock waves +and the slope of the decay of velocity comparable to the ones of the reference data. One can observe that the profiles +calculated in S2 and S3 simulation are closer than the ones computed in S1 and S2, even with a higher ratio of degrees +7 + +2.5D +5D +110D +Centerline(a) S1 simulation. +(b) S2 simulation. +(c) S3 simulation. +Fig. 5 +Contours of mean longitudinal velocity component along cutplanes in 𝑧/𝐷 = 0 for the three simulations +performed. +of freedom, 𝐷𝑂𝐹𝑆3/𝐷𝑂𝐹𝑆2 ≈ 3.42 and 𝐷𝑂𝐹𝑆2/𝐷𝑂𝐹𝑆1 ≈ 2.4. The slight improvement, even with a higher DOF +ratio, can also indicate that the simulation S3 is very close to provide the converged solution for the chosen modeling +approach. +The results presented in Fig. 8b agree well with the results of Fig. 8a. The increase in the simulation resolution +improves the calculation of RMS velocity fluctuation towards the experimental data. Analyzing the results close to the +jet inlet, the increase in the velocity fluctuation occurs further upstream in S3 simulation, which is in agreement with the +contours in Figs. 5 where the shock waves of S3 numerical study appear closer to jet inlet when compared to the S2 and +S1 calculations. However, even with an early increase in the fluctuation levels in S3 computations, the slope of its profile +in Fig. 8b is smoother and closer to the reference than the one calculated in S2 and S1 numerical studies. In the same +image, S3 simulation presents two peaks of velocity fluctuation, with the second one close to the peak indicated in the +experiment. However, its RMS fluctuations are higher than experimental data and S2 simulation solution. The presence +8 + +0.15 +0. 1- +0.05 +1.0 +Y (m)0 +/Ujet +-0.05 +0.2 +-0.1- +0.0 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15- +0. 1- +0.05 +1.0 +Y (m)0 +/Ujet +-0.05 +0.2 +-0.1- +0.0 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15 +0. 1- +0.05 +1.0 +Y (m)0 +/Ujet +-0.05 +-0.1- +0.0 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.9(a) S1 simulation. +(b) S2 simulation. +(c) S3 simulation. +Fig. 6 +Contours of RMS values of longitudinal velocity component fluctuations along cutplanes in 𝑧/𝐷 = 0 for +the three simulations performed. +of small values of velocity fluctuation close to the jet inlet could be related to imposed jet entrance boundary conditions. +Figures 8c and 8d illustrate the profiles of mean and RMS fluctuations of the longitudinal velocity component along +the lipline, respectively. One can notice the improvements in the simulation resolution with S3 and S2 simulations +providing mean profiles closer to the experimental one than the results from the S1 calculation. The mean velocity +oscillations in the vicinity of the inlet jet may be correlated with shock waves. They are also present in the experimental +profile along the lipline. The S1 and S2 simulations present an early reduction of mean velocity when comparing to the +reference profile. The most refined calculation has a mean velocity profile in good agreement with the experimental +data along the lipline. Such behavior is also noticeable along the centerline. The calculations performed in the present +paper have generated fluctuation profiles of the longitudinal velocity fluctuation along the lipline, Fig. 8d, that indicate a +different trend from what is stated in Figs. 8a to 8c. One can notice that, when increasing the simulation resolution, the +peak of velocity fluctuation moves towards the jet inlet, at 𝑥/𝐷 = 2.5 for S1 simulation and ≈ 𝑥/𝐷 = 1 for S2 and S3 +9 + +0.15 +0. 1- +0.05 +0.20 +Y (m)0 +0.15 +uRMS/Ujet +-0.05 +0.10 +0.05 +-0.1- +0.00 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.90.15- +0. 1- +0.05 +0.20 +Y (m)0 +0.15 +uRMS/Ujet +0.10 +-0.05 +0.05 +-0.1- +0.00 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x9%) +0.6 +0.7 +0.8 +0.90.15 +0. 1- +0.05 +0.20 +Y (m)0 +0.15 +uRMS/Ujei +-0.05 +0.10 +0.05 +-0.1- +0.00 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x95) +0.6 +0.7 +0.8 +0.9(a) S1 simulation. +(b) S2 simulation. +(c) S3 simulation. +Fig. 7 +Contours of mean density along cutplanes in 𝑧/𝐷 = 0 for all three simulations. +calculations. This outcome is present in the contours indicated in Fig. 5, where the initial spreading of the mixing layer +in S2 and S3 simulations starts earlier than in the S1 calculation. Such behavior is different from the experimental +RMS profiles along the lipline in which the increase in the velocity fluctuation occurs slowly before reaching a plateau +between 𝑥/𝐷 = 5 to 𝑥/𝐷 = 15. Using a flat hat velocity profile at the jet entrance for the numerical calculations can +explain the divergence between the fluctuations obtained from the numerical approach and the experiments near the inlet +domain. Such boundary condition neglects the turbulent boundary layer effects carried from the nozzle to the jet flow. +10 + +0.15- +0.1 +0.05 +1.05 +1.02 +/rhojet +Y (m)0 +0.99 +-0.05 +0.96 +-0.1 +0.93 +0.90 +-0.15 +0.1 +0.2 +0.3 +0.4 +x9§) +0.6 +0.7 +0.8 +0.90.15 +0. 1 +0.05 +1.05 +1.02 +/rhojet +Y (m)0 +0.99 +-0.05 +0.96 +-0.1 +0.93 +0.90 +-0.15 +0 +0.1 +0.2 +0.3 +0.4 +x9%) +0.6 +0.7 +0.8 +0.90.15 +0. 1 +0.05- +1.05 +1.02 +/rhojet +Y (m)0 +0.99 +-0.05 +0.96 +-0.1 +0.93 +0.90 +-0.15 +0.1 +0.2 +0.3 +0.4 +x9§) +0.6 +0.7 +0.8 +0.9(a) Centerline, mean. +(b) Centerline, RMS values. +(c) Lipline, mean. +(d) Lipline, RMS values. +Fig. 8 +Results for the mean streamwise velocity component distributions (left) and RMS values of streamwise +velocity component fluctuations (right) in the jet centerline, 𝑦/𝐷 = 0 (top), and lipline, 𝑦/𝐷 = 0.5 (bottom). +Table 2 +Summary of potential core length for all simulations. +Simulation +Potential core +Error to +length (𝑥/𝐷) +experimental data (%) +S1 +6.3 +30.0 +S2 +7.8 +13.3 +S3 +8.5 +5.5 +11 + +1.2 +S1 +S2 +S3 +Exp +/U, +0.8 +0.6 +0.4 +0 +5 +10 +15 +20 +x/D0.15 +0.1 +rms +S1 +0.05 +S2 +S3 +Exp +0 +0 +5 +10 +15 +20 +x/D0.8 +S1 +0.7 +S2 +S3 +0.6 +Exp +/U, +0.5 +0.4 +0.3 +0.2 +0 +5 +10 +15 +20 +x/D0.2 +S1 +S2 +S3 +0.15 +Exp +rms +0.1 +u +0.05 +0 +0 +5 +10 +15 +20 +x/DFigure 9 displays different statistical properties of the flow in four streamwise positions, 𝑥/𝐷 = 2.5, 𝑥/𝐷 = 5, +𝑥/𝐷 = 10 and 𝑥/𝐷 = 15. Figures 9a to 9d present the mean longitudinal velocity component, Figs. 9e to 9h illustrate +the RMS of the longitudinal velocity fluctuation, Figs. 9i to 9l introduces the RMS of the radial velocity fluctuation, and +Figs. 9m to 9p indicate the mean shear stress tensor. +The first set of results, in Figs. 9a to 9d, explicit some aspects of the numerical results not investigated by the +comparison of 2-D field of properties. In the first longitudinal position, Fig. 9a, S1 calculations generated mean profiles +that are in better agreement with the experimental data than the ones from S2 and S3 numerical studies, which indicate a +larger spreading of velocity at this position. The early development of the mixing layer from S2 and S3 simulations +reinforces the influence of the choice of boundary conditions imposed. Analyzing the profiles from downstream +positions, Figs. 9b to 9d, it is possible to verify the large spreading of velocity from the S1 simulation, with a reduction +of longitudinal velocity in the jet centerline when compared to the other numerical solutions. Calculations with higher +resolution can better capture the experimental trends, with the simulation S3 getting closer to experimental data. +The profiles of RMS values of streamwise velocity fluctuation are indicated in Figs. 9e to 9h. The numerical results +at 𝑥/𝐷 = 2.5 present a similar profile to the one from the reference. However, the peaks generated by the calculations +are higher than the experimental ones, with the results from the S3 simulation being the closest to experimental data. +The same conclusion can be drawn for 𝑥/𝐷 = 5.0, Fig. 9f. In Figs. 9g and 9h, the all numerical results present a shape +similar to experimental data, with a nearly constant value of velocity fluctuation. In Fig. 9g the experimental data still +present a small level of fluctuation close to the centerline, which is not seen in the numerical profiles. +Profiles of RMS values of radial velocity component fluctuation are presented in Figs. 9i to 9l. In the first two +longitudinal positions, Figs. 9i and 9j, the numerical results present a larger peak of fluctuation than the one from the +experimental data, with the profiles from the S3 simulation getting closer to reference. In Figs. 9k and 9l the RMS +profiles from the calculations are very similar, with the centerline of the experimental data presenting small values of +fluctuation in Fig. 9k. +Mean shear-stress tensor component profiles are presented in Figs. 9m to 9p. In the first two positions, 𝑥/𝐷 = 0.5 +and 𝑥/𝐷 = 2.5, the peak of the shear-stress tensor from numerical calculations is larger than the experimental one. +Moreover, the peak region is wider than the one observed in the experiments. In Fig. 9n the peaks are still larger than +those observed in the reference. The differences between the simulations are smaller and closer to experimental data. +However, the region of the peaks is still wider than the one visualized in the experimental data. In Figs. 9o and 9p, the +differences between numerical results and the experimental data are reduced, and once more, all profiles are nearly +matching. +C. Summary of Discussions +Results from the S2 simulation present significant improvements when compared to the solution from the S1 +calculation. The increase in polynomial order of accuracy in the S3 study brings the results to a good agreement with +the reference experimental data. The most significant point that does not present improvements when increasing the +discretization resolution is related to the mean and fluctuating longitudinal velocity close to jet lipline, Figs. 8c and8d. +Grid refinement yields a velocity fluctuation peak that occurs closer to the jet inlet when compared to the experimental +data. This behavior may be related to the hypothesis used to impose the inlet boundary condition, which neglects the +effects of the jet boundary layer and leads to a not realistic turbulence transition in the vicinity of the nozzle. +The current work provides intel on the resolution requirements to perform large-eddy simulations of supersonic jet +flows. The next step concerns the exploration of alternatives to improve the jet simulation in the lipline close to the jet +inlet condition. A solution to improve the simulation in this region is a better characterization of the flow leaving the +nozzle. The choice of a uniform velocity profile is preliminary, and it does not represent the physics of the flow. The +continuity of the work will focus on options to produce an experiment-like condition in the jet inlet condition. +12 + +(a) 𝑥/𝐷 = 2.5 +(b) 𝑥/𝐷 = 5 +(c) 𝑥/𝐷 = 10 +(d) 𝑥/𝐷 = 15 +(e) 𝑥/𝐷 = 2.5 +(f) 𝑥/𝐷 = 5 +(g) 𝑥/𝐷 = 10 +(h) 𝑥/𝐷 = 15 +(i) 𝑥/𝐷 = 2.5 +(j) 𝑥/𝐷 = 5 +(k) 𝑥/𝐷 = 10 +(l) 𝑥/𝐷 = 15 +(m) 𝑥/𝐷 = 2.5 +(n) 𝑥/𝐷 = 5 +(o) 𝑥/𝐷 = 10 +(p) 𝑥/𝐷 = 15 +Fig. 9 +Profiles of mean streamwise velocity component, RMS of streamwise velocity fluctuation, RMS of radial +velocity fluctuation, and mean shear-stress tensor component (from top to bottom) at four streamwise positions +𝑥/𝐷 = 2.5, 𝑥/𝐷 = 5, 𝑥/𝐷 = 10 and 𝑥/𝐷 = 15 (from left to right). +13 + +1.5 +1 +0.5 +0 +-0.5 +S1 +S2 +-1 +S3 +Exp +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.5 +1 +/U1.5 +1 +0.5 +y/D +0 +-0.5 +S1 +S2 +-1 +S3 +Exp +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +y/D +0 +-0.5 +-1 +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.1 +0.2 +u +/U +rms1.5 +1 +0.5 +y/D +0 +-0.5 +S1 +S2 +-1 +S3 +Exp +-1.5 +0 +0.05 +0.1 +0.15 +V +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.05 +0.1 +0.15 +V +/U +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.05 +0.1 +0.15 +V +/U. +rms1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +0 +0.05 +0.1 +0.15 +V +/U +rms1.5 +1 +0.5 +0 +-0.5 +S1 +S2 +-1 +S3 +Exp +-1.5 +-0.01 +0 +0.01 +/U?1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +-0.01 +0 +0.01 +/U?1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +-0.01 +0 +0.01 +/U?1.5 +1 +0.5 +0 +-0.5 +-1 +-1.5 +-0.01 +0 +0.01 +/U?VI. Concluding Remarks +The current work assesses the effects of mesh and polynomial (hp) refinement using a nodal discontinuous Galerkin +methodology to evaluate the resolution requirements for large eddy simulations of compressible jet flows. The problem +of interest is a perfectly expanded supersonic jet flow with a Mach number of 1.4 and Reynolds number based on the +jet exit diameter of 1.58 × 106. Initially, a mesh with 6.2 × 106 elements is used with interpolation polynomials that +yield second-order spatial accuracy, in order to produce a starting point for the comparisons. Such calculations use, +therefore, the equivalent to approximately 50 × 106 degrees of freedom (DOFs). Afterwards, a new mesh is developed +with some topological improvements and additional refinement, leading to 15.4 × 106 elements. The new mesh is +applied in simulations using second-order and third-order spatial accuracy, resulting in 120 × 106 and 410 × 106 DOFs, +respectively. The results for the simulations are compared to experimental data. +The paper initially investigates the contours of mean velocity, mean pressure, and velocity fluctuations. The +comparison indicates that the mesh/polynomial refinement improves the jet calculations by enhancing the prediction +of the mixing layer and of the series of shock and expansion waves in the jet core. Therefore, as one should expect, +more refined computations lead to an improved ability to predict flow features, as one can see in the present paper by +the comparison of the numerical solutions and the experimental data. Therefore, it is correct to state that the present +paper indicates mesh and discretization parameters for LES-based calculations, using a nodal discontinuous Galerkin +formulation, that provide supersonic jet flow results in good agreement with experiments. +One important aspect that becomes clear in the present calculation results is that the jet inlet boundary condition, +used in the current work, has a significant impact on the ability of representing the very early stages of jet mixing. In +particular, this observation becomes evident by looking at the behavior of RMS values of fluctuating properties near the +jet exit, along the jet lipline. All three simulations have failed to capture the correct mixing behavior, as evidenced by +the comparison with the experimental data. Moreover, the increased numerical resolution, although providing much +better comparisons for the overall solution, does not improve the behavior of fluctuating properties near the jet exit. +Hence, the continuation of the present effort will address possible improvements in the jet inlet boundary conditions. +Acknowledgments +The authors acknowledge the support for the present research provided by Conselho Nacional de Desenvolvimento +Científico e Tecnológico, CNPq, under the Research Grant No. 309985/2013-7. The work is also supported by the +computational resources from the Center for Mathematical Sciences Applied to Industry, CeMEAI, funded by Fundação +de Amparo à Pesquisa do Estado de São Paulo, FAPESP, under the Research Grant No. 2013/07375-0. The authors +further acknowledge the National Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for providing HPC +resources of the SDumont supercomputer. This work was also granted access to the HPC resources of IDRIS under the +allocation 2021-A0112A12067 made by GENCI. The first author acknowledges authorization by his employer, Embraer +S.A., which has allowed his participation in the present research effort. The doctoral scholarship provide by FAPESP to +the third author, under the Research Grant No. 2018/05524-1, is thankfully acknowledged. Additional support to the +fourth author under the FAPESP Research Grant No. 2013/07375-0 is also gratefully acknowledged. +References +[1] Brès, G. A., and Lele, S. K., “Modelling of Jet Noise: A Perspective from Large-Eddy Simulations,” Philosophical Transactions +of the Royal Society A, Vol. 377, No. 2159, 2019, p. 20190081. +[2] Kumar, P., and Mahesh, K., “Large Eddy Simulation of Propeller Wake Instabilities,” Journal of Fluid Mechanics, Vol. 814, +2017, p. 361–396. https://doi.org/10.1017/jfm.2017.20. +[3] Ghate, A. S., Gaetan, K. K., Stich, G.-D., Oliver, M. B., Jeffrey, A. 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A., Jordan, P., Jaunet, V., Rallic, M. L., Cavalieri, A. V. G., Towne, A., Lele, S. K., Colonius, T., and Schmidt, O. T., +“Importance of the Nozzle-Exit Boundary-Layer State in Subsonic Turbulent Jets,” Journal of Fluid Mechanics, Vol. 851, 2018, +pp. 83–124. +[17] Shen, W., and Miller, S. A. E., “Validation of a High-Order Large Eddy Simulation Solver for Acoustic Prediction of Supersonic +Jet Flow,” Journal of Theoretical and Computational Acoustics, Vol. 1950023, 2019. +[18] Corrigan, A., Kercher, A., Liu, J., and Kailasanath, K., “Jet Noise Simulation using a Higher-Order Discontinuous Galerkin +Method,” AIAA Paper No. 2018-1247, Proceedings of the 2018 AIAA Aerospace Sciences Meeting, Kissimmee, FL, 2018. +[19] Bridges, J., and Wernet, M. P., “Turbulence Associated with Broadband Shock Noise in Hot Jets,” AIAA Paper No. 2008-2834, +29th AIAA Aeroacoustics Conference, Vancouver, British Columbia, Canada, 2008. +[20] Vreman, A. W., “Direct and Large Eddy Simulation of the Compressible Turbulent Mixing Layer,” Ph.D. thesis, University of +Twente, Twente, Netherlands, 1995. +[21] Smagorinsky, J., “General Circulation Experiments with the Primitive Equations: I. The Basic Experiment,” Monthly Weather +Review, Vol. 93, No. 3, 1964, pp. 99–164. +[22] Pirozzoli, S., “Numerical Methods for High-Speed Flows,” Annual Review of Fluid Mechanics, Vol. 43, 2011, pp. 163–194. +[23] Gassner, G. J., Winters, A. R., and Kopriva, D. A., “Split Form Nodal Discontinuous Galerkin Schemes with Summation-by-Parts +Property for the Compressible Euler Equations,” Journal of Computational Physics, Vol. 327, 2016, pp. 39–66. +[24] Harten, A., and Hyman, J. M., “Self Adjusting Grid Methods for one Dimensional Hyperbolic Conservation Laws,” Journal of +Computational Physics, Vol. 50, 1983, pp. 253–269. +[25] Bassi, F., and Rebay, S., “A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the +Compressible Navier-Stokes Equations,” Journal of Computational Physics, Vol. 131, 1997, pp. 267–279. +[26] Carpenter, M. H., and Kennedy, C. A., “Fourt-Order 2N-Storage Runge-Kutta Schemes,” Tech. Rep. NASA-TM-109112, NASA +Langley Research Center, 1994. +15 + +[27] Sonntag, M., and Munz, C.-D., “Efficient Parallelization of a Shock Capturing for Discontinuous Galerkin Methods using Finite +Volume Sub-cells,” Journal of Scientific Computing, Vol. 70, 2017, pp. 1262–1289. +[28] Geuzaine, C., and Remacle, J. F., “GMSH: A Three-Dimensional Finite Element Mesh Generator with Built-in Pre- and +Post-Processing Facilities,” International Journal for Numerical Methods in Engineering, Vol. 79, No. 11, 2009, pp. 1309–1331. +[29] Flad, D. G., Frank, H. M., Beck, A. D., and Munz, C.-D., “A Discontinuous Galerkin Spectral Element Method for the Direct +Numerical Simulation of Aeroacoustics,” AIAA Paper No. 2014-2740, Proceedings of the 20th AIAA/CEAS Aeroacoustics +Conference, Atlanta, GA, 2014. +16 + diff --git a/b9AzT4oBgHgl3EQfnf2W/content/tmp_files/load_file.txt b/b9AzT4oBgHgl3EQfnf2W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..67e954cc01f4c516567314cb6f4eb821a32df10b --- /dev/null +++ b/b9AzT4oBgHgl3EQfnf2W/content/tmp_files/load_file.txt @@ -0,0 +1,971 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf,len=970 +page_content='Study on the Resolution of Large-Eddy Simulations for Supersonic Jet Flows Diego F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Abreu∗ Instituto Tecnológico de Aeronáutica, 12228–900, São José dos Campos, SP, Brazil Carlos Junqueira-Junior† Arts et Métiers Institute of Technology, DynFluid, CNAM, HESAM University, 151 Boulevard de l’Hôpital, 75013, Paris, France Eron T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Dauricio‡ Instituto Tecnológico de Aeronáutica, 12228–900, São José dos Campos, SP, Brazil João Luiz F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Azevedo§ Instituto de Aeronáutica e Espaço, 12228–904, São José dos Campos, SP, Brazil The present study is concerned with large-eddy simulations (LES) of supersonic jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The work addresses, in particular, the simulation of a perfectly expanded free jet flow with an exit Mach number of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 and an exit temperature equal to the ambient temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Calcula- tions are performed using a nodal discontinuous Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The present effort studies the effects of mesh and polynomial refinement on the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The present calculations con- sider computational meshes and plynomial orders such that the number of degrees of freedom (DOFs) in the solution ranges from 50 to 410 million.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Mean velocity results and root mean square (RMS) values of velocity fluctuations indicate a better agreement with experimental data as the resolution is increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The generated data provide a good understanding of the effects of increasing the discretization refinement for LES calculations of jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The present results can guide future simulations of similar flow configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Introduction With the progress of computing power in the last years, the large-eddy simulation (LES) formulation appears as an alternative to Reynolds-averaged Navier-Stokes (RANS) methods due to its reasonable cost when compared to the direct numerical simulation (DNS) of the Navier-Stokes equations or even physical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' LES can provide valuable information on complex configurations such as shear layers [1, 2] and detached flows [3, 4] due to its capability to generate unsteady data for flow and temperature fields with high-frequency fluctuations, which are necessary for aerodynamics, acoustics, loads, and heat transfer analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The authors are interested in the LES of jet flows from aircraft and rockets engines [5–7, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' More specifically, on the perfectly expanded configuration, when the jet exit pressure matches the ambient pressure, at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 Mach number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Recent work highlights [8] the effects of structured second-order finite-difference and unstructured nodal discontinuous-Galerkin spatial discretizations [11, 12] on the flow of interest at a fixed number of degrees of freedom (DOF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The results indicate good agreement with experimental and numerical data, where the spatial resolution is sufficient and with the same order of error in the coarser mesh regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Therefore, the current study addresses the effects of refinement on the LES of a supersonic jet flow configuration using the FLEXI framework [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The solver applies an unstructured nodal discontinuous Galerkin spatial discretization that allows evaluating the influence of mesh and polynomial (hp) refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' ∗Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Candidate, Graduate Program in Space Sciences and Technologies, Departamento de Ciência e Tecnologia Aeroespacial, DCTA/ITA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' E-mail: mecabreu@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' †Research Engineer, Arts et Métiers Institute of Technology, DynFluid laboratory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' E-mail: junior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='junqueira@ensam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' ‡Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Candidate, Graduate Program in Space Sciences and Technologies, Departamento de Ciência e Tecnologia Aeroespacial, DCTA/ITA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' E-mail: eron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='tiago90@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' §Senior Research Engineer, Aerodynamics Division, Departamento de Ciência e Tecnologia Aeroespacial, DCTA/IAE/ALA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' E-mail: joaoluiz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='azevedo@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fellow AIAA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='01582v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='flu-dyn] 4 Jan 2023 The literature [14–18] does not agree on the mesh requirements for adequately solving the LES formulation due to employing different numerical methods for solving jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The present paper studies the effects of mesh and polynomial refinement along with mesh topology to identify the minimum mesh requirements for adequately solving the problem of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The research group improved the baseline grid from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' [8] with local mesh refinement in the vicinity of the lipline and with an increase in the number of elements, ranging from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 × 106 to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 × 106 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The jet flow calculations use second-order and third-order polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The simulations present 50 to 410 million DOFs when combining grid and polynomial refinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The generated data for mean velocity and RMS of velocity fluctuations are investigated and compared with experimental data [19] at different regions of the domain where the jet is developing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The paper is organized to introduce the reader to the description of physical and numerical formulation in the second section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Then, one can find details of the experimental configuration and the numerical setup in sections three and four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Finally, the results and the concluding remarks close the work in sections five and six, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Numerical Formulation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Governing Equations The work has the interest in the solution of the filtered Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The filtering strategy is based on a spatial filtering process that separates the flow into a resolved part and a non resolved part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Usually the filter size is obtained from the mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The filtered Navier-Stokes equations in conservative form can be written by 𝜕 ¯Q 𝜕𝑡 + ∇ · F( ¯Q, ∇ ¯Q) = 0, (1) where ¯Q = [ ¯𝜌, ¯𝜌 ˜𝑢, ¯𝜌˜𝑣, ¯𝜌 ˜𝑤, ¯𝜌 ˇ𝐸]𝑇 is the vector of filtered conserved variables and F is the flux vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The flux vector can be divided into the Euler fluxes and the viscous flux, F = F𝑒 − F𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The fluxes with the filtered variables may be written as F𝑒 𝑖 = ������������ ¯𝜌 ˜𝑢𝑖 ¯𝜌 ˜𝑢 ˜𝑢𝑖 + 𝛿1𝑖 ¯𝑝 ¯𝜌˜𝑣 ˜𝑢𝑖 + 𝛿2𝑖 ¯𝑝 ¯𝜌 ˜𝑤 ˜𝑢𝑖 + 𝛿3𝑖 ¯𝑝 ( ¯𝜌 ˇ𝐸 + ¯𝑝) ˜𝑢𝑖 ������������ F𝑣 𝑖 = ������������ 0 𝜏𝑚𝑜𝑑 1𝑖 𝜏𝑚𝑜𝑑 2𝑖 𝜏𝑚𝑜𝑑 3𝑖 ˜𝑢 𝑗𝜏𝑚𝑜𝑑 𝑖 𝑗 − 𝑞𝑚𝑜𝑑 𝑖 ������������ , for 𝑖 = 1, 2, 3, (2) where ˜𝑢𝑖 or ( ˜𝑢, ˜𝑣, ˜𝑤) are the Favre averaged velocity components, ¯𝜌 is the filtered density, ¯𝑝 is the filtered pressure and ¯𝜌 ˇ𝐸 is the filtered total energy per unit volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The terms 𝜏𝑚𝑜𝑑 𝑖 𝑗 and 𝑞𝑚𝑜𝑑 𝑖 are the modified viscous stress tensor and heat flux vector, respectively, and 𝛿𝑖 𝑗 is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The filtered total energy per unit volume, according to the definition proposed by Vreman [20] in its "system I", is given by ¯𝜌 ˇ𝐸 = ¯𝑝 𝛾 − 1 + 1 2 ¯𝜌 ˜𝑢𝑖 ˜𝑢𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (3) The filtered pressure, Favre averaged temperature and filtered density are correlated using the ideal gas equation of state ¯𝑝 = ¯𝜌𝑅 ˜𝑇, and 𝑅 is the gas constant, written as 𝑅 = 𝑐 𝑝 − 𝑐𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The properties 𝑐 𝑝 and 𝑐𝑣 are the specific heat at constant pressure and volume, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The modified viscous stress tensor may be written as 𝜏𝑚𝑜𝑑 𝑖 𝑗 = (𝜇 + 𝜇𝑆𝐺𝑆) � 𝜕 ˜𝑢𝑖 𝜕𝑥 𝑗 + 𝜕 ˜𝑢 𝑗 𝜕𝑥𝑖 � − 2 3 (𝜇 + 𝜇𝑆𝐺𝑆) � 𝜕 ˜𝑢𝑘 𝜕𝑥𝑘 � 𝛿𝑖 𝑗 (4) where 𝜇 is the dynamic viscosity coefficient, calculated by Sutherland’s Law, and 𝜇𝑆𝐺𝑆 is the SGS dynamic viscosity coefficient, which is provided by the subgrid-scale model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The strategy of modeling the subgrid-scale contribution as an additional dynamic viscosity coefficient is based on the Boussinesq hyphotesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The modified heat flux vector, using the same modeling strategy, is given by 𝑞𝑚𝑜𝑑 𝑖 = −(𝑘 + 𝑘𝑆𝐺𝑆) 𝜕 ˜𝑇 𝜕𝑥𝑖 (5) where 𝑘 is the thermal conductivity coefficient of the fluid and 𝑘𝑆𝐺𝑆 is the SGS thermal conductivity coefficient given by 𝑘𝑆𝐺𝑆 = 𝜇𝑆𝐺𝑆𝑐 𝑝 𝑃𝑟𝑆𝐺𝑆 (6) 2 and 𝑃𝑟𝑆𝐺𝑆 is the SGS Prandtl number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The present work employs only the static Smagorinsky model [21] to calculate the subgrid-scale contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Nodal Discontinuous Galerkin Method The nodal discontinuous Galerkin method used in this work is based on the modeling proposed by Kopriva and Gassner [11], and Hindenlang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In this discretization, the domain is divided into multiples hexahedral elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' This choice of elements permits that the interpolating polynomial be defined as a tensor product basis with degree 𝑁 in each space direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The implementation is simpler and improve the computational efficiency of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In this method, the elements from the physical domain are mapped onto a reference unit cube elements 𝐸 = [−1, 1]3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The equations, presented in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (1) need also to be mapped to this new reference domain, leading to 𝐽 𝜕 ¯Q 𝜕𝑡 + ∇𝜉 · ¯F = 0, (7) where ∇𝜉 is the divergence operator with respect to the reference element coordinates, 𝜉 = (𝜉1, 𝜉2, 𝜉3)𝑇 , 𝐽 = |𝜕x/𝜕𝜉| is the Jacobian of the coordinate transformation and ¯F is the contravariant flux vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The discontinuous Galerkin formulation is obtained multiplying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (7) by the test function 𝜓 = 𝜓(𝜉) and integrating over the reference element 𝐸 ∫ 𝐸 𝐽 𝜕 ¯Q 𝜕𝑡 𝜓𝑑𝜉 + ∫ 𝐸 ∇𝜉 · ¯F 𝜓𝑑𝜉 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (8) It is possible to obtain the weak form of the scheme by integrating by parts the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (8) 𝜕 𝜕𝑡 ∫ 𝐸 𝐽 ¯Q𝜓𝑑𝜉 + ∫ 𝜕𝐸 ( ¯F · �𝑁)∗𝜓𝑑𝑆 − ∫ 𝐸 ¯F · (∇𝜉𝜓)𝑑𝜉 = 0, (9) where �𝑁 is the unit normal vector of the reference element faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Because the discontinuous Galerkin scheme allows discontinuities in the interfaces, the surface integral above is ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In this case, a numerical flux, ¯F ∗, is defined, and a Riemann solver is used to compute the value of this flux based on the discontinuous solutions given by the elements sharing the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' For the nodal form of the discontinuous Galerkin formulation, the solution in each element is approximated by a polynomial interpolation of the form ¯Q(𝜉) ≈ 𝑁 ∑︁ 𝑝,𝑞,𝑟=0 ¯Qℎ(𝜉1 𝑝, 𝜉2 𝑞, 𝜉3 𝑟, 𝑡)𝜙𝑝𝑞𝑟 (𝜉), (10) where ¯Qℎ(𝜉1 𝑝, 𝜉2 𝑞, 𝜉3 𝑟, 𝑡) is the value of the vector of conserved variables at each interpolation node in the reference element and 𝜙𝑝𝑞𝑟 (𝜉) is the interpolating polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' For hexahedral elements, the interpolating polynomial is a tensor product basis with degree N in each space direction 𝜙𝑝𝑞𝑟 (𝜉) = 𝑙𝑝(𝜉1)𝑙𝑞(𝜉2)𝑙𝑟 (𝜉3), 𝑙𝑝(𝜉1) = 𝑁𝑝 � 𝑖=0 𝑖≠𝑝 𝜉1 − 𝜉1 𝑖 𝜉1𝑝 − 𝜉1 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (11) The definitions presented are applicable to other two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The numerical scheme used in the simulation additionally presents the split formulation presented by Pirozzoli [22], with the discrete form given by Gassner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' [23], to enhance the stability of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The split formulation is employed to Euler fluxes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The solution and the fluxes are interpolated and integrated at the nodes of a Gauss-Lobatto Legende quadrature, which presents the summation-by parts property, that is necessary to employ the split formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The Riemann solver used in the simulations is a Roe scheme with entropy fix [24] to ensure that second law of thermodynamics is respected, even with the split formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' To be able to adequately handle the viscous flux in the boundaries of the elements, the lifting scheme of Bassi and Rebay [25] is used, which is also known as BR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The time marching method chosen is the five-stage, fourth-order explicit Runge-Kutta scheme of Carpenter and Kennedy [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The shock waves that appear in the simulation are stabilized using the finite-volume sub-cell shock capturing method of Sonntag and Munz [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Even though the methodology used in the simulation solves the discontinuous Galerkin approach, it only handle discontinuities in the interface of the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The shock capturing method permits to stabilize the simulation with shock waves inside the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Experimental Configuration The focus of this work is to investigate the influence of mesh and polynomial refinement on the perfectly expanded jet flow, which is present in many applications, such as supersonic military aircraft and large launch vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The experimental work of Bridges and Wernet [19] provides data flow properties for different jet flow configurations In this work, the interest is to simulate the fully expanded free jet flow configuration with a Mach number of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In this configuration the jet flow has a static pressure in the nozzle exit plane that equals the ambient static pressure with a supersonic velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' For such a flow configuration, the shock waves are weaker when compared to other operating conditions, which reduces the constraints of mesh refinement and, consequently, the computational cost of the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The experimental apparatus for the analysed configuration is composed of a convergent-divergent nozzle designed based on the method of characteristics [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The nozzle exit diameter is 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='8 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The Reynolds number based on nozzle exit diameter is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='58 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The experimental data acquisition applies the Time-Resolved Particle Image Velocimetry (TRPIV) at a 10 kHZ sample rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The investigation uses two sets of cameras, one captures the flow along the nozzle centerline, and the other captures the flow of the mixing layer along the nozzle lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Numerical Setup A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Geometry and Mesh Configuration The geometry used for the calculations in this work presents a divergent shape and axis length of 40𝐷, where 𝐷 is the jet inlet diameter and has external diameters of 16𝐷 and 25𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 1 illustrates a 2-D representation of the computational domain, indicating the inlet surface in red, the farfield region in blue, the lipline in grey, and the centerline in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Two different computational grids are used in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The coarser mesh used here, named M-1 mesh, is the same grid developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The other computational grid, which is termed M-2 mesh here, was developed specifically for the present effort and it represents a considerable improvement over the M-1 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The modifications in the M-2 mesh are both topological and the result of an increase in the number of grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' These modifications result in a much higher refinement level around the jet inlet, encompassing both the original jet as well as the strong mixing region around the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Afterwards, this mesh transitions to a uniform grid point distribution as one moves downstream in the longitudinal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The mesh generation uses a multiblock strategy in order to handle hexahedral cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 1 2-D schematic representation of the computational domain used on the jet flow simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The grid design attempts to capture the jet flow until a distance of 𝑥/𝐷 = 15 from the inlet surface, indicated in red in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Then, the size of elements increases as an attempt to dissipate frequencies that could destabilize the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Previous results [10] support the creation of mesh topology, indicating that a surface with one degree of opening angle would better represent the middle surface of the jet mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' From this surface, two regions rise to increase the resolution of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' They have a geometrical stretching in the section 𝑥/𝐷 = 0 to enable local refinement in the shear region of the flow and a uniform distribution in 𝑥/𝐷 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The internal section of the mixing layer connects to 4 Sponge region neone hexahedral block that forms the core of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' That hexahedral block has its dimensions defined to keep the size of the elements equal to the size of the last cells in the region of the mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The grid refinement in the mixing layer is defined based on the literature [5–7, 9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The grid spacing in the radial and axial directions along the mixing layer is Δ𝑦0/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='001 and Δ𝑥0/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='005, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The centerline presents 651 elements set in geometrical stretching distribution between 𝑥/𝐷 = 0 and 𝑥/𝐷 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Each region of the mixing layer contains 50 cells, and the Azimuthal direction accommodates 180 elements evenly distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 2a presents the radial mesh refinement in two different longitudinal positions, 𝑥/𝐷 = 0 and 𝑥/𝐷 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 2b illustrates the axial mesh refinement along the jet centerline and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 3 exhibits a cutplane of the mesh generated for the current paper and the baseline line mesh used in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The M-1 and M-2 grids have a total of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 million cells, respectively, and they are created with the GMSH [28] mesh generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (a) Radial mesh refinement (Δ𝑦/𝐷) in the longitudinal sections 𝑥/𝐷 = 0 and 𝑥/𝐷 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (b) Longitudinal mesh refinement (Δ𝑥/𝐷) along the jet centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 2 Distribution of grid spacing indicating radial and longitudinal refinement for the M-2 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (a) M-1 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (b) M-2 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 3 Visualization of the half-plane longitudinal cutplanes for the meshes used in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Boundary Conditions Properties on the jet inflow, (·) 𝑗𝑒𝑡, and farfield, (·) 𝑓 𝑓 , surfaces are indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 1 in red and blue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' A weakly enforced solution of a Riemann problem with a Dirichlet condition is enforced at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The flow is characterized as perfectly expanded and isothermal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 𝑝 𝑗𝑒𝑡/𝑝 𝑓 𝑓 = 𝑇𝑗𝑒𝑡/𝑇𝑓 𝑓 = 1, where 𝑝 stands for pressure and 𝑇 5 101 100 10-2 x/D=0 x/D=15 10-3 0 1 2 3 4 5 y/D101 100 △x/D 10-1 10-2 10-3 0 10 20 30 40 x/DY 7for temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The Mach number of the jet at the inlet is 𝑀𝑗𝑒𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 and the Reynolds number based on the diameter of the nozzle is 𝑅𝑒 𝑗𝑒𝑡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='58 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' A small velocity component with 𝑀 𝑓 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='01 in the streamwise direction is imposed at the farfield to avoid numerical issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' A sponge zone [29] is employed around the farfield boundaries, the gray area presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 1, to damp any oscillations that could be reflected back to the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Simulation Settings and DOFs The current work compares the effects of hp refinement using three different calculations: S1, S2, and S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The first simulation uses the M-1 computational grid, while the other two computations apply the M-2 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Both studies, S1 and S2, employ first-order polynomial reconstructions in order to achieve second-order accuracy in spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Calculation S3 uses second-order polynomial reconstructions in order to achieve a third-order accurate spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The simulations, therefore, consider from 50 to 410 million DOFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Table 1 indicates the settings used the three numerical studies performed in the present effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Table 1 Summary of simulations settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Simulation Meshes Order of DOF/cell Cells Total # of DOF Accuracy (106) (106) S1 M-1 2nd order 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 ≈ 50 S2 M-2 2nd order 8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 ≈ 120 S3 M-2 3rd order 27 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 ≈ 410 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Calculation of Statistical Properties The simulation procedure involves three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The first one is to clean off the domain since the computation starts with a static flow initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The simulations run three flow-through times (FTT) to develop the jet flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One FTT is the time required for one particle with the jet velocity to cross the computational domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In the sequence, the simulations run an additional three FTT to produce a statistically steady condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Then, in the last step, data are collected for another three FTT to obtain the statistical properties of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The procedure for developing S3 simulation is slightly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The simulation is a restart from the finished S2 calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The numerical framework FLEXI allows using one solution with different order of accuracy as initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Once the second-order solution was already available, its usage was a short come to initialize the S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Hence, the S3 numerical study runs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 FFT to allow the solution to adapt from second-order accuracy to third-order accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Then it runs an additional 2 FTT to collect data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Different frequencies of data acquisition were employed in each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The S1 case applies 160 kHz, while S2 and S3 cases use 205 and 225 kHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The mean and the root mean square (RMS) fluctuations of properties of the flow are calculated along the centerline, lipline, and different domain surfaces in the streamwise direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The centerline is defined as the line in the center of the geometry 𝑦/𝐷 = 0, whereas the lipline is a surface parallel to the centerline and located at the nozzle diameter, 𝑦/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The results from the lipline are an azimuthal mean from six equally spaced positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The four surfaces in the streamwise positions are 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5, 𝑥/𝐷 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='0, 𝑥/𝐷 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='0, and 𝑥/𝐷 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Surface properties are averaged using six equally spaced positions in the azimuthal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 4 illustrates a Mach number contours snapshot of the jet flow with the lines and surfaces of data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Results The results from S1, S2, and S3 simulations are presented in this section and compared to experimental data [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The focus of this work is to assess the resolution requirements for the correct prediction of supersonic jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' S1 and S2 calculations are performed with the same polynomial order of accuracy, while S3 simulation uses third-order accuracy polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The S1 Numerical study is performed with mesh M-1, with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 × 106 elements, and S2 and S3 calculations are performed with mesh M-2, with 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 × 106 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The simulations have approximately 50, 120, and 410 million DOFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 4 Snapshot of the jet simulation with the two longitudinal lines and three crossflow lines along which data is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Mach number contours are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Velocity and Density Contours Initially, the contours of the mean longitudinal velocity component, RMS of longitudinal velocity fluctuation, and mean density are presented for the three simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 5 presents the contours of the mean longitudinal velocity component on a cut plane at 𝑧/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The contours indicate qualitatively improvement in results when increasing the number of DOFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One can notice the size of the jet core is bigger in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5c than in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5b and 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Moreover, the development of the mixing layer start closer to the jet inlet in S3 simulation than in S2 and S1 calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Furthermore, it is difficult to notice the existence of shock waves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5a, which are more clearly visible in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5b and 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 6 presents the contours of RMS of longitudinal velocity fluctuations on a cut plane at 𝑧/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One can notice the early development of the mixing layer when increasing the number of DOFs in the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 6c the increase in the RMS values for longitudinal velocity fluctuation occurs right after the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The same physical phenomenon occurs farther when decreasing the simulation resolution, which is noticeable when comparing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='s 6b and 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The contours of RMS of longitudinal velocity fluctuation also show that the fluctuation levels get smaller with earlier development of the mixing layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 6c the region of high velocity fluctuation is thinner and presents smaller values than in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 6a, the region with a high level of velocity fluctuation is the largest, and it presents the highest values when compared to the other two results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The last contours presented in this section, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7, compare the mean density results from the three simulations on a cut plane at 𝑧/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' It is possible to observe that each simulation presents different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7a the shock waves are weak, being hardly visible with adequate range scales for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Moreover, one can notice a few shock waves and expansion waves reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The appearance of the first shock waves occurs far from the jet inlet boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 7b presents a different behavior of the flow with shock waves and expansion waves significantly different from those observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The first shock waves appear closer to the jet inlet boundary conditions, and they are visible, which indicates that they are significantly stronger than those from the S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Another interesting observation is the number of shock waves and expansion waves reflection, which is much larger than the presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The final density results are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7c, which is the result from S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' It is possible to observe that the shock and expansion waves are better defined, presenting smaller thickness, which is also evidence of the improvements when increasing the number of DOFs in the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' When comparing the results from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7c with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7b, it is possible to observe that the intensity of the shock waves is stronger in S2 simulation than in S3, even with the larger thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' This is in agreement with results presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5b and 5c, in which the shock waves from S2 calculation where more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The quantity of shock waves reflections in the S2 and S3 test cases are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Velocity Profiles One can better understand the effects of the hp refinement on the numerical solution when comparing the results to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figure 8 presents the mean longitudinal velocity component and the RMS values of the velocity fluctuation distributions along the jet centerline (𝑦/𝐷 = 0) and the jet lipline (𝑦/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Analyzing the results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8a, the improvement in the capacity to capture flow features when increasing the numerical resolution of the calculations is prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One can notice the numerical solution progression of the mean longitudinal velocity component towards the experimental data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The potential core is longer in S3 than in the other cases, which is presented in detail in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Moreover, the most refined numerical study presents the intensity of the shock waves and the slope of the decay of velocity comparable to the ones of the reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One can observe that the profiles calculated in S2 and S3 simulation are closer than the ones computed in S1 and S2, even with a higher ratio of degrees 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5D 5D 110D Centerline(a) S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (b) S2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (c) S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5 Contours of mean longitudinal velocity component along cutplanes in 𝑧/𝐷 = 0 for the three simulations performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' of freedom, 𝐷𝑂𝐹𝑆3/𝐷𝑂𝐹𝑆2 ≈ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='42 and 𝐷𝑂𝐹𝑆2/𝐷𝑂𝐹𝑆1 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The slight improvement, even with a higher DOF ratio, can also indicate that the simulation S3 is very close to provide the converged solution for the chosen modeling approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The results presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8b agree well with the results of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The increase in the simulation resolution improves the calculation of RMS velocity fluctuation towards the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Analyzing the results close to the jet inlet, the increase in the velocity fluctuation occurs further upstream in S3 simulation, which is in agreement with the contours in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5 where the shock waves of S3 numerical study appear closer to jet inlet when compared to the S2 and S1 calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' However, even with an early increase in the fluctuation levels in S3 computations, the slope of its profile in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8b is smoother and closer to the reference than the one calculated in S2 and S1 numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In the same image, S3 simulation presents two peaks of velocity fluctuation, with the second one close to the peak indicated in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' However, its RMS fluctuations are higher than experimental data and S2 simulation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The presence 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 1- 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 x95) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='9(a) S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (b) S2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (c) S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 6 Contours of RMS values of longitudinal velocity component fluctuations along cutplanes in 𝑧/𝐷 = 0 for the three simulations performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' of small values of velocity fluctuation close to the jet inlet could be related to imposed jet entrance boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figures 8c and 8d illustrate the profiles of mean and RMS fluctuations of the longitudinal velocity component along the lipline, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One can notice the improvements in the simulation resolution with S3 and S2 simulations providing mean profiles closer to the experimental one than the results from the S1 calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The mean velocity oscillations in the vicinity of the inlet jet may be correlated with shock waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' They are also present in the experimental profile along the lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The S1 and S2 simulations present an early reduction of mean velocity when comparing to the reference profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The most refined calculation has a mean velocity profile in good agreement with the experimental data along the lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Such behavior is also noticeable along the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The calculations performed in the present paper have generated fluctuation profiles of the longitudinal velocity fluctuation along the lipline, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8d, that indicate a different trend from what is stated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8a to 8c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One can notice that, when increasing the simulation resolution, the peak of velocity fluctuation moves towards the jet inlet, at 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 for S1 simulation and ≈ 𝑥/𝐷 = 1 for S2 and S3 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 1- 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 x95) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='9(a) S1 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (b) S2 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (c) S3 simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 7 Contours of mean density along cutplanes in 𝑧/𝐷 = 0 for all three simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' This outcome is present in the contours indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 5, where the initial spreading of the mixing layer in S2 and S3 simulations starts earlier than in the S1 calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Such behavior is different from the experimental RMS profiles along the lipline in which the increase in the velocity fluctuation occurs slowly before reaching a plateau between 𝑥/𝐷 = 5 to 𝑥/𝐷 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Using a flat hat velocity profile at the jet entrance for the numerical calculations can explain the divergence between the fluctuations obtained from the numerical approach and the experiments near the inlet domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Such boundary condition neglects the turbulent boundary layer effects carried from the nozzle to the 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 x9§) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='9(a) Centerline, mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (b) Centerline, RMS values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (c) Lipline, mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' (d) Lipline, RMS values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8 Results for the mean streamwise velocity component distributions (left) and RMS values of streamwise velocity component fluctuations (right) in the jet centerline, 𝑦/𝐷 = 0 (top), and lipline, 𝑦/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Table 2 Summary of potential core length for all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Simulation Potential core Error to length (𝑥/𝐷) experimental data (%) S1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='3 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='0 S2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='3 S3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 S1 S2 S3 Exp /U, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 0 5 10 15 20 x/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='1 rms S1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='05 S2 S3 Exp 0 0 5 10 15 20 x/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='8 S1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='7 S2 S3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='6 Exp /U, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 0 5 10 15 20 x/D0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 S1 S2 S3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='15 Exp rms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='1 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='05 0 0 5 10 15 20 x/DFigure 9 displays different statistical properties of the flow in four streamwise positions, 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5, 𝑥/𝐷 = 5, 𝑥/𝐷 = 10 and 𝑥/𝐷 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Figures 9a to 9d present the mean longitudinal velocity component, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9e to 9h illustrate the RMS of the longitudinal velocity fluctuation, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9i to 9l introduces the RMS of the radial velocity fluctuation, and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9m to 9p indicate the mean shear stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The first set of results, in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9a to 9d, explicit some aspects of the numerical results not investigated by the comparison of 2-D field of properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In the first longitudinal position, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9a, S1 calculations generated mean profiles that are in better agreement with the experimental data than the ones from S2 and S3 numerical studies, which indicate a larger spreading of velocity at this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The early development of the mixing layer from S2 and S3 simulations reinforces the influence of the choice of boundary conditions imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Analyzing the profiles from downstream positions, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9b to 9d, it is possible to verify the large spreading of velocity from the S1 simulation, with a reduction of longitudinal velocity in the jet centerline when compared to the other numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Calculations with higher resolution can better capture the experimental trends, with the simulation S3 getting closer to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The profiles of RMS values of streamwise velocity fluctuation are indicated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9e to 9h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The numerical results at 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 present a similar profile to the one from the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' However, the peaks generated by the calculations are higher than the experimental ones, with the results from the S3 simulation being the closest to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The same conclusion can be drawn for 𝑥/𝐷 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='0, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9g and 9h, the all numerical results present a shape similar to experimental data, with a nearly constant value of velocity fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9g the experimental data still present a small level of fluctuation close to the centerline, which is not seen in the numerical profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Profiles of RMS values of radial velocity component fluctuation are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9i to 9l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In the first two longitudinal positions, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9i and 9j, the numerical results present a larger peak of fluctuation than the one from the experimental data, with the profiles from the S3 simulation getting closer to reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9k and 9l the RMS profiles from the calculations are very similar, with the centerline of the experimental data presenting small values of fluctuation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Mean shear-stress tensor component profiles are presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9m to 9p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In the first two positions, 𝑥/𝐷 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 and 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5, the peak of the shear-stress tensor from numerical calculations is larger than the experimental one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Moreover, the peak region is wider than the one observed in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9n the peaks are still larger than those observed in the reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The differences between the simulations are smaller and closer to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' However, the region of the peaks is still wider than the one visualized in the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9o and 9p, the differences between numerical results and the experimental data are reduced, and once more, all profiles are nearly matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Summary of Discussions Results from the S2 simulation present significant improvements when compared to the solution from the S1 calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The increase in polynomial order of accuracy in the S3 study brings the results to a good agreement with the reference experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The most significant point that does not present improvements when increasing the discretization resolution is related to the mean and fluctuating longitudinal velocity close to jet lipline, Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 8c and8d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Grid refinement yields a velocity fluctuation peak that occurs closer to the jet inlet when compared to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' This behavior may be related to the hypothesis used to impose the inlet boundary condition, which neglects the effects of the jet boundary layer and leads to a not realistic turbulence transition in the vicinity of the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The current work provides intel on the resolution requirements to perform large-eddy simulations of supersonic jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The next step concerns the exploration of alternatives to improve the jet simulation in the lipline close to the jet inlet condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' A solution to improve the simulation in this region is a better characterization of the flow leaving the nozzle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The choice of a uniform velocity profile is preliminary, and it does not represent the physics of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The continuity of the work will focus on options to produce an experiment-like condition in the jet inlet condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 12 (a) 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 (b) 𝑥/𝐷 = 5 (c) 𝑥/𝐷 = 10 (d) 𝑥/𝐷 = 15 (e) 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 (f) 𝑥/𝐷 = 5 (g) 𝑥/𝐷 = 10 (h) 𝑥/𝐷 = 15 (i) 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 (j) 𝑥/𝐷 = 5 (k) 𝑥/𝐷 = 10 (l) 𝑥/𝐷 = 15 (m) 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5 (n) 𝑥/𝐷 = 5 (o) 𝑥/𝐷 = 10 (p) 𝑥/𝐷 = 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 9 Profiles of mean streamwise velocity component, RMS of streamwise velocity fluctuation, RMS of radial velocity fluctuation, and mean shear-stress tensor component (from top to bottom) at four streamwise positions 𝑥/𝐷 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='5, 𝑥/𝐷 = 5, 𝑥/𝐷 = 10 and 𝑥/𝐷 = 15 (from left to right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 13 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='01 /U?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Concluding Remarks The current work assesses the effects of mesh and polynomial (hp) refinement using a nodal discontinuous Galerkin methodology to evaluate the resolution requirements for large eddy simulations of compressible jet flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The problem of interest is a perfectly expanded supersonic jet flow with a Mach number of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 and Reynolds number based on the jet exit diameter of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='58 × 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Initially, a mesh with 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='2 × 106 elements is used with interpolation polynomials that yield second-order spatial accuracy, in order to produce a starting point for the comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Such calculations use, therefore, the equivalent to approximately 50 × 106 degrees of freedom (DOFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Afterwards, a new mesh is developed with some topological improvements and additional refinement, leading to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='4 × 106 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The new mesh is applied in simulations using second-order and third-order spatial accuracy, resulting in 120 × 106 and 410 × 106 DOFs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The results for the simulations are compared to experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The paper initially investigates the contours of mean velocity, mean pressure, and velocity fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The comparison indicates that the mesh/polynomial refinement improves the jet calculations by enhancing the prediction of the mixing layer and of the series of shock and expansion waves in the jet core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Therefore, as one should expect, more refined computations lead to an improved ability to predict flow features, as one can see in the present paper by the comparison of the numerical solutions and the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Therefore, it is correct to state that the present paper indicates mesh and discretization parameters for LES-based calculations, using a nodal discontinuous Galerkin formulation, that provide supersonic jet flow results in good agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' One important aspect that becomes clear in the present calculation results is that the jet inlet boundary condition, used in the current work, has a significant impact on the ability of representing the very early stages of jet mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' In particular, this observation becomes evident by looking at the behavior of RMS values of fluctuating properties near the jet exit, along the jet lipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' All three simulations have failed to capture the correct mixing behavior, as evidenced by the comparison with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Moreover, the increased numerical resolution, although providing much better comparisons for the overall solution, does not improve the behavior of fluctuating properties near the jet exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Hence, the continuation of the present effort will address possible improvements in the jet inlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Acknowledgments The authors acknowledge the support for the present research provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, under the Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 309985/2013-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The work is also supported by the computational resources from the Center for Mathematical Sciences Applied to Industry, CeMEAI, funded by Fundação de Amparo à Pesquisa do Estado de São Paulo, FAPESP, under the Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 2013/07375-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The authors further acknowledge the National Laboratory for Scientific Computing (LNCC/MCTI, Brazil) for providing HPC resources of the SDumont supercomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' This work was also granted access to the HPC resources of IDRIS under the allocation 2021-A0112A12067 made by GENCI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The first author acknowledges authorization by his employer, Embraer S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=', which has allowed his participation in the present research effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' The doctoral scholarship provide by FAPESP to the third author, under the Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 2018/05524-1, is thankfully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' Additional support to the fourth author under the FAPESP Research Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 2013/07375-0 is also gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' References [1] Brès, G.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 2014-2740, Proceedings of the 20th AIAA/CEAS Aeroacoustics Conference, Atlanta, GA, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/b9AzT4oBgHgl3EQfnf2W/content/2301.01582v1.pdf'} diff --git a/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf b/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d2fa09576fceebd3efc6280a75131170d8d66cbf --- /dev/null +++ b/b9FAT4oBgHgl3EQfXx3V/content/2301.08536v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e757ff748077a1d101d443e9e70af40fe22708b55786bda6a3faffd6f0faac50 +size 2101048 diff --git a/b9FAT4oBgHgl3EQfXx3V/vector_store/index.faiss b/b9FAT4oBgHgl3EQfXx3V/vector_store/index.faiss 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Electrical and +Computer Engineering, The University of Texas at Austin; +Abstract +In the United States, primary open-angle glaucoma (POAG) is the leading cause of blindness, especially among +African American and Hispanic individuals. Deep learning has been widely used to detect POAG using fundus im- +ages as its performance is comparable to or even surpasses diagnosis by clinicians. However, human bias in clinical +diagnosis may be reflected and amplified in the widely-used deep learning models, thus impacting their performance. +Biases may cause (1) underdiagnosis, increasing the risks of delayed or inadequate treatment, and (2) overdiagnosis, +which may increase individuals’ stress, fear, well-being, and unnecessary/costly treatment. In this study, we exam- +ined the underdiagnosis and overdiagnosis when applying deep learning in POAG detection based on the Ocular +Hypertension Treatment Study (OHTS) from 22 centers across 16 states in the United States. Our results show that +the widely-used deep learning model can underdiagnose or overdiagnose under-served populations. The most un- +derdiagnosed group is female younger (< 60 yrs) group, and the most overdiagnosed group is Black older (≥ 60 +yrs) group. Biased diagnosis through traditional deep learning methods may delay disease detection, treatment and +create burdens among under-served populations, thereby, raising ethical concerns about using deep learning models +in ophthalmology clinics. +1 +Introduction +Primary open-angle glaucoma (POAG) is one of the leading causes of blindness in the US and worldwide.1 It has been +projected to affect approximately 111.8 million people by 2040. Among these patients, 5.3 million may be bilaterally +blind.2 In the United States, POAG is the most common form of glaucoma and is the leading cause of blindness among +Black and Hispanic individuals.3,4 POAG is asymptomatic until it reaches an advanced stage whereby peripheral +visual field loss encroaches on central vision. However, early detection and treatment can avoid most blindness +caused by POAG.5 Therefore, timely identification of individuals with glaucoma is critical to guide medical and +surgical treatments and patient monitoring.6,7 Detecting POAG is crucial, yet challenging, due to the high demands +on screening and experiences of ophthalmologists.8 Fundus photography is convenient and inexpensive for recording +optic nerve head structure9. Therefore, developing an automatic deep learning (DL) model to detect POAG with high +accuracy from fundus photographs to address the lack of experienced ophthalmologists is important. +In recent years, developments in artificial intelligence (AI) have provided potential opportunities for automatic POAG +diagnosis using fundus photographs.10–17 While such DL models have increasingly achieved expert-level performance, +there is growing concern that these models may reflect and amplify human bias and reduce the quality of their per- +formance in historically under-served populations such as Black individuals.18–21 The topic of AI-driven underdiag- +nosis and overdiagnosis has been particularly important.21,22 In this work, we define “Underdiagnosis” as the problem +where the model falsely claims that the individual is healthy, increasing the risks of delayed or inadequate treatment, +and “Overdiagnosis” as the problem where the model predicts healthy people wrongly into sick ones, leading to indi- +viduals’ stress, fear, and unnecessary/costly treatment. Therefore, model fairness of POAG diagnosis can be a crucial +concern if used in the clinical pipeline for patient triage. +To our knowledge, the topic of AI-driven underdiagnosis and overdiagnosis of POAG diagnosis has not been ex- +plored before.23 In this study, we will systematically study underdiagnosis and overdiagnosis bias in the AI-based +POAG diagnosis models from a large, cross-sectional dataset obtained from the Ocular Hypertension Treatment Study +(OHTS).24 OHTS is one of the largest longitudinal clinical trials in POAG (1,636 participants and 37,399 images) from +aEqual contributions. +arXiv:2301.11315v1 [cs.CV] 26 Jan 2023 + +22 centers in the United States.25 We chose these subgroups mainly because of the clear history of bias in previous +research.26–29 +2 +Materials and Methods +2.1 +The OHTS dataset +In this study, we perform a study of model bias in POAG diagnosis in a large-scale, longitudinal, and population- +based dataset (Table 1). Previous studies show that the classifier exhibits different performances in different individual +groups stratified by sex, race, and age.29 Inspired by these works, we report results by considering these factors. +The dataset is obtained from the Ocular Hypertension Treatment Study (OHTS). The study protocol was approved +by the Institutional Review Board at each clinical center, and the Weill Cornell Medicine IRB determined that the +protocol does not constitute human subjects research. All risk factors were measured at baseline before the onset of +the disease and collected for approximately 16 years. The participants in this dataset were selected according to both +eligibility and exclusion criteria.24 +Briefly, the eligibility criteria include intraocular pressure (between 24 mm Hg and 32 mm Hg in one eye and between +21 mm Hg and 32 mm Hg in the fellow eye) and age (between 40 and 80 years old). The visual field tests were +interpreted by the Visual Field Reading Center, and the stereoscopic photographs were interpreted by the Optic Disc +Reading Center. Exclusion criteria included previous intraocular surgery, visual acuity worse than 20/40 in either +eye, and diseases that may cause optic disc deterioration and visual field loss (such as diabetic retinopathy). The +gold standard POAG labels were graded at the Optic Disc Reading Center. In brief, two masked certified readers +were instructed to independently detect glaucomatous optic disc deterioration over time. If there was a disagreement +between two readers, a senior reader reviewed the subject in a masked fashion. The POAG diagnosis in a quality +control sample of 86 eyes (50 normal eyes and 36 with progression) showed test-retest agreement at κ = 0.70 (95% +confidence interval [CI], 0.55-0.85). More details of the reading center workflow have been described in Gorden et +al.25 +Total +POAG +No. of images +37,399 +2,327 ( 6.22%) +Sex +Male +16,185 +1,303 ( 8.05%) +Female +21,154 +1,024 ( 4.84%) +Race +Non-Black +28,460 +1,554 ( 5.46%) +Black +8,879 +773 ( 8.71%) +Age +40-49 +4,292 +64 ( 1.49%) +50-59 +11,962 +356 ( 2.98%) +60-69 +11,904 +846 ( 7.11%) +70-79 +7,593 +829 (10.92%) +≥80 +1,588 +232 (14.61%) +Table 1: The characteristics of the OHTS dataset. +2.2 +Definition of POAG underdiagnosis and overdiagnosis +To assess model fairness, we compare underdiagnosis and overdiagnosis rates across subpopulations. Similar to ref,29 +we define the underdiagnosis rate as the false-negative rate (FNR) of the binarized model prediction for the POAG +at the subgroup levels: P(ˆy = non-POAG|y = POAG, A). Here A is the sex, race, or other factors that the model +should be free of bias. For example, the underdiagnosis of female individuals is given by P(ˆy = non-POAG|y = +POAG, female). We then compare these underdiagnosis rates across subpopulations. We say a classifier is fair if the + +Overall population +DenseNet +POAG +Model training +Subpopulation comparisons +Sex +vs +Race +vs +Age +vs +Figure 1: The experimental design. We focus our underdiagnosis and overdiagnosis experiments on subgroups of +race, sex, and age. +individuals in the protected and unprotected groups satisfy the formula: +P(ˆy = non-POAG|y = POAG, female) = P(ˆy = non-POAG|y = POAG, male) +For overdiagnosis, we will measure the false-positive rate (FPR) for the POAG across all subgroups, i.e., P(ˆy = +POAG|y = non-POAG, A). This measure shows that the model fails to diagnose those individuals who would never +develop POAG. Some of the harms caused by overdiagnosis are anxiety and having treatments that are not needed. +Besides single identities, we also examined underdiagnosis and overdiagnosis in intersectional groups - individuals +who belong to two subpopulations.29 For example, the underdiagnosis of Black female individuals is given by P(ˆy = +non-POAG|y = POAG, female, Black). Here, we want to examine if individuals who belong to two subgroups may +have a larger underdiagnosis rate. In other words, not all female individuals are misdiagnosed at the same rate (for +example, Black female individuals are misdiagnosed more than non-Black female individuals). +2.3 +Model development +Figure 1 shows the pipeline of our model. All images are resized to 224 × 224 × 3 and normalized using the mean and +standard deviation of the ImageNet dataset.30 We sequentially apply three augmentation operations on the fly during +training: (1) random rotation between 0◦ and 10◦, (2) random translation: an image was translated randomly along +the x- and y-axes by distances ranging from 0 to 10% of width or height of the image, and (3) random flipping. The +diversity of the dataset could be increased due to these data augmentation techniques, which generate effective and +robust representations. +The input images are then passed through a convolutional neural network to generate the prediction results. In this +study, we used the DenseNet-20131 pre-trained on ImageNet.32 We replaced the last layer with a new randomly +initialized fully-connected layer with 2 output neurons (POAG and normal). We used binary cross-entropy as the loss +function. Since there are only 6.22% of images in the OHTS dataset that has POAG (Table 1, a severe class imbalance +exists for POAG diagnosis, to overcome this problem, we adopted weighted cross-entropy, a commonly used loss +function in classification. The adopted weighted cross-entropy was: +Lθ = − 1 +N +N +� +i=1 +[βyi log(ˆyiDenseNet(xi, θ)) + (1 − β)(1 − yi) log(1 − ˆyiDenseNet(xi, θ))] +(1) +where N is the number of training examples, β is the balancing factor between positive and negative samples, yi is the +observed true label of image xi, ˆyi is the probability predicted by the classifier, and θs represents the parameters of the +DenseNet-201. Here, we used inversely proportional to POAG frequency in the training data. Finally, we fine-tuned +the entire network on the OHTS in an end-to-end manner. + +2.4 +Experimental settings +The model was implemented by Keras with a backend of Tensorflow. The network was optimized using the Adam +optimizer method.33 The learning rate is 5×10−5. The experiments were performed on Intel Core i9-9960 X 16 cores +processor and NVIDIA Quadro RTX 6000 GPU. +We used the five-fold cross-validation in this study. We split the entire dataset randomly into five groups at the +individual level. This ensured that no participant was in more than one group to avoid cross-contamination between +the training and testing datasets. In each fold of the cross-validation, we took one group (20% of total subjects) as the +hold-out test set and the remaining 4 groups as the training set. +3 +Results and discussion +The underdiagnosis (Figure 2) and overdiagnosis (Figure 3) for POAG screening show an inverse relationship in both +subgroups and intersectional groups in the OHTS dataset. As suggested by Seyyed et al,29 this indicates that the model +consistently misclassifies the under-served subpopulations due to potential biases, rather than simple, random errors. +3.1 +Underdiagnosis in subpopulations and intersectional groups +Figure 2A shows that the underdiagnosis rate differs in all subpopulations of sex, race, and age. Specifically, females, +non-Black individuals, and individuals under 60 years old have higher underdiagnosis rates than their counterparts. +In other words, the individuals of these groups are more likely falsely predicted as healthy, preventing them from +receiving appropriate treatments. From Table 1, we can see that the individuals in these groups have a lower prevalence +of POAG. For example, the POAG rate of individuals aged ≥ 60 is around three times more than that of individuals +aged < 60 (9.04% vs. 2.58%). Since these groups may not be adequately represented in the OHTS data, the supervised +machine learning model trained from the data might be biased. +In addition, female individuals have the highest underdiagnosis rate among indicated subgroups. However, the num- +bers of POAG female and male individuals are about the same. This observation suggests that a simple resampling +approach to ensure the dataset is balanced across different groups may not always be a solution. +We also investigate intersectional groups, which means the individuals belong to two subpopulations, e.g., female +Black individuals. As shown in Figure 2B(i)-(iii), intersectional groups also have the problem of underdiagnosis. +Figure 2B(i) shows that female non-Black individuals have a higher underdiagnosis rate than female Black individuals. +Female individuals aged < 60 years have a higher rate than female individuals aged ≥ 60 years. Compared to the +single identities, we observed that the intersectional identities amplify the model bias. For example, the difference +between Black and non-Black individuals is 2.50%, but the difference between female Black and female non-Black is +8.61%. The most underdiagnosed group is young female individuals. +Similarly, Figure 2B(ii) shows that Black females and younger individuals have a higher underdiagnosis rate than +Black males and older adults. Figure 2B(iii) also indicates that female youngers are more heavily underdiagnosed. +There is no significant difference between Black and non-Black younger individuals. This may be partially due to the +small test set sizes (84 cases with the POAG label for individuals aged < 60 years). +3.2 +Overdiagnosis in subpopulations and intersectional groups +Figure 3A shows that healthy males and healthy older adults are more likely to be misclassified as POAG positive. On +the other hand, the Black and non-Black subpopulations in Figure 3A have similar overdiagnosis rates. From Figure +3B, we observed that the intersectional identities are often overdiagnosed more heavily than the group in aggregate. +Specifically, female and Black older adults are more easily overdiagnosed than female and Black younger adults +(Figure 3B(i) and (ii)). + +Figure 2: Underdiagnosis analysis across subgroups of sex, race, and age in the OHTS dataset. A) The underdiagnosis +rates in the subpopulations. B) Intersectional underdiagnosis rates for female individuals, Black individuals, and +individuals aged < 60 years, respectively. The results are average results of the five-fold cross-validation. +Figure 3: Overdiagnosis analysis across subgroups of sex, race, and age in the OHTS dataset. A) The overdiagno- +sis rates in the subpopulations. B) Intersectional overdiagnosis rates for female individuals, Black individuals, and +individuals aged < 60 years, respectively. The results are average results of the five-fold cross-validation. + +A +0.6- + rate +underdiagnosis +Subgroup +0.4 +0.2 +0.0 +Male +Female +Non-black +Black +Age < 60 +Age >= 60 +B +Female +0.8- +Black +0.8- +Age < 60 +0.8- +underdiagnosis rate +0.6- +0.6- +0.6 +T +Intersectional +0.4 - +0.4- +0.4 +T +0.2 - +0.2 - +0.2 +0.0. +0.0 - +0.0 +60 +<60 += 60 +-black +-black +Black +Age +Male +Blac +Age +Non- +Age +(i) +(ii) +(ii)A +0.25- +0.20 +0.15 +0.10 +0.05- +0.00 +Male +Female +Non-black +Black +Age < 60 +Age >= 60 +B +0.20 +Female +0.20 +0.15 +Black +Age < 60 + overdiagnosis rate +0.15 +Intersectional +0.15 +0.10 +0.10 +0.10 - +UI +0.05- +0.05 +0.05 - +0.00 +0.00 +0.00 +60 +.60 +-black +Male +Male +Age +Age +Female +(i) +(ii) +(ili)3.3 +Discussions and Limitations +We have found that underdiagnosis and overdiagnosis exist in the OHTS dataset for POAG diagnosis. The DL model +generated underdiagnosis and overdiagnosis biases in under-served subpopulations, such as female individuals and in- +dividuals under 60 years old. Such biased diagnoses were even greater among individuals with intersectional identities, +including Black females and females aged under 60 years old. Subpopulations like female individuals and individuals +under 60 years old are most affected by POAG in the OHTS dataset, suggesting further attention when applying DL +models in clinical decision-making34. +The DL model performs similarly on Black and non-Black individuals mainly because the number of Black individuals +in our datasets is much smaller than the other group, even though they have a higher prevalence of POAG. Therefore, +the adverse effects on Black individuals are probably mitigated by the small number of their population. This reminds +us that we must also consider the number of subpopulations when constructing the dataset35. +We also found that these differences in underdiagnosis and overdiagnosis exist in other clinical research areas (e.g., +thoracic diseases, heart diseases, and kidney diseases),29,36,37 which means that these disparities may be widespread +in biomedical study. +One limitation of this study is that only one dataset was used. However, the OHTS data is one of the largest longitudinal +clinical trials in POAG from 22 centers in the United States. Therefore, we believe the observations of model bias are +likely to be generalizable. +Another limitation is that we only studied the fairness of a binarized model. Unfortunately, the probabilities predicted +by the model may not be calibrated: probabilities are calibrated where a prediction of POAG with confidence p is +correct p percent of the time.38 That being said, the probabilities predicted by the model may be over-confident in +some cases and under-confident in other cases. Moreover, as we see in Figures 2, the severely imbalanced data may +result in even more bias in the predicted probabilities as they over-favor in predicting the majority class. As such, +we should investigate the relationship between calibration and POAG underdiagnosis/overdiagnosis in the future. In +addition, we plan to develop an efficient method to reduce bias in the future. +4 +Conclusion +In this paper, we systematically study underdiagnosis and overdiagnosis bias in the DL-based POAG diagnosis models +and identify the factors contributing to model fairness. We find deep learning-based underdiagnosis and overdiagnosis +exist among under-served subpopulations in POAG diagnosis on the OHTS dataset. Underdiagnosis will prevent +the individuals from receiving appropriate treatment, while overdiagnosis will let the individuals receive or continue +receiving unnecessary treatments. Bias between the individuals in intersectional subgroups such as females under +60 years and Black females are more severe. This emphasizes that bias mitigation approaches should consider the +combination of characteristics involved in bias rather than a single identity. As deep learning models are implemented +in clinical practice, this problem takes on particular urgency. +Acknowledgment +This work was supported by the National Library of Medicine under Award No. 4R00LM013001, NSF CAREER +Award No. 2145640, and Amazon Research Award. +References +1. Bourne RRA, Stevens GA, White RA, Smith JL, Flaxman SR, Price H, et al. 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Advances in neural +information processing systems. 2017;30. + diff --git a/c9FIT4oBgHgl3EQfnivD/content/tmp_files/load_file.txt b/c9FIT4oBgHgl3EQfnivD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8450454ba1d1e30ce00f64a3f24c87fe64ebc7df --- /dev/null +++ b/c9FIT4oBgHgl3EQfnivD/content/tmp_files/load_file.txt @@ -0,0 +1,450 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf,len=449 +page_content='Evaluate underdiagnosis and overdiagnosis bias of deep learning model on primary open-angle glaucoma diagnosis in under-served populations Mingquan Lin1,a, Yuyun Xiao1,a, Bojian Hou2, Tingyi Wanyan1, Mohit Manoj Sharma1, Zhangyang Wang3, Fei Wang1, Sarah Van Tassel1, Yifan Peng1 1Population Health Sciences, Weill Cornell Medicine, New York, NY, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 2 Department of Biostatistics, Epidemiology and Informatics, University of Pennsylvania;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 3 Electrical and Computer Engineering, The University of Texas at Austin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Abstract In the United States, primary open-angle glaucoma (POAG) is the leading cause of blindness, especially among African American and Hispanic individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Deep learning has been widely used to detect POAG using fundus im- ages as its performance is comparable to or even surpasses diagnosis by clinicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' However, human bias in clinical diagnosis may be reflected and amplified in the widely-used deep learning models, thus impacting their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Biases may cause (1) underdiagnosis, increasing the risks of delayed or inadequate treatment, and (2) overdiagnosis, which may increase individuals’ stress, fear, well-being, and unnecessary/costly treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In this study, we exam- ined the underdiagnosis and overdiagnosis when applying deep learning in POAG detection based on the Ocular Hypertension Treatment Study (OHTS) from 22 centers across 16 states in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Our results show that the widely-used deep learning model can underdiagnose or overdiagnose under-served populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The most un- derdiagnosed group is female younger (< 60 yrs) group, and the most overdiagnosed group is Black older (≥ 60 yrs) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Biased diagnosis through traditional deep learning methods may delay disease detection, treatment and create burdens among under-served populations, thereby, raising ethical concerns about using deep learning models in ophthalmology clinics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 1 Introduction Primary open-angle glaucoma (POAG) is one of the leading causes of blindness in the US and worldwide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='1 It has been projected to affect approximately 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='8 million people by 2040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Among these patients, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='3 million may be bilaterally blind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='2 In the United States, POAG is the most common form of glaucoma and is the leading cause of blindness among Black and Hispanic individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='3,4 POAG is asymptomatic until it reaches an advanced stage whereby peripheral visual field loss encroaches on central vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' However, early detection and treatment can avoid most blindness caused by POAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='5 Therefore, timely identification of individuals with glaucoma is critical to guide medical and surgical treatments and patient monitoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='6,7 Detecting POAG is crucial, yet challenging, due to the high demands on screening and experiences of ophthalmologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='8 Fundus photography is convenient and inexpensive for recording optic nerve head structure9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Therefore, developing an automatic deep learning (DL) model to detect POAG with high accuracy from fundus photographs to address the lack of experienced ophthalmologists is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In recent years, developments in artificial intelligence (AI) have provided potential opportunities for automatic POAG diagnosis using fundus photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='10–17 While such DL models have increasingly achieved expert-level performance, there is growing concern that these models may reflect and amplify human bias and reduce the quality of their per- formance in historically under-served populations such as Black individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='18–21 The topic of AI-driven underdiag- nosis and overdiagnosis has been particularly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='21,22 In this work, we define “Underdiagnosis” as the problem where the model falsely claims that the individual is healthy, increasing the risks of delayed or inadequate treatment, and “Overdiagnosis” as the problem where the model predicts healthy people wrongly into sick ones, leading to indi- viduals’ stress, fear, and unnecessary/costly treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Therefore, model fairness of POAG diagnosis can be a crucial concern if used in the clinical pipeline for patient triage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' To our knowledge, the topic of AI-driven underdiagnosis and overdiagnosis of POAG diagnosis has not been ex- plored before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='23 In this study, we will systematically study underdiagnosis and overdiagnosis bias in the AI-based POAG diagnosis models from a large, cross-sectional dataset obtained from the Ocular Hypertension Treatment Study (OHTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='24 OHTS is one of the largest longitudinal clinical trials in POAG (1,636 participants and 37,399 images) from aEqual contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='11315v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='CV] 26 Jan 2023 22 centers in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='25 We chose these subgroups mainly because of the clear history of bias in previous research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='26–29 2 Materials and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='1 The OHTS dataset In this study, we perform a study of model bias in POAG diagnosis in a large-scale, longitudinal, and population- based dataset (Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Previous studies show that the classifier exhibits different performances in different individual groups stratified by sex, race, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='29 Inspired by these works, we report results by considering these factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The dataset is obtained from the Ocular Hypertension Treatment Study (OHTS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The study protocol was approved by the Institutional Review Board at each clinical center, and the Weill Cornell Medicine IRB determined that the protocol does not constitute human subjects research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' All risk factors were measured at baseline before the onset of the disease and collected for approximately 16 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The participants in this dataset were selected according to both eligibility and exclusion criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='24 Briefly, the eligibility criteria include intraocular pressure (between 24 mm Hg and 32 mm Hg in one eye and between 21 mm Hg and 32 mm Hg in the fellow eye) and age (between 40 and 80 years old).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The visual field tests were interpreted by the Visual Field Reading Center, and the stereoscopic photographs were interpreted by the Optic Disc Reading Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Exclusion criteria included previous intraocular surgery, visual acuity worse than 20/40 in either eye, and diseases that may cause optic disc deterioration and visual field loss (such as diabetic retinopathy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The gold standard POAG labels were graded at the Optic Disc Reading Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In brief, two masked certified readers were instructed to independently detect glaucomatous optic disc deterioration over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' If there was a disagreement between two readers, a senior reader reviewed the subject in a masked fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The POAG diagnosis in a quality control sample of 86 eyes (50 normal eyes and 36 with progression) showed test-retest agreement at κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='70 (95% confidence interval [CI], 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='55-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' More details of the reading center workflow have been described in Gorden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='25 Total POAG No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' of images 37,399 2,327 ( 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='22%) Sex Male 16,185 1,303 ( 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='05%) Female 21,154 1,024 ( 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='84%) Race Non-Black 28,460 1,554 ( 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='46%) Black 8,879 773 ( 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='71%) Age 40-49 4,292 64 ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='49%) 50-59 11,962 356 ( 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='98%) 60-69 11,904 846 ( 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='11%) 70-79 7,593 829 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='92%) ≥80 1,588 232 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='61%) Table 1: The characteristics of the OHTS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='2 Definition of POAG underdiagnosis and overdiagnosis To assess model fairness, we compare underdiagnosis and overdiagnosis rates across subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Similar to ref,29 we define the underdiagnosis rate as the false-negative rate (FNR) of the binarized model prediction for the POAG at the subgroup levels: P(ˆy = non-POAG|y = POAG, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Here A is the sex, race, or other factors that the model should be free of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' For example, the underdiagnosis of female individuals is given by P(ˆy = non-POAG|y = POAG, female).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We then compare these underdiagnosis rates across subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We say a classifier is fair if the Overall population DenseNet POAG Model training Subpopulation comparisons Sex vs Race vs Age vs Figure 1: The experimental design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We focus our underdiagnosis and overdiagnosis experiments on subgroups of race, sex, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' individuals in the protected and unprotected groups satisfy the formula: P(ˆy = non-POAG|y = POAG, female) = P(ˆy = non-POAG|y = POAG, male) For overdiagnosis, we will measure the false-positive rate (FPR) for the POAG across all subgroups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=', P(ˆy = POAG|y = non-POAG, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' This measure shows that the model fails to diagnose those individuals who would never develop POAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Some of the harms caused by overdiagnosis are anxiety and having treatments that are not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Besides single identities, we also examined underdiagnosis and overdiagnosis in intersectional groups - individuals who belong to two subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='29 For example, the underdiagnosis of Black female individuals is given by P(ˆy = non-POAG|y = POAG, female, Black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Here, we want to examine if individuals who belong to two subgroups may have a larger underdiagnosis rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In other words, not all female individuals are misdiagnosed at the same rate (for example, Black female individuals are misdiagnosed more than non-Black female individuals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='3 Model development Figure 1 shows the pipeline of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' All images are resized to 224 × 224 × 3 and normalized using the mean and standard deviation of the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='30 We sequentially apply three augmentation operations on the fly during training: (1) random rotation between 0◦ and 10◦, (2) random translation: an image was translated randomly along the x- and y-axes by distances ranging from 0 to 10% of width or height of the image, and (3) random flipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The diversity of the dataset could be increased due to these data augmentation techniques, which generate effective and robust representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The input images are then passed through a convolutional neural network to generate the prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In this study, we used the DenseNet-20131 pre-trained on ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='32 We replaced the last layer with a new randomly initialized fully-connected layer with 2 output neurons (POAG and normal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We used binary cross-entropy as the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Since there are only 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='22% of images in the OHTS dataset that has POAG (Table 1, a severe class imbalance exists for POAG diagnosis, to overcome this problem, we adopted weighted cross-entropy, a commonly used loss function in classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The adopted weighted cross-entropy was: Lθ = − 1 N N � i=1 [βyi log(ˆyiDenseNet(xi, θ)) + (1 − β)(1 − yi) log(1 − ˆyiDenseNet(xi, θ))] (1) where N is the number of training examples, β is the balancing factor between positive and negative samples, yi is the observed true label of image xi, ˆyi is the probability predicted by the classifier, and θs represents the parameters of the DenseNet-201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Here, we used inversely proportional to POAG frequency in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Finally, we fine-tuned the entire network on the OHTS in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='4 Experimental settings The model was implemented by Keras with a backend of Tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The network was optimized using the Adam optimizer method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='33 The learning rate is 5×10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The experiments were performed on Intel Core i9-9960 X 16 cores processor and NVIDIA Quadro RTX 6000 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We used the five-fold cross-validation in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We split the entire dataset randomly into five groups at the individual level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' This ensured that no participant was in more than one group to avoid cross-contamination between the training and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In each fold of the cross-validation, we took one group (20% of total subjects) as the hold-out test set and the remaining 4 groups as the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 3 Results and discussion The underdiagnosis (Figure 2) and overdiagnosis (Figure 3) for POAG screening show an inverse relationship in both subgroups and intersectional groups in the OHTS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' As suggested by Seyyed et al,29 this indicates that the model consistently misclassifies the under-served subpopulations due to potential biases, rather than simple, random errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='1 Underdiagnosis in subpopulations and intersectional groups Figure 2A shows that the underdiagnosis rate differs in all subpopulations of sex, race, and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Specifically, females, non-Black individuals, and individuals under 60 years old have higher underdiagnosis rates than their counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In other words, the individuals of these groups are more likely falsely predicted as healthy, preventing them from receiving appropriate treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' From Table 1, we can see that the individuals in these groups have a lower prevalence of POAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' For example, the POAG rate of individuals aged ≥ 60 is around three times more than that of individuals aged < 60 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='04% vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='58%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Since these groups may not be adequately represented in the OHTS data, the supervised machine learning model trained from the data might be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In addition, female individuals have the highest underdiagnosis rate among indicated subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' However, the num- bers of POAG female and male individuals are about the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' This observation suggests that a simple resampling approach to ensure the dataset is balanced across different groups may not always be a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We also investigate intersectional groups, which means the individuals belong to two subpopulations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=', female Black individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' As shown in Figure 2B(i)-(iii), intersectional groups also have the problem of underdiagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Figure 2B(i) shows that female non-Black individuals have a higher underdiagnosis rate than female Black individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Female individuals aged < 60 years have a higher rate than female individuals aged ≥ 60 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Compared to the single identities, we observed that the intersectional identities amplify the model bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' For example, the difference between Black and non-Black individuals is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='50%, but the difference between female Black and female non-Black is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='61%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The most underdiagnosed group is young female individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Similarly, Figure 2B(ii) shows that Black females and younger individuals have a higher underdiagnosis rate than Black males and older adults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Figure 2B(iii) also indicates that female youngers are more heavily underdiagnosed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' There is no significant difference between Black and non-Black younger individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' This may be partially due to the small test set sizes (84 cases with the POAG label for individuals aged < 60 years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='2 Overdiagnosis in subpopulations and intersectional groups Figure 3A shows that healthy males and healthy older adults are more likely to be misclassified as POAG positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' On the other hand, the Black and non-Black subpopulations in Figure 3A have similar overdiagnosis rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' From Figure 3B, we observed that the intersectional identities are often overdiagnosed more heavily than the group in aggregate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Specifically, female and Black older adults are more easily overdiagnosed than female and Black younger adults (Figure 3B(i) and (ii)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Figure 2: Underdiagnosis analysis across subgroups of sex, race, and age in the OHTS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' A) The underdiagnosis rates in the subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' B) Intersectional underdiagnosis rates for female individuals, Black individuals, and individuals aged < 60 years, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The results are average results of the five-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Figure 3: Overdiagnosis analysis across subgroups of sex, race, and age in the OHTS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' A) The overdiagno- sis rates in the subpopulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' B) Intersectional overdiagnosis rates for female individuals, Black individuals, and individuals aged < 60 years, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The results are average results of the five-fold cross-validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='6- rate underdiagnosis Subgroup 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='0 Male Female Non-black Black Age < 60 Age >= 60 B Female 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='8- Black 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='8- Age < 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='8- underdiagnosis rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='6 T Intersectional 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='4 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='05 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='00 60 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='60 black Male Male Age Age Female (i) (ii) (ili)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='3 Discussions and Limitations We have found that underdiagnosis and overdiagnosis exist in the OHTS dataset for POAG diagnosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The DL model generated underdiagnosis and overdiagnosis biases in under-served subpopulations, such as female individuals and in- dividuals under 60 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Such biased diagnoses were even greater among individuals with intersectional identities, including Black females and females aged under 60 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Subpopulations like female individuals and individuals under 60 years old are most affected by POAG in the OHTS dataset, suggesting further attention when applying DL models in clinical decision-making34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' The DL model performs similarly on Black and non-Black individuals mainly because the number of Black individuals in our datasets is much smaller than the other group, even though they have a higher prevalence of POAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Therefore, the adverse effects on Black individuals are probably mitigated by the small number of their population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' This reminds us that we must also consider the number of subpopulations when constructing the dataset35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We also found that these differences in underdiagnosis and overdiagnosis exist in other clinical research areas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=', thoracic diseases, heart diseases, and kidney diseases),29,36,37 which means that these disparities may be widespread in biomedical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' One limitation of this study is that only one dataset was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' However, the OHTS data is one of the largest longitudinal clinical trials in POAG from 22 centers in the United States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Therefore, we believe the observations of model bias are likely to be generalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Another limitation is that we only studied the fairness of a binarized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Unfortunately, the probabilities predicted by the model may not be calibrated: probabilities are calibrated where a prediction of POAG with confidence p is correct p percent of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content='38 That being said, the probabilities predicted by the model may be over-confident in some cases and under-confident in other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Moreover, as we see in Figures 2, the severely imbalanced data may result in even more bias in the predicted probabilities as they over-favor in predicting the majority class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' As such, we should investigate the relationship between calibration and POAG underdiagnosis/overdiagnosis in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' In addition, we plan to develop an efficient method to reduce bias in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 4 Conclusion In this paper, we systematically study underdiagnosis and overdiagnosis bias in the DL-based POAG diagnosis models and identify the factors contributing to model fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' We find deep learning-based underdiagnosis and overdiagnosis exist among under-served subpopulations in POAG diagnosis on the OHTS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Underdiagnosis will prevent the individuals from receiving appropriate treatment, while overdiagnosis will let the individuals receive or continue receiving unnecessary treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Bias between the individuals in intersectional subgroups such as females under 60 years and Black females are more severe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' This emphasizes that bias mitigation approaches should consider the combination of characteristics involved in bias rather than a single identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' As deep learning models are implemented in clinical practice, this problem takes on particular urgency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' Acknowledgment This work was supported by the National Library of Medicine under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} +page_content=' 4R00LM013001, NSF CAREER Award No.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf'} diff --git a/ctE3T4oBgHgl3EQfeAqc/content/tmp_files/2301.04540v1.pdf.txt b/ctE3T4oBgHgl3EQfeAqc/content/tmp_files/2301.04540v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ea0301c36ea898592b871f906eb3ec80be655d20 --- /dev/null +++ b/ctE3T4oBgHgl3EQfeAqc/content/tmp_files/2301.04540v1.pdf.txt @@ -0,0 +1,1955 @@ +Towards a unified nonlocal, peridynamics framework for the coarse-graining +of molecular dynamics data with fractures +Huaiqian Youa, Xiao Xub, Yue Yua,∗, Stewart Sillingc, Marta D’Eliad, John Fosterb +aDepartment of Mathematics, Lehigh University, Bethlehem, PA; +bDepartment of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX; +cCenter for Computing Research, Sandia National Laboratories, Albuquerque,NM; +dCenter for Computing Research, Sandia National Laboratories, Livermore, CA; +Abstract +Molecular dynamics (MD) has served as a powerful tool for designing materials with reduced reliance on +laboratory testing. However, the use of molecular dynamics directly to treat the deformation and failure +of materials at the mesoscale is still largely beyond reach. In this work, we propose a learning framework +to extract a peridynamic model as a mesoscale continuum surrogate from MD simulated material fracture +data sets. Firstly, we develop a novel coarse-graining method, to automatically handle the material fracture +and its corresponding discontinuities in MD displacement data set. Inspired by the Weighted Essentially +Non-Oscillatory (WENO) scheme, the key idea lies at an adaptive procedure to automatically choose the +locally smoothest stencil, then reconstruct the coarse-grained material displacement field as piecewise smooth +solutions containing discontinuities. Then, based on the coarse-grained MD data, a two-phase optimization- +based learning approach is proposed to infer the optimal peridynamics model with damage criterion. In the +first phase, we identify the optimal nonlocal kernel function from data sets without material damage, to +capture the material stiffness properties. Then, in the second phase, the material damage criterion is learnt +as a smoothed step function from the data with fractures. As a result, a peridynamics surrogate is obtained. +As a continuum model, our peridynamics surrogate model can be employed in further prediction tasks +with different grid resolutions from training, and hence allows for substantial reductions in computational +cost compared with molecular dynamics. We illustrate the efficacy of the proposed approach with several +numerical tests for single layer graphene. Our tests show that the proposed data-driven model is robust and +generalizable, in the sense that it is capable in modeling the initialization and growth of fractures under +discretization and loading settings that are different from the ones used during training. +Keywords: +Nonlocal Models, Machine Learning, Homogenization, Peridynamics, Material Fracture. +Contents +1 +Introduction +2 +∗Corresponding author +Email address: yuy214@lehigh.edu (Yue Yu) +arXiv:2301.04540v1 [cond-mat.mtrl-sci] 11 Jan 2023 + +2 +Coarse-Graining of Molecular Dynamics Displacement with Damage +4 +3 +A Peridynamics Model with Brittle Fracture +8 +3.1 +Linear Peridynamic Solid (LPS) Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +3.2 +Peridynamics Formulation for Brittle Fractures . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +4 +Learning Algorithm +15 +5 +Application to single-layer graphene +18 +5.1 +Data Generation and Learning results +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +5.2 +Extrapolation to Longer Time Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +5.3 +Generalization to Different Body Forces and Crack Patterns . . . . . . . . . . . . . . . . . . . +23 +5.4 +Generalization to Different Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +6 +Conclusions +25 +1. Introduction +Detection and prediction of material damage progression attract lots of interests to the broad scientific +and engineering community [1–10]. Physically, the propagation of cracks results from a long-term physical +process with its origin in the atomistic scale, which often requires a micro-structural model such as molec- +ular dynamics (MD). However, although MD has made enormous advances in capabilities through better +algorithms, better interatomic potentials, and improvements in computational power, its direct employment +in treating the deformation and failure of materials at the mesoscale is still largely beyond reach. At the +mesoscale and above, a continuum model of mechanics is often required in practice. This fact creates the +need for homogenized models that act at larger scales and that, combined with new advanced architectures, +allow for fast and accurate predictions of material deformation and fracture [11–21]. +In this paper, we aim to address the question of how to extract coarse-grained measurements and a +homogenized surrogate model from MD simulations, that is able to capture material deformation and the +nucleation and growth of fractures. Nonlocal models are among the best candidates for this task [22]. In the +context of homogenization, nonlocal models are characterized by integral operators (as opposed to differen- +tiable operators) that embed time and length scales in their definition. Therefore, they are able to capture +long-range effects that classical PDE models fail to describe [23], which makes nonlocal models viable alter- +natives to partial differential equation (PDE) models when the effects of the small-scale behavior of a system +affect its global state [22, 24–31]. For monitoring and predicting material fractures, because the nonlocal +viewpoint avoids classical notions like deformation gradient, nonlocal models allow a natural description of +processes requiring reduced regularity in the relevant solution [32, 33]. As such, the nonlocal continuum +mechanics model, in the form of peridynamics [31, 34–42], provides a unified modeling of continuum media +where continuity and complex material damage modes can be captured autonomously. +2 + +In peridynamics and the general nonlocal models, constitutive laws take the form of integrand functions, +whose functional form is often justified a posteriori, which makes rigorous calibration and validation chal- +lenging and time consuming. On the other hand, although the nonlocal constitutive laws must be consistent +with the classical effective properties, they contain information about the small-scale response of the system +and must be chosen to reproduce this response with the greatest fidelity. Therefore, it is desired to extract +an optimal integrand function from small-scale data, such that the calibrated nonlocal model reproduces +the material responses and can further serve as a homogenized surrogate for future material deformation +and fracture prediction tasks [43, 44]. Recently, with the explosion of machine learning, optimized nonlocal +models were designed with the purpose of accurately reproducing observed coarse-grained behavior and pre- +dicting unseen behavior with the learnt model. We refer the reader to [45–50] for several examples of the use +of optimization-based machine learning for the design of homogenized nonlocal operators and the rigorous +analysis of its learning theory [50, 51]. +Although successful in providing optimal nonlocal surrogates to the homogenization problem, to the +authors’ best knowledge, none of these approaches addresses the challenge of capturing the main features +of dynamic fracture that are seen in small-scale data. Fundamental challenges are still present, mainly due +to the two difficulties. Firstly, when mapping the MD measurements onto a coarser grid, coarse graining +methods can use the mean atomic velocities weighted by a smoothing function [52]. In a nonlocal setting, +a smoothed displacement field can be shown to evolve according to the peridynamic linear momentum +balance [44, 47]. However, once the material fracture occurs, such a weighted average approach might overly +smooth the displacement field and introduce errors near cracks in the coarse-grained data set. Secondly, in +peridynamics the material damage is often described by breaking bonds. Therefore, the integrand functions +present jumps near the damage criterion, which results in nonsmooth losses in the optimization problem +and hinders the application of a suite of continuous optimization techniques. Herein, we address these two +challenges and present a complete workflow demonstrating how to obtain large-scale nonlocal descriptions +that capture MD behavior with fractures. +To accomplish this, we develop a novel coarse-graining method which is inspired by the Weighted Essen- +tially Non-Oscillatory (WENO) scheme, and extend the machine learning technique in our previous work +[47] to identify a smoothed damage criterion together with optimal nonlocal kernel functions. We summarize +below our main contributions. +• We develop a novel coarse-grained approach from micro-scale fracture measurements, to automatically +choose a locally smoothest stencil and capture the displacement discontinuities. +As such, coarse- +graining measurements are obtained without overly smoothing the crack pattern. +• We propose a two-step optimization strategy, and identify the best upscaled surrogate in the form of +peridynamics. Without prior knowledge of the material properties, the resultant peridynamics model +describes the material deformation together with the nucleation and growth of fractures. +3 + +Figure 1: An example of the vanilla coarse-graining method developed in [44] and our proposed approach, in handling the +MD measurements with a crack. Small blue points represent the MD particles and red dots stand for coarse-grained points. +Left: results from the vanilla coarse-graining method with weight function ω. Right: results from our proposed coarse-graining +method with an adaptive weight function ˆω. +• The optimal nonlocal model generalizes well to fracture patterns that are substantially different from +the ones used for training. The optimal model also enables extrapolation to longer time simulations +and a multiscale capability to predictions across resolutions. +Outline of the paper. Section 2 shows how to obtain an adaptive stencil in the form of a smoothness indicator +function, to extract coarse-grained measurements from MD displacements with fractures. In Section 3 we +summarizes the linear peridynamic solid (LPS) model, the treatment of material fracture and the handling +of free surfaces, and the discretization technique used in this work. Section 4 presents our two-step learning +approach consisting of a kernel learning step and a damage criterion learning step. Section 5 verifies the +learning technique for MD displacements and studies the generalizability of the resultant model. On a single +layered graphene, we demonstrate efficacy of our workflow by identifying an optimal two-dimensional nonlocal +model and employing this model in complex prediction tasks. In particular, we illustrate several properties +including generalization with respect to loadings, domain settings, crack shapes, and grid resolutions. Section +6 summarizes our contributions and provides future research ideas. +2. Coarse-Graining of Molecular Dynamics Displacement with Damage +In this section, we introduce the coarse-graining method to map the displacement field data from MD +simulations into a larger-scale discretized data cloud. For materials without fracture, in [44, 47] a nonlocal +coarse-graining method was proposed. In this method, the coarse grained displacement for each particle is +defined as a weighted average of the microscale displacements in its neighborhood. As such, a smoothed dis- +placement field is obtained, which preserves a linear momentum balance as a consequence of the momentum +balance for the atoms. +However, this coarse-graining method hides a pitfall: once the material fracture occurs introducing discon- +tinuities in the displacement field, the weighted average approach would overly smooth the displacement field +4 + +40 +20 +0 +-20 +-40 +-60-40-20 +0 +20 +40 +6040 +20 +0 +-20 +-40 +60 +-40 +-20 +0 +20 +40 +60Figure 2: Schematic and examples of calculation for the smoothness indicator function α(x, Xε). (a): A demonstration of the +projection of MD points between x and Xε. (b): When the projected displacement field is continuous, the smoothness indicator +α(x, Xε) stays close to 1 since ϵ(D1, D2) is close to ϵ0. (c): when material fracture occurs and the projected displacement field +is discontinuous (with a jump at d = 0.5), ϵ(D1, D2) +ϵ0 +reaches the minimum when D1 and D2 both consist of a smooth curve, +and we have α(x, Xε) ≈ 0. +and smudge the crack pattern. To resolve this challenge, in this section, we will extend the coarse-graining +method to an adaptive approach, so as to automatically handle the material fracture and its corresponding +discontinuities in MD displacement data set. +To introduce the coarse-graining method, we consider the MD data set as an assembly of S mutually +interacting particles. Then, we define the mass of each mutually interacting particles as Mε, ε = 1, 2, ..., S, +the reference positions of these particles as Xε, and the displacement vectors as Uε(t). Each particle is +subjected to a prescribed external force, Bε(t). +The coarse-grained measurements can be defined by choosing a compactly supported function ω(x, ·) for +each material point x ∈ Rd, such that +� +Rd ω(x, Xε)dx = 1, +ω(x, Xε) = 0 +if +|x − Xε| > R. +(2.1) +Then, the smoothed material density and body force density are expressed as +ρ(x) = +S +� +ε=1 +ω(x, Xε)Mε, +b(x, t) = +S +� +ε=1 +ω(x, Xε)Bε. +(2.2) +Correspondingly, the smoothed displacement field at material point x is obtained by +u(x, t) = +1 +ρ(x) +S +� +ε=1 +ω(x, Xε)MεUε(t). +(2.3) +5 + +U ↑e(D1, D2) +U +e(D1, D2) +U +(b) +e(D1, D2) +(a) +0.82 +0.84 +0.79 +03 +03 +03 +4 +4 +d +0 +4 +d +U +U+ +(c) +e(D1, D2) +e(D1, D2) +e(D1, D2) +0 +～ 0.51 + 0.51 +03 +E0 +03 +1 +1 +1 +0 +d +Regression line of D, +D +Regression line of D +-- Regression line of D2 +D.In [47], the authors proposed to employ a general cone-shaped weighted function for all material points, by +defining +ω(xi, Xε) := +τ(xi, Xε) +� +j τ(xj, Xε), where τ(x, X) = max{0, R − |X − x|}. +(2.4) +Here, R is a pre-chosen hyperparameter, representing the coarse-graining radius. Such an approach was +found to be effective for materials without fracture, where the displacement field is continuous. Then in [47] +a data-driven surrogate model was built from this coarse-grained displacement field, which has successfully +captured the material properties as well as a constitutive law acting at a larger scale. +However, once the material fracture occurs, the smooth weight function such as the one in (2.4) would +smudge the crack pattern and hence may compromise the reliability of the resultant surrogate model. As +shown in the left plot of Figure 1: coarse-grained points appear on the middle of the crack, showing the +effect of overly-smoothing the displacement field. +To provide coarse-grained displacement field for both damaged and undamaged material regions, we +propose to adjust the smoothing function ω(x, Xε) when the material point x is close to the crack. Intuitively, +the weight function should choose the locally smoothest stencil and avoid crossing discontinuities in the +averaging procedure as much as possible. +Similar idea was employed in the Essentially Non-Oscillatory +(ENO) and Weighted Essentially Non-Oscillatory (WENO) methods [53, 54], to develop finite difference +schemes for PDE problems with piecewise smooth solutions containing discontinuities. +Inspired by the +WENO methods, our key idea is to assign an additional smoothness indicator function α(x, Xε) to each pair +of continuum material point x and MD particle Xε, and modify the weight function ω(x, Xε) as: +ˆω(x, Xε) = +ω(x, Xε)α(x, Xε) +� +Rd ω(x, Xε)α(x, Xε)dx. +(2.5) +When the crack intersects with the bond between x and Xε, the displacement field between x and Xε +contains discontinuity. One should use less information from Xε to calculate the weighted average on x, by +taking the smoothness indicator function α(x, Xε) ≈ 0. That means, an adaptive procedure is required, to +detect if there is a displacement jump between x and Xε. +As demonstrated in Fig 2, we construct the smoothness indicator function α(x, Xε) with the following +procedure. First, we project all the MD particles within a distance of R from x to the line segment that +connects x and Xε. +For each MD particle Xi, we denote the projected point as �Xi, and calculate its +projected position variable, di, as the distance from x to �Xi. The displacement vector Ui(t) on Xi is also +projected, and we denote its component along segment x − Xε as Ui. Next, we select all the particles with +their projections lie between x and Xε, to form a set of data pairs D = {(di, Ui)}. When plotting Ui as +a function of di, the curve will present a jump when there is a crack intersecting the segment between x +and Xε, and such a jump would naturally divide the set D as two sets, with each set representing a smooth +curve. Therefore, our goal is then to identify the discontinuity of U(d) and define the smoothness indicator +function according to it. Numerically, we loop over all possible combinations of splitting the data pair set D +6 + +into two sets, D1 and D2, such that D1 +� D2 = D and D1 +� D2 = ∅. Then, we perform linear regressions on +D1 and D2: +kβ, bβ = argmin +k,b +� +(di,Ui)∈Dβ +|kdi + b − Ui|2, +β = 1, 2, +(2.6) +to obtain a fitted line for each set. +In the meantime, we also perform a linear regression on the entire +displacement data set D, and obtain the fitted parameter set (k, b). +Denoting the total squared error +ϵ(D1, D2) associated with D1 and D2 as: +ϵ(D1, D2) := +2 +� +β=1 +� +{(di,Ui)}∈Dβ +(kβdi + bβ − Ui)2, +(2.7) +and a squared error for the whole data set D as +ϵ0 := +� +{(di,Ui)}∈D +(kdi + b − Ui)2, +(2.8) +we define the smoothness indicator function α(x, Xε) as +α(x, Xε) := +min +(D1,D2)ϵ(D1, D2) +ϵ0 +. +(2.9) +Intuitively, when there is no material fracture and hence U(d) is a smooth curve without discontinuity, we +anticipate to have (kβ, bβ) ≈ (k, b) for β = 1, 2. As a result, we have ϵ(D1, D2) ≈ ϵ0 and α(x, Xε) ≈ 1. +In this case, the adjusted weight function ˆω(x, Xε) will stay the same as the original weight function, and +hence our smoothness indicator function will not alter the coarse-graining approach for materials without +fracture. Figure 2(b) shows an example with continuous displacement. It can be observed that ϵ(D1, D2) +stays roughly the same and close to ϵ0 for different partitions. On the other hand, when fracture occurs, +ϵ(D1, D2) would achieve its minimum when both D1 and D2 consist of a smooth curve. That means, when D1 +and D2 are separated by the crack. In this case, we will have +min +(D1,D2)ϵ(D1, D2) < ϵ0 and hence α(x, Xε) < 1. +Therefore, the adjusted weight function ˆω(x, Xε) would automatically reduce the weights of those particle +points crossing discontinuities, so as to reduce the overly smoothing near cracks. Figure 2(c) presents an +example where the displacement is piecewise constant with a jump at d = 0.5, demonstrating that the +smooth indicator would reach its minimum when neither D1 or D2 contains the displacement jump. +Once the adjusted weight functions are obtained, the smoothed mass density, body force density, and +7 + +displacement can be calculated as +ρ(x) = +S +� +ε=1 +ˆω(x, Xε)Mε, +(2.10) +b(x, t) = +S +� +ε=1 +ˆω(x, Xε)Bε, +(2.11) +u(x, t) = +1 +ρ(x) +S +� +ε=1 +ˆω(x, Xε)MεUε(t). +(2.12) +The result of this modified weight function is demonstrated in the right plot of Figure 1. One can see that +the coarse-grained points (red dots) are almost aligned with the crack interface, verified the efficacy of our +modified coarse-graining method in handling MD data set with cracks. +Similar as the derivation in [47], we point out that our coarse-grained formulation naturally induces a +nonlocal equation of u. The goal of the present work is therefore to identify an optimal nonlocal model in +the form of peridynamics, which faithfully represents given MD displacements under a given set of loading +conditions, and is generalizable to further prediction tasks for analysis of material deformation and crack +propagation phenomena. +3. A Peridynamics Model with Brittle Fracture +In the previous section, a data set of function trios, T := {(ρm, um, bm)}M +m=1, were derived from our +coarse-grained formulation, such that each trio contains a coarse-grained density field ρm(x), a body force +density bm(x, t), and their corresponding displacement field um(x, t) for material point x ∈ Ωm ⊂ Rd and +t ∈ [0, T m]. Herein, we note that each sample may have different spatial domain Ωm and observation range +T m. We propose to learn a nonlocal momentum balance equation based on these function trios in the form +of peridynamic equation of motion [34], to provide a continuum model with direct description of fracture +within the basic field equations. In peridynamics, each x interacts through bond forces with other material +points y within a neighborhood with radius δ known as the family of x, denoted by Bδ(x). Here, the horizon +δ determines the extent of the nonlocal interactions. The equation of motion for material point x is given as +ρ(x)∂2u(x, t) +∂t2 += +� +Bδ(x) +f(y, x, t) dy + b(x, t). +(3.1) +A material model in peridynamics supplies values of f(y, x, t) in terms of the deformations of the families +of x and y and any other relevant variables such as temperature [55]. Peridynamics can model fracture +because the equation of motion (3.1) is an integro-differential equation that does not involve the partial +derivatives of displacement with respect to position, which leads to a lower requirement of the solution +regularity. Moreover, many material models have been developed for peridynamics, and any material model +from the local theory can be translated into peridynamic form [56]. For a more thorough introduction and +8 + +review about peridynamics, we refer interested readers to [31]. +In this work, we aim to addresses the question of how to use MD to obtain a peridynamic material model +that is able to treat material deformation and the nucleation and growth of fractures. With a purpose of +demonstration and without loss of generality, we consider a 2D simulation problem (d = 2) in single-layered +graphene, and concern small deformations in the linear regime of material response, although the algorithm +may be generalized to finite deformations and 3D cases. +In [47], a data-driven two-dimensional linear +peridynamic solid (LPS) model[55] under plane-stress assumption was found to adequately represent the +material response from MD simulations on a graphene sheet without fracture. Inspired by such preliminary +results, in this work we also employ the LPS model as the base model, and further consider learning of +material failure from coarse-grained MD data sets. In this section, we first briefly introduce the LPS model +without material fracture in Section 3.1. Then, we extend the model to describe material fracture and handle +the imposition of traction loads as fracture surfaces open up in Section 3.2. Then, in the next section we +will describe our learning algorithm. +3.1. Linear Peridynamic Solid (LPS) Model +In this section, we summarize the mathematical formulation for the LPS model [57–59]. The LPS model +is a prototypical state-based model which can be seen as a nonlocal extension of the linear elasticity model. +It is suitable to describe isotropic elastic materials under infinitesimal deformation. Comparing with the +previously developed bond-based peridynamic models [34, 60], the LPS model has advantages in that it is +not restricted to a Poisson’s ratio of 1/4, which is important for our application since the Poisson ratio of +graphene was found to be negative from MD and molecular statistics simulations [61, 62]. +Consider a body occupying the domain Ω ⊂ Rd, and let θ be the nonlocal dilatation, generalizing +the local divergence of the displacement. In this section, we consider the material without damage, with +fully prescribed Dirichlet type boundary conditions, and will further extend the discussions to more general +boundary conditions and brittle fractures in Section 3.2. Here we note that in nonlocal problems, unless +otherwise stated, the boundary conditions should no longer be prescribed on the sharp interface, ∂Ω, but on +a collar of thickness of at least δ surrounding the domain Ω, which we denote as: +BΩ := {x /∈ Ω|dist(x, ∂Ω) < 2δ} . +Given nonlocal boundary conditions prescribed on the nonlocal volumetric boundary domain (or simply +nonlocal boundary): +u(x, t) := uD(x, t) +x ∈ BΩ, t ∈ [0, T], +and the initial velocity φ(x) and displacement ψ(x) for x ∈ Ω � BΩ at t = 0, the peridynamic operator in +9 + +the LPS model is given by +LK[u](x, t) := − C1 +m +� +Bδ(x) +(λ − µ) K(|y − x|) (y − x) (θ(x, t) + θ(y, t)) dy +− C2 +m +� +Bδ(x) +µK(|y − x|)(y − x) ⊗ (y − x) +|y − x|2 +(u(y, t) − u(x, t)) dy, +(3.2) +and the nonlocal dilatation is defined via +θ(x, t) := d +m +� +Bδ(x) +K(|y − x|)(y − x) · (u(y, t) − u(x, t)) dy, +(3.3) +where m := +� +Bδ(0) K(|z|)|z|2dz is the weighted volume, λ is Lam´e’s first parameter, and µ is the shear +modulus. To recover parameters for 3D linear elasticity, one should take C1 = 3, C2 = 30; whereas for 2D +problems, C1 = 2, C2 = 16. Here we note that m is determined by the horizon size δ and the influence function +K. We use the subscript K in the nonlocal operator LK[u](x) to emphasize the operator’s dependence on +the influence function K. Then the time-dependent LPS problem is given by +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ρ(x)∂2u(x, t) +∂t2 ++ LK[u](x, t) = b(x, t), +(x, t) ∈ Ω × [0, T]; +u(x, t) = uD(x, t), +(x, t) ∈ BΩ × [0, T]; +u(x, 0) = ψ(x), +x ∈ Ω � BΩ; +˙u(x, 0) = φ(x), +x ∈ Ω � BΩ. +(3.4) +Note that our learning algorithm is compatible with other type of boundary conditions in BΩ. Here we focus +on Dirchlet type of boundary condition in this work for its simplicity. +To discretize the above LPS model, we employ the optimization-based meshfree quadrature rule developed +in [63–70]. Suppose that the values of function trios ρ(x), u(x, t), and b(x, t) are provided on a set1 of coarse- +grained material points χ := {xi}I +i=1 ⊂ Ω � BΩ and time instances tn = n∆t, n = 0, · · · , T/∆t. We write +the discretized approximation of LK as +Lh +K[u](xi, tn) := −C1 +mi +� +xj∈Bδ(xi) � χ +(λ − µ) Kij (xj − xi) +� +θh(xi, tn) + θh(xj, tn) +� +Wj,i +−C2 +mi +� +xj∈Bδ(xi) � χ +µKij +(xj − xi) ⊗ (xj − xi) +|xj − xi|2 +(u(xj, tn) − u(xi, tn)) Wj,i, +(3.5) +θh(xi, tn) := d +mi +� +xj∈Bδ(xi) � χ +Kij(xj − xi) · (u(xj, tn) − u(xi, tn)) Wj,i, +(3.6) +1Although the machine learning algorithm as well as the quadrature rule is compatible with the general non-uniform grids, +in this work we consider the uniform grids with grid size h and uniform time steps with size ∆t, for simplicity. +10 + +where Kij := K(|xj − xi|) and mi := +� +xj∈Bδ(xi) � χ +Kij|xj −xi|2Wj,i. The quadrature weights Wj,i are associ- +ated with a local neighborhood of particles for each discretization point xi, generated by local optimizations +to make the approximation rule exact for certain classes of functions. For each xi ∈ χ � Ω we solve for Wj,i +via +argmin +{ωj,i} +� +xj∈χ � Bδ(xi)\{xi} +W 2 +j,i +s.t., +� +xj∈Bδ(xi) +q(xi, xj)Wj,i = +� +Bδ(xi) +q(xi, y)dy +∀ q ∈ Vxi, +(3.7) +where Vxi denotes the space of functions which should be integrated exactly. Following [64], in this work we +take Vxi := +� +q(y − xi) = p(y−xi) +|y−xi|3 | p ∈ P5(Rd) such that +� +Bδ(0) q(y)dy < ∞ +� +and P5(Rd) denotes the space +of quintic polynomials. As the horizon size δ vanishes, this discretization preserves the consistency in the +limit to the local solution [63, 64]. Moreover, we point out that the quadrature weights, Wj,i, only depend +on the grid set χ and it is invariant of the influence K. Hence, in our learning algorithm one only need to +generate the quadrature weights and solve the local optimization problem (3.7) once in the preprocessing +step. +For the dynamic peridynamics model, to discretize in time we apply the central difference time stepping +scheme. With time step size ∆t, at the (n + 1)−th time step one can solve for the displacement un+1 +i +≈ +u(xi, tn+1) following: +� +� +� +ρ(xi)¨un +i + Lh +K[u](xi, tn) = b(xi, n∆t), +for xi in Ω � χ, +un+1 +i += uD(xi, (n + 1)∆t), +for xi in BΩ � χ, +(3.8) +where Lh +K is the discretized nonlocal operator as defined in (3.5), and the acceleration ¨un +i is estimated via +the central difference scheme: +¨un +i := un+1 +i +− 2un +i + un−1 +i +∆t2 +. +(3.9) +As the initial conditions, we set u0 +i = ψ(xi) and u1 +i − u0 +i +∆t += φ(xi) for xi ∈ (BΩ � Ω) � χ. +3.2. Peridynamics Formulation for Brittle Fractures +One of the main appeals of peridynamics is to handle fracture problems, where free surfaces are associated +with the evolution of a fracture surface. In this section, we consider the LPS model with free surfaces, then +apply it to the treatment of brittle fractures. +To describe the free surfaces associated with the time evolution of a fracture surface, we now consider +general mixed boundary conditions: ∂Ω = ∂ΩD +� ∂ΩN and (∂ΩD)o �(∂ΩN)o = ∅. Here ∂ΩD and ∂ΩN are +both curves. ∂ΩN is the (possibly time-dependent) sharp crack surface evolving with the material fractures, +and a free surface boundary condition is applied on it. To define a Dirichlet-type constraint, we denote +BΩD := {x /∈ Ω|dist(x, ∂ΩD) < 2δ}, +11 + +and assume that the value of u(x, t) = uD(x, t) is given on x ∈ BΩD. For notation simplicity, we denote +ΩD := Ω ∪ BΩD. To apply the free surface boundary condition, we denote +IΩN := {x ∈ Ω|dist(x, ∂ΩN) < δ}, IΩ := {x ∈ Ω|dist(x, ∂Ω) < δ}. +Unless stated otherwise, in this paper we further assume sufficient regularity in the boundary region IΩ +that there exists a unique orthogonal projection of x onto ∂Ω, which is the closest point on ∂Ω to x, and +we denote this projection as x. Then, one has x − x = sxn(x) for x ∈ IΩN, where 0 < sx < δ. Here n +denotes the normal direction pointing out of the domain for each x ∈ IΩN, and let p denote the tangential +direction. In our numerical solver, we treat x with the free surface boundary condition if the projection of +x is in ∂ΩN. Otherwise, we use the Dirichlet-type boundary condition at x. +In peridynamics, material damage is incorporated into the constitutive model by allowing the bonds of +material points to break irreversibly. To model brittle fracture in the LPS model, we employ a smoothed +critical stretch criterion, where weakening occurs when a bond is extended beyond some predetermined +critical bond deformed length [40, 64, 71]. In particular, a scalar state function γ(x, y, t) is defined and takes +values in the interval [0, 1], to describe the bond weakening and breakage through the crack growing: +γ(x, y, t) := 1 +2 +� +− tanh +�maxτ∈[0,t] S(x, y, τ) − s0 +η +� ++ 1 +� +, +(3.10) +where +S(x, y, τ) := |x − y + u(x, τ) − u(y, τ)| +|x − y| +− 1 +(3.11) +and s0 is the critical stretch criterion depending on the material. γ(x, y, t) is a history-dependent function, +i.e., a bond can never recover once it exceeds the critical stretch criterion. +An illustration of γ can be +visualized in Figure 3, where a hyperparameter η ≪ 1 can be tuned to control the level of smoothness. +When γ(x, y, t) = 1, the bond between material points x and y are considered “intact” and the change of +displacement on material point y may have an impact on the displacement at x. When the stretch S(x, y, τ) +exceeds the critical criterion s0 for some time τ < t, the material gets damaged and we have γ(x, y, t) < 1. +As the stretch further increases, finally γ(x, y, t) = 0 and we consider the bonds between x and y as fully +“broken”. Instead of defining γ as a step function following [64], in (3.10) we allow the weakening of force +scalar within small ranges of excessive bond stretch values and set γ as a smoothed step function. As shown +in [40], such a smoothed state function would impose the continuity of the learnt bond force f(x, y, t) in our +peridynamic model, and guarantee the well-posedness of the peridynamic model as a dynamic system. On +the other hand, a continuous formulation of the damage factor γ would result in a continuous optimization +problem, and allows generic optimization routines to be used in the training procedure. +With the state function γ, we treat the time-evolving fracture as free surfaces and employ the following +12 + +max + < t S(x i,x j, ) +0 +0.5 +1 +1.5 +(x i,x j,t) + = 10 -8 + = 0.05 +S0 +Figure 3: An illustration of the smoothed scalar state function γ, with the tunable parameter η = {0.05, 10−8}. +formulation: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +ρ(x)∂2u(x, t) +∂t2 ++ LKN[u](x, t) = b(x, t), +(x, t) ∈ Ω × [0, T]; +u(x, t) = uD(x, t), +(x, t) ∈ BΩD × [0, T]; +u(x, 0) = ψ(x), +x ∈ Ω � BΩ; +˙u(x, 0) = φ(x), +x ∈ Ω � BΩ. +(3.12) +Here, the modified LPS operator LKN follows the formulation in [64, 70]: +LKN[u](x, t) := −C1 +m +� +Bδ(x) +(λ − µ) K(|y − x|)γ(x, y, t) (y − x) (θcorr(x, t) + θcorr(y, t)) dy +− C2 +m +� +Bδ(x) +µK(|y − x|)γ(x, y, t)(y − x) ⊗ (y − x) +|y − x|2 +(u(y, t) − u(x, t)) dy +− 2C1θcorr(x, t) +m +� +Bδ(x) +(λ − µ) K(|y − x|)(1 − γ(x, y, t)) (y − x) dy +− C2θcorr(x, t) +2m +� +Bδ(x) +(λ + 2µ)K(|y − x|)(1 − γ(x, y, t))[(y − x) · n][(y − x) · p]2 +|y − x|2 +ndy ++ C2θcorr(x, t) +2m +� +Bδ(x) +λK(|y − x|)(1 − γ(x, y, t))[(y − x) · n]3 +|y − x|2 +ndy +(3.13) +with +θcorr(x, t) := d +m +� +Bδ(x) +K(|y − x|)γ(x, y, t) (y − x) · M(x) · (u(y, t) − u(x, t)) dy, +(3.14) +M(x, t) := +� +d +m +� +Bδ(x) +K(|y − x|)γ(x, y, t) (y − x) ⊗ (y − x) dy +�−1 +. +(3.15) +As such, the LPS model provides an approximation for the corresponding linear elastic model with free +surfaces in the case of linear displacement fields. We notice that when all bonds in Bδ(x) are intact, i.e., the +material point x is sufficiently far away from the free surface, we have γ(x, y, t) = 1 for all y ∈ Bδ(x). Then +(3.13) yields LKN = LK and the original momentum balance and nonlocal dilatation formulation in the +13 + +LPS model are obtained. Therefore, (3.13) provides a unified mathematical framework which automatically +captures material deformation and the evolution of cracks as free surfaces. +We now extend the optimization-based quadrature rule and the central difference time-stepping method +introduced in Section 3.1, to the LPS model (3.13) with fracture. Particularly, at the (n + 1)−th time step +we approximate the state function γ(xi, xj, tn) via +γn +ij := 1 +2 +� +�− tanh +� +� +max +0≤m≤nSm +ij − s0 +η +� +� + 1 +� +� , where Sm +ij := |xi − xj + um +i − um +j | +|xi − xj| +− 1. +(3.16) +Then the approximated displacement field un+1 +i +≈ u(xi, tn+1) can be solved via the following formulation: +� +� +� +ρ(xi)¨un +i + Lh +KN[u](xi, tn) = b(xi, n∆t), +for xi in Ω � χ, +un+1 +i += uD(xi, (n + 1)∆t), +for xi in BΩD +� χ, +(3.17) +where +Lh +KN[u](xi, tn) := −C1 +mi +� +xj∈Bδ(xi) � χ +(λ − µ) Kijγn +ij (xj − xi) +� +(θcorr)n +i + (θcorr)n +j +� +Wj,i +− C2 +mi +� +xj∈Bδ(xi) � χ +µKijγn +ij +(xj − xi) ⊗ (xj − xi) +|xj − xi|2 +� +un +j − un +i +� +Wj,i +− 2C1(θcorr)n +i +mi +� +xj∈Bδ(xi) � χ +(λ − µ) Kij(1 − γn +ij) (xj − xi) Wj,i +− C2(θcorr)n +i +2mi +� +xj∈Bδ(xi) � χ +(λ + 2µ)Kij(1 − γn +ij)[(xj − xi) · nn +i ][(xj − xi) · pn +i ]2 +|xj − xi|2 +nn +i Wj,i ++ C2(θcorr)n +i +2mi +� +xj∈Bδ(xi) � χ +λKij(1 − γn +ij)[(xj − xi) · nn +i ]3 +|xj − xi|2 +nn +i Wj,i +(3.18) +with +(θcorr)n +i := d +mi +� +xj∈Bδ(xi) � χ +Kijγn +ij (xj − xi) · Mn +i · +� +un +j − un +i +� +Wj,i, +(3.19) +Mn +i := +� +� d +mi +� +xj∈Bδ(xi) � χ +Kijγn +ij (xj − xi) ⊗ (xj − xi) Wj,i +� +� +−1 +. +(3.20) +Here we note that the free surface ∂ΩN as well as the normal vector n(x) on free surfaces both change as +the fracture evolves. To numerically approximate n(xi, tn) at each time step, we updated it via +nn +i = − +� +xj∈χ∩Bδ(xi) +(xj − xi)Wj,iγn +ij +����� +� +xj∈χ∩Bδ(xi) +(xj − xi)Wj,iγn +ij +����� +, +(3.21) +14 + +and the tangential vector pn +i is calculated as the orthogonal direction to nn +i . The correction tensor should be +invertible to ensure that the correction dilitation can be computed. This holds as long as the bonds in the +horizon are non-colinear. For fracture case resulting in bond break, leaving an isolated particle, we replace +the matrix inverse with the pseudo-inverse. +4. Learning Algorithm +Algorithm 1 Workflow for learning the LPS model from MD data sets. +1: To obtain samples without material fracture, generate relatively small MD displacements on fine grids +{Xm +ε } using different external forces and domain configurations, then group the samples into two data +sets, MDNon-Frac +train +for training the nonlocal kernel and MDNon-Frac +val +for hyperparameter tuning: +MDNon-Frac +train/val := {M m +ε , Um +ε (t), Bm +ε (t)}, m = 1, · · · , M Non-Frac +train/val . +2: Generate MD displacements samples with material fracture, on fine grids { �Xm +ε } using different external +forces and domain configurations, then group the samples into two data sets, MDFrac +train for training the +damage criterion and MDFrac +test for test: +MDFrac +train/test := {� +M m +ε , �Um +ε (t), �Bm +ε (t)}, m = 1, · · · , M Frac +train/test. +3: Coarse grain the data sets MDNon-Frac +train/val and MDFrac +train/test, then evaluate the coarse grained data at coarser +grids χm to obtain the functio trio sets +T Non-Frac +train/val := {ρm(xi), um(xi, tn), bm(xi, tn)}, m = 1, · · · , M Non-Frac +train/val , +T Frac +train/test := {�ρm(xi), �um(xi, tn), �bm(xi, tn)}, m = 1, · · · , M Frac +train/test. +4: (Kernel learning step): Solve the optimization problem based on the non-fracture data set T Non-Frac +train +: +� +� +� +(λ∗, µ∗, D∗, α∗) = argmin +λ,µ,D,α +Res(T Non-Frac +train +) +subject to solvability constraints (4.4), +and tune the hyperparameters δ∗ and P ∗, to minimize the test errors on the validation data set T Non-Frac +val +. +5: (Damage criterion learning step): With fixed parameters (λ∗, µ∗, D∗, α∗, δ∗, P ∗), train for the opti- +mal fracture criterion parameter based on the fracture data sets T Frac +train: +s∗ +0 = argmin +s0 +� +Res(T Frac). +6: To study the generalizability on unseen external forces and fracture scenarios, use the learnt LPS model +to predict the material deformation and fracture on T Frac +test . +Let T := {ρm(xi,m), um(xi,m, tn +m), bm(xi,m, tn +m)}, m = 1, · · · , M, be coarse-grained function trios avail- +able at xi,m ∈ χm and tn +m = n∆tm, n = 1, · · · , Nm, our goal is to identify an optimal constitutive relation +on the basis of MD data sets. Here, we use χm and ∆tm to highlight the fact that in our learning algorithm, +each sample can be of different spatial/temporal domain and resolutions. In the following content, we will +skip the subscript m and denote the function trios as ρm(xi), um(xi, tn), and bm(xi, tn) for simplicity. Let +LKN be the LPS operator defined in (3.18), we aim to learn an optimal continuum model in the form of +15 + +LPS models, where the optimal model consists of the influence function K, which may be sign-changing, +and parameters λ, µ and s0, such that the action of LKN most closely satisfy (3.18) for all s. Formally, the +optimal influence function and parameters, (λ∗, µ∗, s∗ +0, K∗), are the solution of the following optimization +problem: +(λ∗, µ∗, s∗ +0, K∗) = argmin +λ,µ,s0,K +1 +M +M +� +m=1 +Nm−1 +� +n=1 +∆tm +��ρm(xi)(¨um)n +i + Lh +KN[um](xi, tn) − bm(xi, tn) +��2 +ℓ2(χm). +(4.1) +The influence function K(|x − y|) will now be parameterized. Following [72], In this work, the interacting +kernel function K(|x − y|) is taken as a radial function compactly supported on the δ-ball Bδ(x) with α-th +order singularity: +K(|x − y|) = +P +� +k=0 +Dk +|x − y|α Bk,P +�|x − y| +δ +� +. +(4.2) +Here the Bernstein polynomials are defined as +Bk,P (r) = +� +�P +k +� +� rk(1 − r)P −k, +for 0 ≤ r ≤ 1. +(4.3) +Following the arguments in [47, 65], in the learning algorithm we require the fractional order α to be bounded +by 3 and allow Dk ∈ R for all k with sufficient well-posedness conditions embedded for the discretized +operator. Here, we note that in the samples with material fracture, some particles might become isolated +due to fragmentation, and hence it would be impossible to require solvability constraints. Therefore, we only +apply the solvability constraints to the model without fracture. With the analysis in [47], given a tolerance +parameter ζ > 0 we apply the following solvability constraints: +� +� +� +� +� +� +� +� +� +� +� +λ + µ > 0, µ > 0, α < 3, Λmin(Γ(α,D,δ,P )) ≥ ζ, +Λmin(Φ(α,D,δ,P )Γ† +(α,D,δ,P )Φt +(α,D,δ,P )) ≥ ζ, +Λmin(Γ(α,D,δ,P ) − 2Φt +(α,D,δ,P )Φ(α,D,δ,P )) ≥ 0. +(4.4) +Here Γ and Φ are the matrices that correspond to the deviatoric and dilatation contributions of the defor- +mation, and Λmin(A) denotes the smallest nonzero eigenvalue of a matrix A. +The overall formulation of the constrained optimization problem is as follows. +Given a collection of +training samples {ρm(xi), um(xi, tn), bm(xi, tn)}, m = 1, · · · , M, we seek to learn the parameters λ, µ, the +Bernstein polynomial coefficients D = [D0, · · · , DP ] ∈ RP +1, the order α, the horizon δ, the polynomial +16 + +order P, and the damage criterion s0, by minimizing the mean square loss (MSL) of the LPS equation: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +(λ∗, µ∗, D∗, α∗, δ∗, P ∗, s∗ +0) = +argmin +λ,µ,D,α,δ,P,s0 +1 +M +M +� +m=1 +Nm−1 +� +n=1 +∆tm +��ρm(xi)(¨um)n +i + Lh +KN[um](xi, tn) +− bm(xi, tn) +��2 +ℓ2(χm) +subject to solvability constraints (4.4). +(4.5) +However, numerically solving the constraint optimization problem (4.4) could be time-consuming and +possibly unstable, due to three factors. First, as shown in Figure 3, when s0 is away from the optimal +value, its impact on the loss function would be relatively flattened, causing the vanishing gradient issue in +optimizers. Second, the update of s0 would induce the change of correction operator (3.18), which increases +the computational cost on each epoch. Lastly, the imposition of solvability constraints (4.4) would also be +expensive, since it involves additional calculations (such as with the projection method) and/or subiterations +(such as with the augmented Lagrangian method), together with the evaluation of eigenvalues at each epoch. +To make the optimization algorithm more efficient and robust, we propose to separate the solving procedure +of the damage criterion, s0, with other parameters, and propose a “two-stage” strategy. Key components +are summarized in Algorithm 1. In particular, we notice that the correction operator (3.18) and the damage +criterion, s0, are only associated with samples with material fractures, while the influence function K and +other material parameters can be inferred from samples without fracture. Therefore, we divide the training +data set into two sets: +T Non-Frac := {ρm(xi), um(xi, tn), bm(xi, tn)}, m = 1, · · · , M Non-Frac, +which includes all samples without fracture, and +T Frac := {�ρm(xi), �um(xi, tn), �bm(xi, tn)}, m = 1, · · · , M Frac +for training samples with fracture. Then the optimization problem (4.5) is also split, into a non-fracture +kernel learning step and a damage criterion learning step. +For the kernel learning step, we infer the influence function K and the Lam´e moduli λ and µ by solving +a constraint optimization problem from T Non-Frac: +� +� +� +� +� +(λ∗, µ∗, D∗, α∗, δ∗, P ∗) = argmin +λ,µ,D,α,δ,P +Res(T Non-Frac) +subject to solvability constraints (4.4), +(4.6) +17 + +where +Res(T Non-Frac) := +1 +M Non-Frac +M Non-Frac +� +m=1 +Nm−1 +� +n=1 +∆tm +��ρm(xi)(¨um)n +i + Lh +K[um](xi, tn) − bm(xi, tn) +��2 +ℓ2(χm). (4.7) +As such, one only has to evaluate the nonlocal operator without fracture following (3.5), which is computa- +tionally more efficient. In this step, we treat δ and P as hyperparameters to be separately tuned, to achieve +the best learning accuracy without overfitting. For each combination of δ and P, the Adam optimizer in +PyTorch is employed, together with the augmented Lagrangian method to impose the inequality constraints. +For further details of the optimization algorithm and settings, we refer interested readers to [47]. +In the damage criterion learning step, we fix the learnt parameters (λ∗, µ∗, D∗, α∗, δ∗, P ∗) and search for +the optimal s0 by considering a unconstraint optimization problem on T Frac: +s∗ +0 = argmin +s0 +� +Res(T Frac), +(4.8) +where +� +Res(T Frac) := +1 +M Frac +M Frac +� +m=1 +Nm−1 +� +n=1 +∆tm +���ρm(xi)(¨�u +m)n +i + Lh +NK[�um](xi, tn) − �bm(xi, tn) +��2 +ℓ2(χm) +(4.9) +In all tests, we set the smoothing parameter η = 0.05, and employ the bisection method to solve for s∗ +0. +5. Application to single-layer graphene +To illustrate the capability of our method in obtaining an optimal surrogate material damage model from +coarse-grained MD displacements, we consider single layer graphene sheets as the application. Graphene is +a single layer of carbon atoms, tightly bound in a hexagonal honeycomb lattice. Up to now, much of what +is known about the mechanical and electronic properties of graphene is based on models on the atomistic +scale, such as the MD simulations. However, the use of MD directly to treat the deformation and failure +of materials at the mesoscale is still largely beyond reach. Hence, we aim to learn a peridynamic model by +upscaling from MD to the continuum scale. +For the present study, an MD model was created using the Tersoff interatomic potential [73], a widely used +potential in the MD community for graphene [44]. Unstressed graphene nominally has an interatomic spacing +of 1.46˚A. Without otherwise stated, in this study values of the coarse-grained data trios are evaluated on a +square lattice of nodes with spacing h=5.0˚A. The only exception is in Section 5.4, where we also consider +an additional, finer data set generated with spacing 3.17˚A, to assess the generalization properties of the +proposed learning approach to different grids. Without the loss of generality, in this work we consider MD +simulations on temperature 0K. In all cases, external loading is applied to the atoms in the MD grid. For +the non-fracture data sets, the magnitude of the loading is chosen so that the bond strains are no larger than +18 + +Figure 4: Contours of exemplar U1 displacement in typical MD simulations at zero temperature for the four data sets. From +left to right: (a) Non-fracture training data set MDNon-Frac +train +, for the kernel learning step; (b) Non-fracture validation data set +MDNon-Frac +val +for the kernel learning step; (c) Fracture training data set MDFrac +train, for the damage criterion learning step; (d) +Fracture test data set MDFrac +test , to study the efficacy and generalizability of the overall workflow. +2%, which is less than the strains at which nonlinear effects appear. In all MD experiments, the atoms are +initialized with positions on a hexagonal lattice in the x1-x2 plane with an interatomic spacing of 1.46˚A. The +mass of each atom is 2.0E-26kg, or 12amu. For purposes of computing stresses, the thickness of the lattice +is set to 3.35˚A, which is the approximate distance between layers in multilayer graphene. On quasi-static +data sets, we smooth the MD simulation results in time as described in [47]. For the dynamic data sets, the +MD time step size is set as 4.95E-14s, or 49.5fs. +5.1. Data Generation and Learning results +In this section, we apply the learning algorithm described in Section 4, to extract a coarse-grained model +from MD simulations of a graphene sheet at 0K. For the purpose of training, validation and test, we generate +the following four groups of MD simulations, with exemplar images showing contours of U1, the component +of atomic displacement in the x1 direction, provided in Figure 4. +1) Non-fracture training data set (MDNon-Frac +train +, with 70 quasi-static MD simulation samples): The MD +domain is a 100˚A×100˚A square, and, for k1, k2 ∈ {0, π +50, 2π +50 , . . . , 5π +50 }, the prescribed external loadings are +given by +b(x1, x2) = (C1 +k1,k2 cos(k1x1) cos(k2x2), 0), or b(x1, x2) = (0, C2 +k1,k2 cos(k1x1) cos(k2x2)). +(5.1) +The constant C1 +k1,k2 and C2 +k1,k2 are adjusted so that the bond strains are no larger than 2%, so the deformation +remains in the linear range of material response. A periodic boundary condition is employed for all samples +in this data set. +2) Non-fracture validation data set (MDNon-Frac +val +, with 10 quasi-static MD simulation samples). For the same +MD grid and coarse-grained nodes as in the non-fracture training data set, the applied loads in the validation +19 + +(a) Non-Fracture Training +(b) Non-Fracture Validation +(c) Fracture Training +(d) Fracture Test +Data Set MIDNon-Frac +Data Set MDNon-Frac +rain +valk +C1 +k +C2 +k +pk +Rk +1 +0.001 +0 +0 +25 +2 +0 +0.001 +0 +25 +3 +0.001 +0 +0 +15 +4 +0 +0.001 +0 +15 +5 +0.001 +0 +0 +10 +6 +0.001 +0 +1 +25 +7 +0 +0.001 +1 +25 +8 +0.001 +0 +1 +15 +9 +0 +0.001 +1 +15 +10 +0.001 +0 +1 +10 +Table 1: Parameters used in the MD loading in the 10 validation tests. +data set are as follows: +b(x1, x2) = (C1 +k, C2 +k) +1 +� +j=−1 +(−1)j cos +�π +2 min +� +1, rj,k +Rk +�� +(5.2) +where +rj,k = +� +(x1 − (1 − pk)Lj)2 + (x2 − pkLj)2 +(5.3) +where L=50 and the values of the parameters C1 +k, C2 +k, pk and Rk are given in Table 1. In each case, loads +are applied to the atoms within three disks of radius Rk with centers at the center of the grid and at the +left and right boundaries (if pk = 0) or the upper and lower boundaries (if pk = 1). The loads in all cases +are self-equilibrated and periodic. +3) Fracture training data set (MDFrac +train, with 1 dynamic MD simulation sample): The domain of the graphene +sheet is set as a square: [−50˚A, 50˚A] × [−50˚A, 50˚A]. The MD grid initially contains a slit (edge crack) of +length 25˚A oriented vertically extending from the lower surface. The vertical edges of the MD grid have +prescribed velocities in the x1 direction that tend to open the crack. To help maintain stable crack growth, the +prescribed velocities decrease linearly with x2, thus tending to limit the crack growth velocity. A schematic +of the crack pattern at the 40-th time step in the MD simulation can be find in Figure 4(c). For the purpose +of validation on different grid resolutions, the density, displacement and external loading are computed at +two sets of coarse-grained nodes, which are spaced 5˚A or 3.17˚A, apart on a square lattice, respectively. +4) Fracture test data set (MDFrac +test , with 1 dynamic MD simulation sample): To demonstrate that the learned +material model applies to different loading scenarios and crack patterns, one additional test case is considered. +Here, the MD region and the pre-existing slit are the same as in the fracture training data set. However, +instead of hard loading along the vertical edges, a non-zero body force is applied to the atoms in the MD +grid as: +b1 = b0 +� +e−t/tr(1 − e−t/tr) +� +sin +�πx1 +L +� +e−(1/2+x2/L) +where L = 100˚A is the edge length of the sample, tr is a constant pulse duration time, and b0 is a positive +20 + +Figure 5: Learning results on a single layer graphene sheet. Left: The optimal influence function K for the LPS model. Right: +The optimal damage criterion s∗ +0 is obtained at 0.11. +constant. (The origin is at the center of the sample.) This loading exerts a pulse that tends to open the +crack. The resulting crack pattern, which includes branching, is substantially different from that which +occurs in the training data. A view of the crack pattern at the 40-th time step in the MD simulation can be +found in Figure 4(d). +As metrics of accuracy on tests, we compare the prediction from the learnt peridynamics model with the +ground-truth data from coarse-grained MD measurements. Solution contours are provided as a qualitative +validation. With the purpose of providing a quantitative comparison, we also calculate the averaged (in time) +mean square errors (MSEs) of the displacement field and the damage field. To provide a fair comparison +between different sets, all these qualitative accuracy metrics are normalized with respect to the ground-truth +data. +For the kernel learning step, we have followed a similar procedure as in [47]. +The learned influence +function K is plotted in the left plot of Figure 5, and the optimal material parameter are obtained as +λ = −0.4796(TPA), µ = 0.7978(TPA), with the Poisson’s ratio ν = −0.4297, and the horizon size δ = 20.0˚A. +Then, for the damage criterion learning step, since the crack initiates at the 5-th time steps, we use the +fracture training data set from the 5-th time step till the 20-th time step to learn the damage criterion s0, +then solve for the optimal s0 by minimizing the loss � +Res(T Frac) in (4.8). Note that when calculating the +loss function, we apply Dirichlet-type boundary conditions on a layer of particles near the boundary of our +square domain, and hence only the particles in [−50˚A+2δ, 50˚A−2δ]×[−50˚A+2δ, 50−2δ] are considered in +(4.8). This setting differs from the settings in non-fracture data sets, where periodic boundary conditions are +considered for all samples. This is due to the fact that it is generally non-realistic to prescribe the periodic +boundary condition in the problem with a crack, since the crack itself does not satisfy the periodic condition. +21 + +1e-4 +1.2 +1.0 +0.8 +K/m(6) +0.6 +0.4 +0.2 +0.0 +-0.2 +6 +8 +10 +12 +14 +16 +18 +20 +Bond Length (A)1e-4 +3.50 +3.45 +3.40 +Loss +3.35 +3.30 +3.25 +3.20 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +So +so = 0.11Figure 6: Comparison of the prediction and the ground truth measurement from the MD data set at time steps 20, 30, and 40, +on the fracture training data set where the graphene sheet is subject to zero body force. Here, the first 20 steps were used for +training, then we use the learnt model to predict for the next 20 steps. (a) Comparison on the displacement field, where the +color of the particles represents the horizontal displacement. (b) Comparison on the damage field. +This fact also highlights the generalizability of the proposed approach: our homogenized surrogate model +can handle data sets with different domains, loadings, and also boundary conditions. A demonstration of +the loss function for different values of s0 is provided in the right plot of Figure 5. The optimal damage +criterion is obtained as s∗ +0 = 0.11, which is consistent with the results s0 = 0.145 inferred directly from MD +data set in [44]. +5.2. Extrapolation to Longer Time Simulations +Next, we validate the learnt model, by using it in a longer term simulation on the fracture training data +set, to predict the material deformation and crack propagation upto the 40-th step. Note that we have used +the data upto the 20-th time step for the purpose of training, and therefore this test can be seen as an +investigation on the long-term extrapolation capability of our coarse-grained surrogate model. To solve for +the displacement field from LPS model, at the n-th time step, we first assume there is no broken bond and +solve for the displacement field ˆun+1, then we update the bond-stretch for each connecting bond, and we +keep solving for the displacement until there is no new bond breaking. Then, we define the damage profile +at each particle xi at time step n as +φ(xi)n = 1 − +� +xj∈Bδ(xi) γn(xi, xj) +� +xj∈Bδ(xi) 1 +. +(5.4) +22 + +(a) Displacement Fiel +(b) Damage Field +Data +Prediction +Data +Prediction +0.40 +40 +20 +20 +Step 20 +o +0.35 +40 +20 +20 +0.30 +-50 +25 +25 +50 +-50 +25 +0 +25 +50 +20 +40 +40 +0.25 +20 +20 +Step 30 +0 +0 +0.20 +20 +20 +40 +40 +0.15 +-50 +-25 +25 +50 +-5025 +0 +25 +50 +20 +40 +40 +0.10 +20 +20 +Step 40 +0 +40 +0.05 +20 +20 +-40 +40 +-50-250 +0.00 +25 +50 +50 +0 +50Figure 7: Comparison of the prediction and the ground truth measurement from the MD data set at time steps 20, 30, and +40, on the fracture test data set where the graphene sheet is subject to unseen and nonzero body force. (a) Comparison on the +displacement field, where the color of the particles represents the horizontal displacement. (b) Comparison on the damage field. +Figure 6 shows the comparison of displacement and damage fields at time steps 20, 30, and 40. It is observed +that the prediction not only matches the data within the training set (step 20) but also exhibits a good +agreement at steps 30 and 40, which are not included in the training set. This result suggests that our +learned damage criterion s0 is applicable to longer term simulations out of the training data set. For the +first 40 steps, we have obtained 27% relative error for the prediction of displacement field and 9% relative +error for damage field. +5.3. Generalization to Different Body Forces and Crack Patterns +In this Section, we use the learned LPS surrogate to model the same graphene sheet subject to a different +body force load as described in the fracture test data set. Differs from the settings in the training data set, +in this data set the graphene sheet is subject to nonzero body load, with its crack pattern at the 40-th time +step illustrated Figure 4(d). Compared with the crack pattern in the training data set (see Figure 4(c)), the +crack path in this test data set is less symmetric and bifurcates at the middle of the domain. Hence, with this +example we not only investigate the extrapolation capability of the learnt model by making a longer time +(40 steps) predictions, but also aim to verify its generalizability, since both the loading scenario and crack +pattern from this test data set are not covered in the training data. All these factors make the validation +more challenging. In Figure 7 we show the prediction of displacement and damage fields from the learnt +LPS model upto time step 40. Visually good agreements are observed between the coarse-grained data and +23 + +a) Displacement Field +(b)Damage Field +Data +Prediction +Data +Prediction +0.40 +40 +40 +20 +20 +09 +Step 20 +0.35 +20 +20 +40 +40 +-40 +0.30 +-50 +0 +25 + 50 +50 +25 +25 +50 +20 +40 +40 +0.25 +20 +20 +Step 30 +0 +0 +0.20 +20 +20 +-40 +40 +20 +0.15 +50 +0 +50 +50 +50 +40 +40 +0.10 +20 +20 +Step 40 +0.05 +20 +60 +20 +0.00 +50 +0 +50 +50 +50Figure 8: Comparison of the prediction and the ground truth measurement from the MD data set at time steps 20, 30, and +40, on the fracture training data set with fine grids where the graphene sheet is subject to zero body force. Here, we used +coarser grids data in the first 20 steps for training, then we use the learnt model to predict for the next 20 steps on a finer +resolution. (a) Comparison on the displacement field, where the color of the particles represents the horizontal displacement. +(b) Comparison on the damage field. +LPS predictions. This example has qualitatively validated that the learned material damage model can be +directly applied to problems with different body force. For the first 40 steps, we have obtained 58% relative +error for the prediction of displacement field and 15% relative error for damage field. +5.4. Generalization to Different Resolutions +Last but not least, we study the resolution generalizability of our learning algorithm. Specifically, we +use the same MD data as the training data, but evaluate the density, displacement and force loading on a +coarse-grained grid with smaller grid size h = 3.17˚A. Since all training data sets are with a fixed grid size +h = 5˚A, with this study we aim to investigate if the learnt surrogate model allows the grid size to be rescaled, +providing a multiscale capability and allowing for flexible solver resolution and reductions in computational +cost. As suggested by [47], we scale the horizon size δ proportionally with the grid size h to provide a fixed +horizon/grid size ratio. In particular, we take δ = 4h = 12.68˚A. Then, the optimal damage criterion is also +scaled correspondingly to guarantee a consistent critical release rate. As proved in [71], the damage criterion +and horizon size should satisfy the relation s0 ∝ +1 +√ +δ in the LPS model. Thus we use s0 := 0.11 +� +5 +3.17 for our +fine scale simulation. In Figure 8 we show the displacement and and damage field prediction results upto +time step 40, demonstrating a qualitative agreement between the coarse-grained MD data and our numerical +predictions. On the displacement field, we have obtained 19% relative error in average for the first 40 steps, +24 + +(a) Displacement Field +(b) Damage Field +Data +Prediction +Data +Prediction +0.5 +40 +40- +20 +20 +Step 20 +40 +20 +20 1 +0.4 +-40 +40 +5025 +25 +50 +5025 +0 +25 +20 +40 +40 +0.3 +20 +20 +Step 30 +o- +20 +20 +40 +40 +0.2 +-50-25 +25 +50 +50-25 +25 +20 +1 +40 +40 +20 +20 - +0.1 +Step 40 +0 +40 +20 +20 +40 +40 +50-25 +0 +2550 +50 +0 +50 +0.0which is even smaller than the prediction error without resolution alternation on the same data set (27% as +shown in Section 5.2). For the damage field, one can see that the crack pattern predicted by our surrogate +model grows faster than the crack from MD data set. Therefore, a larger prediction error, 30% average for +damage field in the first 40 steps, is obtained. This example suggests that the surrogate model can provide +qualitatively consistent displacement predictions on different resolutions. On the other hand, the prediction +on damage field is sub-optimal, possibly due to the fact that the material crack originates from microscale +phenomena, and hence is more sensitive to the prediction scales. To improve the prediction accuracy on +the damage field across different resolutions, practitioners might consider performing the damage criterion +learning step on the new resolution, to provide a correction for the damage criterion. +6. Conclusions +In this paper, we demonstrate a data-driven workflow to extract a coarse-grained surrogate model from +MD data with fracture. +Firstly, to handle the discontinuities induced by material fracture in the MD +displacement measurements, a smoothness indicator function is introduced, to automatically choose the +locally smoothest stencil from the neighborhood of each coarse grained grid. As such, the coarse-graining +measurements are built based on this adaptive stencil, to automatically handle the discontinuities in MD +displacement data set without overly smoothing the crack pattern. It is shown that this novel adaptive +procedure significantly improves the capability of capturing the location of crack interfaces. Then, based on +the coarse-grained data set we proposed to extract a peridynamics surrogate, which is a continuum mechanics +model that allows a natural treatment of discontinuities by replacing spatial derivatives of stress tensors +with integrals of force density functions. By learning the kernel function of the integral and the damage +criterion with a two-step optimization approach, we obtain a linear peridynamic solid model which provides +good agreement with nanoscale test data while being capable to provide further material deformation and +fracture predictions under unseen domain settings, loading scenarios, and even different grid resolutions. +These features greatly reducing the cost of the calculation in comparison with MD, especially when used +together with different discretization resolutions. +Although the present work focuses on relatively small deformations and a linear peridynamics model, +the results suggest that this method may impact a broader range of materials and applications. As another +natural follow-up work, one may further combine the nonlocal model with the approximation power of neural +networks, to obtain a nonlinear peridynamics model in the form of integral neural operators [74–77]. +Acknowledgements +H. You and Y. Yu would like to acknowledge support by the National Science Foundation under award +DMS-1753031 and the AFOSR grant FA9550-22-1-0197. Portions of this research were conducted on Lehigh +University’s Research Computing infrastructure partially supported by NSF Award 2019035. +25 + +S. Silling and M. D’Elia would like to acknowledge the support of the Sandia National Laboratories (SNL) +Laboratory-directed Research and Development program and by the U.S. Department of Energy (DOE), +Office of Advanced Scientific Computing Research (ASCR) under the Collaboratory on Mathematics and +Physics-Informed Learning Machines for Multiscale and Multiphysics Problems (PhILMs) project. +This +article has been authored by an employee of National Technology and Engineering Solutions of Sandia, LLC +under Contract No. DE-NA0003525 with the U.S. Department of Energy (DOE). The employee owns all +right, title and interest in and to the article and is solely responsible for its contents. 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Yu, Learning deep implicit fourier neural operators (ifnos) +with applications to heterogeneous material modeling, Computer Methods in Applied Mechanics and +Engineering 398 (2022) 115296. +31 + diff --git a/ctE3T4oBgHgl3EQfeAqc/content/tmp_files/load_file.txt b/ctE3T4oBgHgl3EQfeAqc/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5ea6db73322e589519f6ba320bdcdb472e50740e --- /dev/null +++ b/ctE3T4oBgHgl3EQfeAqc/content/tmp_files/load_file.txt @@ -0,0 +1,1191 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf,len=1190 +page_content='Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures Huaiqian Youa, Xiao Xub, Yue Yua,∗, Stewart Sillingc, Marta D’Eliad, John Fosterb aDepartment of Mathematics, Lehigh University, Bethlehem, PA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' bDepartment of Petroleum and Geosystems Engineering, The University of Texas at Austin, Austin, TX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' cCenter for Computing Research, Sandia National Laboratories, Albuquerque,NM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' dCenter for Computing Research, Sandia National Laboratories, Livermore, CA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Abstract Molecular dynamics (MD) has served as a powerful tool for designing materials with reduced reliance on laboratory testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' However, the use of molecular dynamics directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this work, we propose a learning framework to extract a peridynamic model as a mesoscale continuum surrogate from MD simulated material fracture data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Firstly, we develop a novel coarse-graining method, to automatically handle the material fracture and its corresponding discontinuities in MD displacement data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Inspired by the Weighted Essentially Non-Oscillatory (WENO) scheme, the key idea lies at an adaptive procedure to automatically choose the locally smoothest stencil, then reconstruct the coarse-grained material displacement field as piecewise smooth solutions containing discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, based on the coarse-grained MD data, a two-phase optimization- based learning approach is proposed to infer the optimal peridynamics model with damage criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In the first phase, we identify the optimal nonlocal kernel function from data sets without material damage, to capture the material stiffness properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, in the second phase, the material damage criterion is learnt as a smoothed step function from the data with fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As a result, a peridynamics surrogate is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As a continuum model, our peridynamics surrogate model can be employed in further prediction tasks with different grid resolutions from training, and hence allows for substantial reductions in computational cost compared with molecular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We illustrate the efficacy of the proposed approach with several numerical tests for single layer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Our tests show that the proposed data-driven model is robust and generalizable, in the sense that it is capable in modeling the initialization and growth of fractures under discretization and loading settings that are different from the ones used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Keywords: Nonlocal Models, Machine Learning, Homogenization, Peridynamics, Material Fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Contents 1 Introduction 2 ∗Corresponding author Email address: yuy214@lehigh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='edu (Yue Yu) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='04540v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='mtrl-sci] 11 Jan 2023 2 Coarse-Graining of Molecular Dynamics Displacement with Damage 4 3 A Peridynamics Model with Brittle Fracture 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1 Linear Peridynamic Solid (LPS) Model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2 Peridynamics Formulation for Brittle Fractures .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 11 4 Learning Algorithm 15 5 Application to single-layer graphene 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1 Data Generation and Learning results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 22 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='3 Generalization to Different Body Forces and Crack Patterns .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4 Generalization to Different Resolutions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 24 6 Conclusions 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Introduction Detection and prediction of material damage progression attract lots of interests to the broad scientific and engineering community [1–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Physically, the propagation of cracks results from a long-term physical process with its origin in the atomistic scale, which often requires a micro-structural model such as molec- ular dynamics (MD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' However, although MD has made enormous advances in capabilities through better algorithms, better interatomic potentials, and improvements in computational power, its direct employment in treating the deformation and failure of materials at the mesoscale is still largely beyond reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' At the mesoscale and above, a continuum model of mechanics is often required in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This fact creates the need for homogenized models that act at larger scales and that, combined with new advanced architectures, allow for fast and accurate predictions of material deformation and fracture [11–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this paper, we aim to address the question of how to extract coarse-grained measurements and a homogenized surrogate model from MD simulations, that is able to capture material deformation and the nucleation and growth of fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Nonlocal models are among the best candidates for this task [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In the context of homogenization, nonlocal models are characterized by integral operators (as opposed to differen- tiable operators) that embed time and length scales in their definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, they are able to capture long-range effects that classical PDE models fail to describe [23], which makes nonlocal models viable alter- natives to partial differential equation (PDE) models when the effects of the small-scale behavior of a system affect its global state [22, 24–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For monitoring and predicting material fractures, because the nonlocal viewpoint avoids classical notions like deformation gradient, nonlocal models allow a natural description of processes requiring reduced regularity in the relevant solution [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As such, the nonlocal continuum mechanics model, in the form of peridynamics [31, 34–42], provides a unified modeling of continuum media where continuity and complex material damage modes can be captured autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 2 In peridynamics and the general nonlocal models, constitutive laws take the form of integrand functions, whose functional form is often justified a posteriori, which makes rigorous calibration and validation chal- lenging and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' On the other hand, although the nonlocal constitutive laws must be consistent with the classical effective properties, they contain information about the small-scale response of the system and must be chosen to reproduce this response with the greatest fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, it is desired to extract an optimal integrand function from small-scale data, such that the calibrated nonlocal model reproduces the material responses and can further serve as a homogenized surrogate for future material deformation and fracture prediction tasks [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Recently, with the explosion of machine learning, optimized nonlocal models were designed with the purpose of accurately reproducing observed coarse-grained behavior and pre- dicting unseen behavior with the learnt model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We refer the reader to [45–50] for several examples of the use of optimization-based machine learning for the design of homogenized nonlocal operators and the rigorous analysis of its learning theory [50, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Although successful in providing optimal nonlocal surrogates to the homogenization problem, to the authors’ best knowledge, none of these approaches addresses the challenge of capturing the main features of dynamic fracture that are seen in small-scale data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Fundamental challenges are still present, mainly due to the two difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Firstly, when mapping the MD measurements onto a coarser grid, coarse graining methods can use the mean atomic velocities weighted by a smoothing function [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In a nonlocal setting, a smoothed displacement field can be shown to evolve according to the peridynamic linear momentum balance [44, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' However, once the material fracture occurs, such a weighted average approach might overly smooth the displacement field and introduce errors near cracks in the coarse-grained data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Secondly, in peridynamics the material damage is often described by breaking bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, the integrand functions present jumps near the damage criterion, which results in nonsmooth losses in the optimization problem and hinders the application of a suite of continuous optimization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Herein, we address these two challenges and present a complete workflow demonstrating how to obtain large-scale nonlocal descriptions that capture MD behavior with fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To accomplish this, we develop a novel coarse-graining method which is inspired by the Weighted Essen- tially Non-Oscillatory (WENO) scheme, and extend the machine learning technique in our previous work [47] to identify a smoothed damage criterion together with optimal nonlocal kernel functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We summarize below our main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We develop a novel coarse-grained approach from micro-scale fracture measurements, to automatically choose a locally smoothest stencil and capture the displacement discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As such, coarse- graining measurements are obtained without overly smoothing the crack pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We propose a two-step optimization strategy, and identify the best upscaled surrogate in the form of peridynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Without prior knowledge of the material properties, the resultant peridynamics model describes the material deformation together with the nucleation and growth of fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 3 Figure 1: An example of the vanilla coarse-graining method developed in [44] and our proposed approach, in handling the MD measurements with a crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Small blue points represent the MD particles and red dots stand for coarse-grained points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Left: results from the vanilla coarse-graining method with weight function ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Right: results from our proposed coarse-graining method with an adaptive weight function ˆω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The optimal nonlocal model generalizes well to fracture patterns that are substantially different from the ones used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The optimal model also enables extrapolation to longer time simulations and a multiscale capability to predictions across resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Outline of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Section 2 shows how to obtain an adaptive stencil in the form of a smoothness indicator function, to extract coarse-grained measurements from MD displacements with fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In Section 3 we summarizes the linear peridynamic solid (LPS) model, the treatment of material fracture and the handling of free surfaces, and the discretization technique used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Section 4 presents our two-step learning approach consisting of a kernel learning step and a damage criterion learning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Section 5 verifies the learning technique for MD displacements and studies the generalizability of the resultant model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' On a single layered graphene, we demonstrate efficacy of our workflow by identifying an optimal two-dimensional nonlocal model and employing this model in complex prediction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In particular, we illustrate several properties including generalization with respect to loadings, domain settings, crack shapes, and grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Section 6 summarizes our contributions and provides future research ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Coarse-Graining of Molecular Dynamics Displacement with Damage In this section, we introduce the coarse-graining method to map the displacement field data from MD simulations into a larger-scale discretized data cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For materials without fracture, in [44, 47] a nonlocal coarse-graining method was proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this method, the coarse grained displacement for each particle is defined as a weighted average of the microscale displacements in its neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As such, a smoothed dis- placement field is obtained, which preserves a linear momentum balance as a consequence of the momentum balance for the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' However, this coarse-graining method hides a pitfall: once the material fracture occurs introducing discon- tinuities in the displacement field, the weighted average approach would overly smooth the displacement field 4 40 20 0 20 40 60-40-20 0 20 40 6040 20 0 20 40 60 40 20 0 20 40 60Figure 2: Schematic and examples of calculation for the smoothness indicator function α(x, Xε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (a): A demonstration of the projection of MD points between x and Xε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (b): When the projected displacement field is continuous, the smoothness indicator α(x, Xε) stays close to 1 since ϵ(D1, D2) is close to ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (c): when material fracture occurs and the projected displacement field is discontinuous (with a jump at d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5), ϵ(D1, D2) ϵ0 reaches the minimum when D1 and D2 both consist of a smooth curve, and we have α(x, Xε) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' and smudge the crack pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To resolve this challenge, in this section, we will extend the coarse-graining method to an adaptive approach, so as to automatically handle the material fracture and its corresponding discontinuities in MD displacement data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To introduce the coarse-graining method, we consider the MD data set as an assembly of S mutually interacting particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, we define the mass of each mutually interacting particles as Mε, ε = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=', S, the reference positions of these particles as Xε, and the displacement vectors as Uε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Each particle is subjected to a prescribed external force, Bε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The coarse-grained measurements can be defined by choosing a compactly supported function ω(x, ·) for each material point x ∈ Rd, such that � Rd ω(x, Xε)dx = 1, ω(x, Xε) = 0 if |x − Xε| > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1) Then, the smoothed material density and body force density are expressed as ρ(x) = S � ε=1 ω(x, Xε)Mε, b(x, t) = S � ε=1 ω(x, Xε)Bε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2) Correspondingly, the smoothed displacement field at material point x is obtained by u(x, t) = 1 ρ(x) S � ε=1 ω(x, Xε)MεUε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='3) 5 U ↑e(D1, D2) U e(D1, D2) U (b) e(D1, D2) (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='82 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='79 03 03 03 4 4 d 0 4 d U U+ (c) e(D1, D2) e(D1, D2) e(D1, D2) 0 ～ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='51 03 E0 03 1 1 1 0 d Regression line of D, D Regression line of D -- Regression line of D2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='In [47], the authors proposed to employ a general cone-shaped weighted function for all material points, by defining ω(xi, Xε) := τ(xi, Xε) � j τ(xj, Xε), where τ(x, X) = max{0, R − |X − x|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4) Here, R is a pre-chosen hyperparameter, representing the coarse-graining radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Such an approach was found to be effective for materials without fracture, where the displacement field is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then in [47] a data-driven surrogate model was built from this coarse-grained displacement field, which has successfully captured the material properties as well as a constitutive law acting at a larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' However, once the material fracture occurs, the smooth weight function such as the one in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4) would smudge the crack pattern and hence may compromise the reliability of the resultant surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As shown in the left plot of Figure 1: coarse-grained points appear on the middle of the crack, showing the effect of overly-smoothing the displacement field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To provide coarse-grained displacement field for both damaged and undamaged material regions, we propose to adjust the smoothing function ω(x, Xε) when the material point x is close to the crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Intuitively, the weight function should choose the locally smoothest stencil and avoid crossing discontinuities in the averaging procedure as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Similar idea was employed in the Essentially Non-Oscillatory (ENO) and Weighted Essentially Non-Oscillatory (WENO) methods [53, 54], to develop finite difference schemes for PDE problems with piecewise smooth solutions containing discontinuities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Inspired by the WENO methods, our key idea is to assign an additional smoothness indicator function α(x, Xε) to each pair of continuum material point x and MD particle Xε, and modify the weight function ω(x, Xε) as: ˆω(x, Xε) = ω(x, Xε)α(x, Xε) � Rd ω(x, Xε)α(x, Xε)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5) When the crack intersects with the bond between x and Xε, the displacement field between x and Xε contains discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' One should use less information from Xε to calculate the weighted average on x, by taking the smoothness indicator function α(x, Xε) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' That means, an adaptive procedure is required, to detect if there is a displacement jump between x and Xε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As demonstrated in Fig 2, we construct the smoothness indicator function α(x, Xε) with the following procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' First, we project all the MD particles within a distance of R from x to the line segment that connects x and Xε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For each MD particle Xi, we denote the projected point as �Xi, and calculate its projected position variable, di, as the distance from x to �Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The displacement vector Ui(t) on Xi is also projected, and we denote its component along segment x − Xε as Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Next, we select all the particles with their projections lie between x and Xε, to form a set of data pairs D = {(di, Ui)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' When plotting Ui as a function of di, the curve will present a jump when there is a crack intersecting the segment between x and Xε, and such a jump would naturally divide the set D as two sets, with each set representing a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, our goal is then to identify the discontinuity of U(d) and define the smoothness indicator function according to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Numerically, we loop over all possible combinations of splitting the data pair set D 6 into two sets, D1 and D2, such that D1 � D2 = D and D1 � D2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, we perform linear regressions on D1 and D2: kβ, bβ = argmin k,b � (di,Ui)∈Dβ |kdi + b − Ui|2, β = 1, 2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='6) to obtain a fitted line for each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In the meantime, we also perform a linear regression on the entire displacement data set D, and obtain the fitted parameter set (k, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Denoting the total squared error ϵ(D1, D2) associated with D1 and D2 as: ϵ(D1, D2) := 2 � β=1 � {(di,Ui)}∈Dβ (kβdi + bβ − Ui)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='7) and a squared error for the whole data set D as ϵ0 := � {(di,Ui)}∈D (kdi + b − Ui)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='8) we define the smoothness indicator function α(x, Xε) as α(x, Xε) := min (D1,D2)ϵ(D1, D2) ϵ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='9) Intuitively, when there is no material fracture and hence U(d) is a smooth curve without discontinuity, we anticipate to have (kβ, bβ) ≈ (k, b) for β = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As a result, we have ϵ(D1, D2) ≈ ϵ0 and α(x, Xε) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this case, the adjusted weight function ˆω(x, Xε) will stay the same as the original weight function, and hence our smoothness indicator function will not alter the coarse-graining approach for materials without fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Figure 2(b) shows an example with continuous displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' It can be observed that ϵ(D1, D2) stays roughly the same and close to ϵ0 for different partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' On the other hand, when fracture occurs, ϵ(D1, D2) would achieve its minimum when both D1 and D2 consist of a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' That means, when D1 and D2 are separated by the crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this case, we will have min (D1,D2)ϵ(D1, D2) < ϵ0 and hence α(x, Xε) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, the adjusted weight function ˆω(x, Xε) would automatically reduce the weights of those particle points crossing discontinuities, so as to reduce the overly smoothing near cracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Figure 2(c) presents an example where the displacement is piecewise constant with a jump at d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5, demonstrating that the smooth indicator would reach its minimum when neither D1 or D2 contains the displacement jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Once the adjusted weight functions are obtained, the smoothed mass density, body force density, and 7 displacement can be calculated as ρ(x) = S � ε=1 ˆω(x, Xε)Mε, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='10) b(x, t) = S � ε=1 ˆω(x, Xε)Bε, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='11) u(x, t) = 1 ρ(x) S � ε=1 ˆω(x, Xε)MεUε(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='12) The result of this modified weight function is demonstrated in the right plot of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' One can see that the coarse-grained points (red dots) are almost aligned with the crack interface, verified the efficacy of our modified coarse-graining method in handling MD data set with cracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Similar as the derivation in [47], we point out that our coarse-grained formulation naturally induces a nonlocal equation of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The goal of the present work is therefore to identify an optimal nonlocal model in the form of peridynamics, which faithfully represents given MD displacements under a given set of loading conditions, and is generalizable to further prediction tasks for analysis of material deformation and crack propagation phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' A Peridynamics Model with Brittle Fracture In the previous section, a data set of function trios, T := {(ρm, um, bm)}M m=1, were derived from our coarse-grained formulation, such that each trio contains a coarse-grained density field ρm(x), a body force density bm(x, t), and their corresponding displacement field um(x, t) for material point x ∈ Ωm ⊂ Rd and t ∈ [0, T m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Herein, we note that each sample may have different spatial domain Ωm and observation range T m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We propose to learn a nonlocal momentum balance equation based on these function trios in the form of peridynamic equation of motion [34], to provide a continuum model with direct description of fracture within the basic field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In peridynamics, each x interacts through bond forces with other material points y within a neighborhood with radius δ known as the family of x, denoted by Bδ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here, the horizon δ determines the extent of the nonlocal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The equation of motion for material point x is given as ρ(x)∂2u(x, t) ∂t2 = � Bδ(x) f(y, x, t) dy + b(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1) A material model in peridynamics supplies values of f(y, x, t) in terms of the deformations of the families of x and y and any other relevant variables such as temperature [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Peridynamics can model fracture because the equation of motion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1) is an integro-differential equation that does not involve the partial derivatives of displacement with respect to position, which leads to a lower requirement of the solution regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Moreover, many material models have been developed for peridynamics, and any material model from the local theory can be translated into peridynamic form [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For a more thorough introduction and 8 review about peridynamics, we refer interested readers to [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this work, we aim to addresses the question of how to use MD to obtain a peridynamic material model that is able to treat material deformation and the nucleation and growth of fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' With a purpose of demonstration and without loss of generality, we consider a 2D simulation problem (d = 2) in single-layered graphene, and concern small deformations in the linear regime of material response, although the algorithm may be generalized to finite deformations and 3D cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In [47], a data-driven two-dimensional linear peridynamic solid (LPS) model[55] under plane-stress assumption was found to adequately represent the material response from MD simulations on a graphene sheet without fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Inspired by such preliminary results, in this work we also employ the LPS model as the base model, and further consider learning of material failure from coarse-grained MD data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this section, we first briefly introduce the LPS model without material fracture in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, we extend the model to describe material fracture and handle the imposition of traction loads as fracture surfaces open up in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, in the next section we will describe our learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Linear Peridynamic Solid (LPS) Model In this section, we summarize the mathematical formulation for the LPS model [57–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The LPS model is a prototypical state-based model which can be seen as a nonlocal extension of the linear elasticity model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' It is suitable to describe isotropic elastic materials under infinitesimal deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Comparing with the previously developed bond-based peridynamic models [34, 60], the LPS model has advantages in that it is not restricted to a Poisson’s ratio of 1/4, which is important for our application since the Poisson ratio of graphene was found to be negative from MD and molecular statistics simulations [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Consider a body occupying the domain Ω ⊂ Rd, and let θ be the nonlocal dilatation, generalizing the local divergence of the displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this section, we consider the material without damage, with fully prescribed Dirichlet type boundary conditions, and will further extend the discussions to more general boundary conditions and brittle fractures in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here we note that in nonlocal problems, unless otherwise stated, the boundary conditions should no longer be prescribed on the sharp interface, ∂Ω, but on a collar of thickness of at least δ surrounding the domain Ω, which we denote as: BΩ := {x /∈ Ω|dist(x, ∂Ω) < 2δ} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Given nonlocal boundary conditions prescribed on the nonlocal volumetric boundary domain (or simply nonlocal boundary): u(x, t) := uD(x, t) x ∈ BΩ, t ∈ [0, T], and the initial velocity φ(x) and displacement ψ(x) for x ∈ Ω � BΩ at t = 0, the peridynamic operator in 9 the LPS model is given by LK[u](x, t) := − C1 m � Bδ(x) (λ − µ) K(|y − x|) (y − x) (θ(x, t) + θ(y, t)) dy − C2 m � Bδ(x) µK(|y − x|)(y − x) ⊗ (y − x) |y − x|2 (u(y, t) − u(x, t)) dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2) and the nonlocal dilatation is defined via θ(x, t) := d m � Bδ(x) K(|y − x|)(y − x) · (u(y, t) − u(x, t)) dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='3) where m := � Bδ(0) K(|z|)|z|2dz is the weighted volume, λ is Lam´e’s first parameter, and µ is the shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To recover parameters for 3D linear elasticity, one should take C1 = 3, C2 = 30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' whereas for 2D problems, C1 = 2, C2 = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here we note that m is determined by the horizon size δ and the influence function K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We use the subscript K in the nonlocal operator LK[u](x) to emphasize the operator’s dependence on the influence function K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then the time-dependent LPS problem is given by � � � � � � � � � � � � � � � � � � � ρ(x)∂2u(x, t) ∂t2 + LK[u](x, t) = b(x, t), (x, t) ∈ Ω × [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' u(x, t) = uD(x, t), (x, t) ∈ BΩ × [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' u(x, 0) = ψ(x), x ∈ Ω � BΩ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' ˙u(x, 0) = φ(x), x ∈ Ω � BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4) Note that our learning algorithm is compatible with other type of boundary conditions in BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here we focus on Dirchlet type of boundary condition in this work for its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To discretize the above LPS model, we employ the optimization-based meshfree quadrature rule developed in [63–70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Suppose that the values of function trios ρ(x), u(x, t), and b(x, t) are provided on a set1 of coarse- grained material points χ := {xi}I i=1 ⊂ Ω � BΩ and time instances tn = n∆t, n = 0, · · · , T/∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We write the discretized approximation of LK as Lh K[u](xi, tn) := −C1 mi � xj∈Bδ(xi) � χ (λ − µ) Kij (xj − xi) � θh(xi, tn) + θh(xj, tn) � Wj,i −C2 mi � xj∈Bδ(xi) � χ µKij (xj − xi) ⊗ (xj − xi) |xj − xi|2 (u(xj, tn) − u(xi, tn)) Wj,i, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5) θh(xi, tn) := d mi � xj∈Bδ(xi) � χ Kij(xj − xi) · (u(xj, tn) − u(xi, tn)) Wj,i, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='6) 1Although the machine learning algorithm as well as the quadrature rule is compatible with the general non-uniform grids, in this work we consider the uniform grids with grid size h and uniform time steps with size ∆t, for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 10 where Kij := K(|xj − xi|) and mi := � xj∈Bδ(xi) � χ Kij|xj −xi|2Wj,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The quadrature weights Wj,i are associ- ated with a local neighborhood of particles for each discretization point xi, generated by local optimizations to make the approximation rule exact for certain classes of functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For each xi ∈ χ � Ω we solve for Wj,i via argmin {ωj,i} � xj∈χ � Bδ(xi)\\{xi} W 2 j,i s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=', � xj∈Bδ(xi) q(xi, xj)Wj,i = � Bδ(xi) q(xi, y)dy ∀ q ∈ Vxi, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='7) where Vxi denotes the space of functions which should be integrated exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Following [64], in this work we take Vxi := � q(y − xi) = p(y−xi) |y−xi|3 | p ∈ P5(Rd) such that � Bδ(0) q(y)dy < ∞ � and P5(Rd) denotes the space of quintic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As the horizon size δ vanishes, this discretization preserves the consistency in the limit to the local solution [63, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Moreover, we point out that the quadrature weights, Wj,i, only depend on the grid set χ and it is invariant of the influence K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Hence, in our learning algorithm one only need to generate the quadrature weights and solve the local optimization problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='7) once in the preprocessing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the dynamic peridynamics model, to discretize in time we apply the central difference time stepping scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' With time step size ∆t, at the (n + 1)−th time step one can solve for the displacement un+1 i ≈ u(xi, tn+1) following: � � � ρ(xi)¨un i + Lh K[u](xi, tn) = b(xi, n∆t), for xi in Ω � χ, un+1 i = uD(xi, (n + 1)∆t), for xi in BΩ � χ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='8) where Lh K is the discretized nonlocal operator as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5), and the acceleration ¨un i is estimated via the central difference scheme: ¨un i := un+1 i − 2un i + un−1 i ∆t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='9) As the initial conditions, we set u0 i = ψ(xi) and u1 i − u0 i ∆t = φ(xi) for xi ∈ (BΩ � Ω) � χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Peridynamics Formulation for Brittle Fractures One of the main appeals of peridynamics is to handle fracture problems, where free surfaces are associated with the evolution of a fracture surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this section, we consider the LPS model with free surfaces, then apply it to the treatment of brittle fractures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To describe the free surfaces associated with the time evolution of a fracture surface, we now consider general mixed boundary conditions: ∂Ω = ∂ΩD � ∂ΩN and (∂ΩD)o �(∂ΩN)o = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here ∂ΩD and ∂ΩN are both curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' ∂ΩN is the (possibly time-dependent) sharp crack surface evolving with the material fractures, and a free surface boundary condition is applied on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To define a Dirichlet-type constraint, we denote BΩD := {x /∈ Ω|dist(x, ∂ΩD) < 2δ}, 11 and assume that the value of u(x, t) = uD(x, t) is given on x ∈ BΩD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For notation simplicity, we denote ΩD := Ω ∪ BΩD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To apply the free surface boundary condition, we denote IΩN := {x ∈ Ω|dist(x, ∂ΩN) < δ}, IΩ := {x ∈ Ω|dist(x, ∂Ω) < δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Unless stated otherwise, in this paper we further assume sufficient regularity in the boundary region IΩ that there exists a unique orthogonal projection of x onto ∂Ω, which is the closest point on ∂Ω to x, and we denote this projection as x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, one has x − x = sxn(x) for x ∈ IΩN, where 0 < sx < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here n denotes the normal direction pointing out of the domain for each x ∈ IΩN, and let p denote the tangential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In our numerical solver, we treat x with the free surface boundary condition if the projection of x is in ∂ΩN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Otherwise, we use the Dirichlet-type boundary condition at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In peridynamics, material damage is incorporated into the constitutive model by allowing the bonds of material points to break irreversibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To model brittle fracture in the LPS model, we employ a smoothed critical stretch criterion, where weakening occurs when a bond is extended beyond some predetermined critical bond deformed length [40, 64, 71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In particular, a scalar state function γ(x, y, t) is defined and takes values in the interval [0, 1], to describe the bond weakening and breakage through the crack growing: γ(x, y, t) := 1 2 � − tanh �maxτ∈[0,t] S(x, y, τ) − s0 η � + 1 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='10) where S(x, y, τ) := |x − y + u(x, τ) − u(y, τ)| |x − y| − 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='11) and s0 is the critical stretch criterion depending on the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' γ(x, y, t) is a history-dependent function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=', a bond can never recover once it exceeds the critical stretch criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' An illustration of γ can be visualized in Figure 3, where a hyperparameter η ≪ 1 can be tuned to control the level of smoothness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' When γ(x, y, t) = 1, the bond between material points x and y are considered “intact” and the change of displacement on material point y may have an impact on the displacement at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' When the stretch S(x, y, τ) exceeds the critical criterion s0 for some time τ < t, the material gets damaged and we have γ(x, y, t) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As the stretch further increases, finally γ(x, y, t) = 0 and we consider the bonds between x and y as fully “broken”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Instead of defining γ as a step function following [64], in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='10) we allow the weakening of force scalar within small ranges of excessive bond stretch values and set γ as a smoothed step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As shown in [40], such a smoothed state function would impose the continuity of the learnt bond force f(x, y, t) in our peridynamic model, and guarantee the well-posedness of the peridynamic model as a dynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' On the other hand, a continuous formulation of the damage factor γ would result in a continuous optimization problem, and allows generic optimization routines to be used in the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' With the state function γ, we treat the time-evolving fracture as free surfaces and employ the following 12 max < t S(x i,x j, ) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5 (x i,x j,t) = 10 -8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='05 S0 Figure 3: An illustration of the smoothed scalar state function γ, with the tunable parameter η = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='05, 10−8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' formulation: � � � � � � � � � � � � � � � � � � � ρ(x)∂2u(x, t) ∂t2 + LKN[u](x, t) = b(x, t), (x, t) ∈ Ω × [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' u(x, t) = uD(x, t), (x, t) ∈ BΩD × [0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' u(x, 0) = ψ(x), x ∈ Ω � BΩ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' ˙u(x, 0) = φ(x), x ∈ Ω � BΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='12) Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' the modified LPS operator LKN follows the formulation in [64,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 70]: LKN[u](x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t) := −C1 m � Bδ(x) (λ − µ) K(|y − x|)γ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t) (y − x) (θcorr(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t) + θcorr(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t)) dy − C2 m � Bδ(x) µK(|y − x|)γ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t)(y − x) ⊗ (y − x) |y − x|2 (u(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t) − u(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t)) dy − 2C1θcorr(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t) m � Bδ(x) (λ − µ) K(|y − x|)(1 − γ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t)) (y − x) dy − C2θcorr(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t) 2m � Bδ(x) (λ + 2µ)K(|y − x|)(1 − γ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t))[(y − x) · n][(y − x) · p]2 |y − x|2 ndy + C2θcorr(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t) 2m � Bδ(x) λK(|y − x|)(1 − γ(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' t))[(y − x) · n]3 |y − x|2 ndy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='13) with θcorr(x, t) := d m � Bδ(x) K(|y − x|)γ(x, y, t) (y − x) · M(x) · (u(y, t) − u(x, t)) dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='14) M(x, t) := � d m � Bδ(x) K(|y − x|)γ(x, y, t) (y − x) ⊗ (y − x) dy �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='15) As such, the LPS model provides an approximation for the corresponding linear elastic model with free surfaces in the case of linear displacement fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We notice that when all bonds in Bδ(x) are intact, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=', the material point x is sufficiently far away from the free surface, we have γ(x, y, t) = 1 for all y ∈ Bδ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='13) yields LKN = LK and the original momentum balance and nonlocal dilatation formulation in the 13 LPS model are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='13) provides a unified mathematical framework which automatically captures material deformation and the evolution of cracks as free surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' We now extend the optimization-based quadrature rule and the central difference time-stepping method introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1, to the LPS model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='13) with fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Particularly, at the (n + 1)−th time step we approximate the state function γ(xi, xj, tn) via γn ij := 1 2 � �− tanh � � max 0≤m≤nSm ij − s0 η � � + 1 � � , where Sm ij := |xi − xj + um i − um j | |xi − xj| − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='16) Then the approximated displacement field un+1 i ≈ u(xi, tn+1) can be solved via the following formulation: � � � ρ(xi)¨un i + Lh KN[u](xi, tn) = b(xi, n∆t), for xi in Ω � χ, un+1 i = uD(xi, (n + 1)∆t), for xi in BΩD � χ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='17) where Lh KN[u](xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' tn) := −C1 mi � xj∈Bδ(xi) � χ (λ − µ) Kijγn ij (xj − xi) � (θcorr)n i + (θcorr)n j � Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='i − C2 mi � xj∈Bδ(xi) � χ µKijγn ij (xj − xi) ⊗ (xj − xi) |xj − xi|2 � un j − un i � Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='i − 2C1(θcorr)n i mi � xj∈Bδ(xi) � χ (λ − µ) Kij(1 − γn ij) (xj − xi) Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='i − C2(θcorr)n i 2mi � xj∈Bδ(xi) � χ (λ + 2µ)Kij(1 − γn ij)[(xj − xi) · nn i ][(xj − xi) · pn i ]2 |xj − xi|2 nn i Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='i + C2(θcorr)n i 2mi � xj∈Bδ(xi) � χ λKij(1 − γn ij)[(xj − xi) · nn i ]3 |xj − xi|2 nn i Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='i (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='18) with (θcorr)n i := d mi � xj∈Bδ(xi) � χ Kijγn ij (xj − xi) · Mn i · � un j − un i � Wj,i, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='19) Mn i := � � d mi � xj∈Bδ(xi) � χ Kijγn ij (xj − xi) ⊗ (xj − xi) Wj,i � � −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='20) Here we note that the free surface ∂ΩN as well as the normal vector n(x) on free surfaces both change as the fracture evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To numerically approximate n(xi, tn) at each time step, we updated it via nn i = − � xj∈χ∩Bδ(xi) (xj − xi)Wj,iγn ij ����� � xj∈χ∩Bδ(xi) (xj − xi)Wj,iγn ij ����� , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='21) 14 and the tangential vector pn i is calculated as the orthogonal direction to nn i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The correction tensor should be invertible to ensure that the correction dilitation can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This holds as long as the bonds in the horizon are non-colinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For fracture case resulting in bond break, leaving an isolated particle, we replace the matrix inverse with the pseudo-inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Learning Algorithm Algorithm 1 Workflow for learning the LPS model from MD data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 1: To obtain samples without material fracture, generate relatively small MD displacements on fine grids {Xm ε } using different external forces and domain configurations, then group the samples into two data sets, MDNon-Frac train for training the nonlocal kernel and MDNon-Frac val for hyperparameter tuning: MDNon-Frac train/val := {M m ε , Um ε (t), Bm ε (t)}, m = 1, · · · , M Non-Frac train/val .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 2: Generate MD displacements samples with material fracture, on fine grids { �Xm ε } using different external forces and domain configurations, then group the samples into two data sets, MDFrac train for training the damage criterion and MDFrac test for test: MDFrac train/test := {� M m ε , �Um ε (t), �Bm ε (t)}, m = 1, · · · , M Frac train/test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 3: Coarse grain the data sets MDNon-Frac train/val and MDFrac train/test, then evaluate the coarse grained data at coarser grids χm to obtain the functio trio sets T Non-Frac train/val := {ρm(xi), um(xi, tn), bm(xi, tn)}, m = 1, · · · , M Non-Frac train/val , T Frac train/test := {�ρm(xi), �um(xi, tn), �bm(xi, tn)}, m = 1, · · · , M Frac train/test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 4: (Kernel learning step): Solve the optimization problem based on the non-fracture data set T Non-Frac train : � � � (λ∗, µ∗, D∗, α∗) = argmin λ,µ,D,α Res(T Non-Frac train ) subject to solvability constraints (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4), and tune the hyperparameters δ∗ and P ∗, to minimize the test errors on the validation data set T Non-Frac val .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 5: (Damage criterion learning step): With fixed parameters (λ∗, µ∗, D∗, α∗, δ∗, P ∗), train for the opti- mal fracture criterion parameter based on the fracture data sets T Frac train: s∗ 0 = argmin s0 � Res(T Frac).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 6: To study the generalizability on unseen external forces and fracture scenarios, use the learnt LPS model to predict the material deformation and fracture on T Frac test .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Let T := {ρm(xi,m), um(xi,m, tn m), bm(xi,m, tn m)}, m = 1, · · · , M, be coarse-grained function trios avail- able at xi,m ∈ χm and tn m = n∆tm, n = 1, · · · , Nm, our goal is to identify an optimal constitutive relation on the basis of MD data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here, we use χm and ∆tm to highlight the fact that in our learning algorithm, each sample can be of different spatial/temporal domain and resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In the following content, we will skip the subscript m and denote the function trios as ρm(xi), um(xi, tn), and bm(xi, tn) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Let LKN be the LPS operator defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='18), we aim to learn an optimal continuum model in the form of 15 LPS models, where the optimal model consists of the influence function K, which may be sign-changing, and parameters λ, µ and s0, such that the action of LKN most closely satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='18) for all s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Formally, the optimal influence function and parameters, (λ∗, µ∗, s∗ 0, K∗), are the solution of the following optimization problem: (λ∗, µ∗, s∗ 0, K∗) = argmin λ,µ,s0,K 1 M M � m=1 Nm−1 � n=1 ∆tm ��ρm(xi)(¨um)n i + Lh KN[um](xi, tn) − bm(xi, tn) ��2 ℓ2(χm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1) The influence function K(|x − y|) will now be parameterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Following [72], In this work, the interacting kernel function K(|x − y|) is taken as a radial function compactly supported on the δ-ball Bδ(x) with α-th order singularity: K(|x − y|) = P � k=0 Dk |x − y|α Bk,P �|x − y| δ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2) Here the Bernstein polynomials are defined as Bk,P (r) = � �P k � � rk(1 − r)P −k, for 0 ≤ r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='3) Following the arguments in [47, 65], in the learning algorithm we require the fractional order α to be bounded by 3 and allow Dk ∈ R for all k with sufficient well-posedness conditions embedded for the discretized operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here, we note that in the samples with material fracture, some particles might become isolated due to fragmentation, and hence it would be impossible to require solvability constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, we only apply the solvability constraints to the model without fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' With the analysis in [47], given a tolerance parameter ζ > 0 we apply the following solvability constraints: � � � � � � � � � � � λ + µ > 0, µ > 0, α < 3, Λmin(Γ(α,D,δ,P )) ≥ ζ, Λmin(Φ(α,D,δ,P )Γ† (α,D,δ,P )Φt (α,D,δ,P )) ≥ ζ, Λmin(Γ(α,D,δ,P ) − 2Φt (α,D,δ,P )Φ(α,D,δ,P )) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4) Here Γ and Φ are the matrices that correspond to the deviatoric and dilatation contributions of the defor- mation, and Λmin(A) denotes the smallest nonzero eigenvalue of a matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The overall formulation of the constrained optimization problem is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Given a collection of training samples {ρm(xi),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' um(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' tn),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' bm(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' tn)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' m = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' we seek to learn the parameters λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' the Bernstein polynomial coefficients D = [D0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' DP ] ∈ RP +1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' the order α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' the horizon δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' the polynomial 16 order P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' and the damage criterion s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' by minimizing the mean square loss (MSL) of the LPS equation: � � � � � � � � � � � � � � � (λ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' µ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' D∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' α∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' δ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' P ∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' s∗ 0) = argmin λ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='s0 1 M M � m=1 Nm−1 � n=1 ∆tm ��ρm(xi)(¨um)n i + Lh KN[um](xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' tn) − bm(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' tn) ��2 ℓ2(χm) subject to solvability constraints (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5) However, numerically solving the constraint optimization problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4) could be time-consuming and possibly unstable, due to three factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' First, as shown in Figure 3, when s0 is away from the optimal value, its impact on the loss function would be relatively flattened, causing the vanishing gradient issue in optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Second, the update of s0 would induce the change of correction operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='18), which increases the computational cost on each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Lastly, the imposition of solvability constraints (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4) would also be expensive, since it involves additional calculations (such as with the projection method) and/or subiterations (such as with the augmented Lagrangian method), together with the evaluation of eigenvalues at each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To make the optimization algorithm more efficient and robust, we propose to separate the solving procedure of the damage criterion, s0, with other parameters, and propose a “two-stage” strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Key components are summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In particular, we notice that the correction operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='18) and the damage criterion, s0, are only associated with samples with material fractures, while the influence function K and other material parameters can be inferred from samples without fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, we divide the training data set into two sets: T Non-Frac := {ρm(xi), um(xi, tn), bm(xi, tn)}, m = 1, · · · , M Non-Frac, which includes all samples without fracture, and T Frac := {�ρm(xi), �um(xi, tn), �bm(xi, tn)}, m = 1, · · · , M Frac for training samples with fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then the optimization problem (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5) is also split, into a non-fracture kernel learning step and a damage criterion learning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the kernel learning step, we infer the influence function K and the Lam´e moduli λ and µ by solving a constraint optimization problem from T Non-Frac: � � � � � (λ∗, µ∗, D∗, α∗, δ∗, P ∗) = argmin λ,µ,D,α,δ,P Res(T Non-Frac) subject to solvability constraints (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='6) 17 where Res(T Non-Frac) := 1 M Non-Frac M Non-Frac � m=1 Nm−1 � n=1 ∆tm ��ρm(xi)(¨um)n i + Lh K[um](xi, tn) − bm(xi, tn) ��2 ℓ2(χm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='7) As such, one only has to evaluate the nonlocal operator without fracture following (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5), which is computa- tionally more efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In this step, we treat δ and P as hyperparameters to be separately tuned, to achieve the best learning accuracy without overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For each combination of δ and P, the Adam optimizer in PyTorch is employed, together with the augmented Lagrangian method to impose the inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For further details of the optimization algorithm and settings, we refer interested readers to [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In the damage criterion learning step, we fix the learnt parameters (λ∗, µ∗, D∗, α∗, δ∗, P ∗) and search for the optimal s0 by considering a unconstraint optimization problem on T Frac: s∗ 0 = argmin s0 � Res(T Frac), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='8) where � Res(T Frac) := 1 M Frac M Frac � m=1 Nm−1 � n=1 ∆tm ���ρm(xi)(¨�u m)n i + Lh NK[�um](xi, tn) − �bm(xi, tn) ��2 ℓ2(χm) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='9) In all tests, we set the smoothing parameter η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='05, and employ the bisection method to solve for s∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Application to single-layer graphene To illustrate the capability of our method in obtaining an optimal surrogate material damage model from coarse-grained MD displacements, we consider single layer graphene sheets as the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Graphene is a single layer of carbon atoms, tightly bound in a hexagonal honeycomb lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Up to now, much of what is known about the mechanical and electronic properties of graphene is based on models on the atomistic scale, such as the MD simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' However, the use of MD directly to treat the deformation and failure of materials at the mesoscale is still largely beyond reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Hence, we aim to learn a peridynamic model by upscaling from MD to the continuum scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the present study, an MD model was created using the Tersoff interatomic potential [73], a widely used potential in the MD community for graphene [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Unstressed graphene nominally has an interatomic spacing of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='46˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Without otherwise stated, in this study values of the coarse-grained data trios are evaluated on a square lattice of nodes with spacing h=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='0˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The only exception is in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4, where we also consider an additional, finer data set generated with spacing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='17˚A, to assess the generalization properties of the proposed learning approach to different grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Without the loss of generality, in this work we consider MD simulations on temperature 0K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In all cases, external loading is applied to the atoms in the MD grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the non-fracture data sets, the magnitude of the loading is chosen so that the bond strains are no larger than 18 Figure 4: Contours of exemplar U1 displacement in typical MD simulations at zero temperature for the four data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' From left to right: (a) Non-fracture training data set MDNon-Frac train , for the kernel learning step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (b) Non-fracture validation data set MDNon-Frac val for the kernel learning step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (c) Fracture training data set MDFrac train, for the damage criterion learning step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (d) Fracture test data set MDFrac test , to study the efficacy and generalizability of the overall workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 2%, which is less than the strains at which nonlinear effects appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In all MD experiments, the atoms are initialized with positions on a hexagonal lattice in the x1-x2 plane with an interatomic spacing of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='46˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The mass of each atom is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='0E-26kg, or 12amu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For purposes of computing stresses, the thickness of the lattice is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='35˚A, which is the approximate distance between layers in multilayer graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' On quasi-static data sets, we smooth the MD simulation results in time as described in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the dynamic data sets, the MD time step size is set as 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='95E-14s, or 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Data Generation and Learning results In this section, we apply the learning algorithm described in Section 4, to extract a coarse-grained model from MD simulations of a graphene sheet at 0K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the purpose of training, validation and test, we generate the following four groups of MD simulations, with exemplar images showing contours of U1, the component of atomic displacement in the x1 direction, provided in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 1) Non-fracture training data set (MDNon-Frac train , with 70 quasi-static MD simulation samples): The MD domain is a 100˚A×100˚A square, and, for k1, k2 ∈ {0, π 50, 2π 50 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' , 5π 50 }, the prescribed external loadings are given by b(x1, x2) = (C1 k1,k2 cos(k1x1) cos(k2x2), 0), or b(x1, x2) = (0, C2 k1,k2 cos(k1x1) cos(k2x2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1) The constant C1 k1,k2 and C2 k1,k2 are adjusted so that the bond strains are no larger than 2%, so the deformation remains in the linear range of material response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' A periodic boundary condition is employed for all samples in this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 2) Non-fracture validation data set (MDNon-Frac val , with 10 quasi-static MD simulation samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the same MD grid and coarse-grained nodes as in the non-fracture training data set, the applied loads in the validation 19 (a) Non-Fracture Training (b) Non-Fracture Validation (c) Fracture Training (d) Fracture Test Data Set MIDNon-Frac Data Set MDNon-Frac rain valk C1 k C2 k pk Rk 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 0 25 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 25 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 0 15 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 15 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 0 10 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 1 25 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 1 25 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 1 15 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 1 15 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='001 0 1 10 Table 1: Parameters used in the MD loading in the 10 validation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' data set are as follows: b(x1, x2) = (C1 k, C2 k) 1 � j=−1 (−1)j cos �π 2 min � 1, rj,k Rk �� (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2) where rj,k = � (x1 − (1 − pk)Lj)2 + (x2 − pkLj)2 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='3) where L=50 and the values of the parameters C1 k, C2 k, pk and Rk are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In each case, loads are applied to the atoms within three disks of radius Rk with centers at the center of the grid and at the left and right boundaries (if pk = 0) or the upper and lower boundaries (if pk = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The loads in all cases are self-equilibrated and periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 3) Fracture training data set (MDFrac train, with 1 dynamic MD simulation sample): The domain of the graphene sheet is set as a square: [−50˚A, 50˚A] × [−50˚A, 50˚A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The MD grid initially contains a slit (edge crack) of length 25˚A oriented vertically extending from the lower surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The vertical edges of the MD grid have prescribed velocities in the x1 direction that tend to open the crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To help maintain stable crack growth, the prescribed velocities decrease linearly with x2, thus tending to limit the crack growth velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' A schematic of the crack pattern at the 40-th time step in the MD simulation can be find in Figure 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the purpose of validation on different grid resolutions, the density, displacement and external loading are computed at two sets of coarse-grained nodes, which are spaced 5˚A or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='17˚A, apart on a square lattice, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 4) Fracture test data set (MDFrac test , with 1 dynamic MD simulation sample): To demonstrate that the learned material model applies to different loading scenarios and crack patterns, one additional test case is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here, the MD region and the pre-existing slit are the same as in the fracture training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' However, instead of hard loading along the vertical edges, a non-zero body force is applied to the atoms in the MD grid as: b1 = b0 � e−t/tr(1 − e−t/tr) � sin �πx1 L � e−(1/2+x2/L) where L = 100˚A is the edge length of the sample, tr is a constant pulse duration time, and b0 is a positive 20 Figure 5: Learning results on a single layer graphene sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Left: The optimal influence function K for the LPS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Right: The optimal damage criterion s∗ 0 is obtained at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (The origin is at the center of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=') This loading exerts a pulse that tends to open the crack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The resulting crack pattern, which includes branching, is substantially different from that which occurs in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' A view of the crack pattern at the 40-th time step in the MD simulation can be found in Figure 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As metrics of accuracy on tests, we compare the prediction from the learnt peridynamics model with the ground-truth data from coarse-grained MD measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Solution contours are provided as a qualitative validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' With the purpose of providing a quantitative comparison, we also calculate the averaged (in time) mean square errors (MSEs) of the displacement field and the damage field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To provide a fair comparison between different sets, all these qualitative accuracy metrics are normalized with respect to the ground-truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the kernel learning step, we have followed a similar procedure as in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The learned influence function K is plotted in the left plot of Figure 5, and the optimal material parameter are obtained as λ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4796(TPA), µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='7978(TPA), with the Poisson’s ratio ν = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4297, and the horizon size δ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='0˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, for the damage criterion learning step, since the crack initiates at the 5-th time steps, we use the fracture training data set from the 5-th time step till the 20-th time step to learn the damage criterion s0, then solve for the optimal s0 by minimizing the loss � Res(T Frac) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Note that when calculating the loss function, we apply Dirichlet-type boundary conditions on a layer of particles near the boundary of our square domain, and hence only the particles in [−50˚A+2δ, 50˚A−2δ]×[−50˚A+2δ, 50−2δ] are considered in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This setting differs from the settings in non-fracture data sets, where periodic boundary conditions are considered for all samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This is due to the fact that it is generally non-realistic to prescribe the periodic boundary condition in the problem with a crack, since the crack itself does not satisfy the periodic condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 21 1e-4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='8 K/m(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2 6 8 10 12 14 16 18 20 Bond Length (A)1e-4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='40 Loss 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='20 So so = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='11Figure 6: Comparison of the prediction and the ground truth measurement from the MD data set at time steps 20, 30, and 40, on the fracture training data set where the graphene sheet is subject to zero body force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here, the first 20 steps were used for training, then we use the learnt model to predict for the next 20 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (a) Comparison on the displacement field, where the color of the particles represents the horizontal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (b) Comparison on the damage field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This fact also highlights the generalizability of the proposed approach: our homogenized surrogate model can handle data sets with different domains, loadings, and also boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' A demonstration of the loss function for different values of s0 is provided in the right plot of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The optimal damage criterion is obtained as s∗ 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='11, which is consistent with the results s0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='145 inferred directly from MD data set in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Extrapolation to Longer Time Simulations Next, we validate the learnt model, by using it in a longer term simulation on the fracture training data set, to predict the material deformation and crack propagation upto the 40-th step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Note that we have used the data upto the 20-th time step for the purpose of training, and therefore this test can be seen as an investigation on the long-term extrapolation capability of our coarse-grained surrogate model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To solve for the displacement field from LPS model, at the n-th time step, we first assume there is no broken bond and solve for the displacement field ˆun+1, then we update the bond-stretch for each connecting bond, and we keep solving for the displacement until there is no new bond breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, we define the damage profile at each particle xi at time step n as φ(xi)n = 1 − � xj∈Bδ(xi) γn(xi, xj) � xj∈Bδ(xi) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4) 22 (a) Displacement Fiel (b) Damage Field Data Prediction Data Prediction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='40 40 20 20 Step 20 o 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='35 40 20 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='30 50 25 25 50 50 25 0 25 50 20 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='25 20 20 Step 30 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='20 20 20 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='15 50 25 25 50 5025 0 25 50 20 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='10 20 20 Step 40 0 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='05 20 20 40 40 50-250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='00 25 50 50 0 50Figure 7: Comparison of the prediction and the ground truth measurement from the MD data set at time steps 20, 30, and 40, on the fracture test data set where the graphene sheet is subject to unseen and nonzero body force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (a) Comparison on the displacement field, where the color of the particles represents the horizontal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (b) Comparison on the damage field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Figure 6 shows the comparison of displacement and damage fields at time steps 20, 30, and 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' It is observed that the prediction not only matches the data within the training set (step 20) but also exhibits a good agreement at steps 30 and 40, which are not included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This result suggests that our learned damage criterion s0 is applicable to longer term simulations out of the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the first 40 steps, we have obtained 27% relative error for the prediction of displacement field and 9% relative error for damage field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Generalization to Different Body Forces and Crack Patterns In this Section, we use the learned LPS surrogate to model the same graphene sheet subject to a different body force load as described in the fracture test data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Differs from the settings in the training data set, in this data set the graphene sheet is subject to nonzero body load, with its crack pattern at the 40-th time step illustrated Figure 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Compared with the crack pattern in the training data set (see Figure 4(c)), the crack path in this test data set is less symmetric and bifurcates at the middle of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Hence, with this example we not only investigate the extrapolation capability of the learnt model by making a longer time (40 steps) predictions, but also aim to verify its generalizability, since both the loading scenario and crack pattern from this test data set are not covered in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' All these factors make the validation more challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In Figure 7 we show the prediction of displacement and damage fields from the learnt LPS model upto time step 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Visually good agreements are observed between the coarse-grained data and 23 a) Displacement Field (b)Damage Field Data Prediction Data Prediction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='40 40 40 20 20 09 Step 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='35 20 20 40 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='30 50 0 25 50 50 25 25 50 20 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='25 20 20 Step 30 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='20 20 20 40 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='15 50 0 50 50 50 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='10 20 20 Step 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='05 20 60 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='00 50 0 50 50 50Figure 8: Comparison of the prediction and the ground truth measurement from the MD data set at time steps 20, 30, and 40, on the fracture training data set with fine grids where the graphene sheet is subject to zero body force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Here, we used coarser grids data in the first 20 steps for training, then we use the learnt model to predict for the next 20 steps on a finer resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (a) Comparison on the displacement field, where the color of the particles represents the horizontal displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' (b) Comparison on the damage field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' LPS predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This example has qualitatively validated that the learned material damage model can be directly applied to problems with different body force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the first 40 steps, we have obtained 58% relative error for the prediction of displacement field and 15% relative error for damage field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Generalization to Different Resolutions Last but not least, we study the resolution generalizability of our learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Specifically, we use the same MD data as the training data, but evaluate the density, displacement and force loading on a coarse-grained grid with smaller grid size h = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='17˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Since all training data sets are with a fixed grid size h = 5˚A, with this study we aim to investigate if the learnt surrogate model allows the grid size to be rescaled, providing a multiscale capability and allowing for flexible solver resolution and reductions in computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As suggested by [47], we scale the horizon size δ proportionally with the grid size h to provide a fixed horizon/grid size ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In particular, we take δ = 4h = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='68˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, the optimal damage criterion is also scaled correspondingly to guarantee a consistent critical release rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As proved in [71], the damage criterion and horizon size should satisfy the relation s0 ∝ 1 √ δ in the LPS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Thus we use s0 := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='11 � 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='17 for our fine scale simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' In Figure 8 we show the displacement and and damage field prediction results upto time step 40, demonstrating a qualitative agreement between the coarse-grained MD data and our numerical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' On the displacement field, we have obtained 19% relative error in average for the first 40 steps, 24 (a) Displacement Field (b) Damage Field Data Prediction Data Prediction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='5 40 40- 20 20 Step 20 40 20 20 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='4 40 40 5025 25 50 5025 0 25 20 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='3 20 20 Step 30 o- 20 20 40 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2 50-25 25 50 50-25 25 20 1 40 40 20 20 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='1 Step 40 0 40 20 20 40 40 50-25 0 2550 50 0 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='0which is even smaller than the prediction error without resolution alternation on the same data set (27% as shown in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' For the damage field, one can see that the crack pattern predicted by our surrogate model grows faster than the crack from MD data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Therefore, a larger prediction error, 30% average for damage field in the first 40 steps, is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This example suggests that the surrogate model can provide qualitatively consistent displacement predictions on different resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' On the other hand, the prediction on damage field is sub-optimal, possibly due to the fact that the material crack originates from microscale phenomena, and hence is more sensitive to the prediction scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' To improve the prediction accuracy on the damage field across different resolutions, practitioners might consider performing the damage criterion learning step on the new resolution, to provide a correction for the damage criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Conclusions In this paper, we demonstrate a data-driven workflow to extract a coarse-grained surrogate model from MD data with fracture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Firstly, to handle the discontinuities induced by material fracture in the MD displacement measurements, a smoothness indicator function is introduced, to automatically choose the locally smoothest stencil from the neighborhood of each coarse grained grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As such, the coarse-graining measurements are built based on this adaptive stencil, to automatically handle the discontinuities in MD displacement data set without overly smoothing the crack pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' It is shown that this novel adaptive procedure significantly improves the capability of capturing the location of crack interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Then, based on the coarse-grained data set we proposed to extract a peridynamics surrogate, which is a continuum mechanics model that allows a natural treatment of discontinuities by replacing spatial derivatives of stress tensors with integrals of force density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' By learning the kernel function of the integral and the damage criterion with a two-step optimization approach, we obtain a linear peridynamic solid model which provides good agreement with nanoscale test data while being capable to provide further material deformation and fracture predictions under unseen domain settings, loading scenarios, and even different grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' These features greatly reducing the cost of the calculation in comparison with MD, especially when used together with different discretization resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Although the present work focuses on relatively small deformations and a linear peridynamics model, the results suggest that this method may impact a broader range of materials and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' As another natural follow-up work, one may further combine the nonlocal model with the approximation power of neural networks, to obtain a nonlinear peridynamics model in the form of integral neural operators [74–77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Acknowledgements H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' You and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Yu would like to acknowledge support by the National Science Foundation under award DMS-1753031 and the AFOSR grant FA9550-22-1-0197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Portions of this research were conducted on Lehigh University’s Research Computing infrastructure partially supported by NSF Award 2019035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' 25 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Silling and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' D’Elia would like to acknowledge the support of the Sandia National Laboratories (SNL) Laboratory-directed Research and Development program and by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Department of Energy (DOE), Office of Advanced Scientific Computing Research (ASCR) under the Collaboratory on Mathematics and Physics-Informed Learning Machines for Multiscale and Multiphysics Problems (PhILMs) project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' This article has been authored by an employee of National Technology and Engineering Solutions of Sandia, LLC under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' DE-NA0003525 with the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' Department of Energy (DOE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The employee owns all right, title and interest in and to the article and is solely responsible for its contents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this article or allow others to do so, for United States Government purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctE3T4oBgHgl3EQfeAqc/content/2301.04540v1.pdf'} +page_content=' The DOE 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b/f9E0T4oBgHgl3EQfpAES/content/tmp_files/2301.02532v1.pdf.txt @@ -0,0 +1,764 @@ +Better Balance in Informatics: +An Honest Discussion with Students⋆ +Elisavet Kozyri1, Mariel Evelyn Markussen Ellingsen2, Ragnhild Abel Grape1, +and Letizia Jaccheri3 +1 UiT The Arctic University of Norway +2 Woid AS +3 Norwegian University of Science and Technology +Abstract. In recent years, there has been considerable effort to pro- +mote gender balance in the academic environment of Computer Science +(CS). However, there is still a gender gap at all CS academic levels: from +students, to PhD candidates, to faculty members. This general trend is +followed by the Department of Computer Science at UiT The Arctic Uni- +versity of Norway. To combat this trend within the CS environment at +UiT, we embarked on structured discussions with students of our depart- +ment. After analyzing the data collected from these discussions, we were +able to identify action items that could mitigate the existing gender gap +at our department. In particular, these discussions elucidated ways to +achieve (i) a balanced flow of students into CS undergraduate program, +(ii) a balanced CS study environment, and (iii) a balanced flow of grad- +uates into higher levels of the CS academia (e.g., PhD program). This +paper presents the results of the discussions and the subsequent recom- +mendations that we made to the administration of the department. We +also provide a road-map that other institutions could follow to organize +similar events as part of their gender-balance action plan. +Keywords: Gender balance · computer science · diversity · inclusion · +student study +1 +Introduction +Innovations in Computer Science shape the lives of everyone in our society. To +create innovative solutions tailored to everyone, it is important that all groups of +society are represented in the creation of these solutions. However, this is still not +the case in the field of Computer Science (CS). Having an awareness of the lack +of representation and the different barriers people face in CS are fundamental +in helping the field target those challenges and becoming more equitable and +inclusive [8]. +Statistics from Europe show that women are still highly underrepresented in +CS. According to Eurostat [4], the percentage of female specialists in Information +and Communications Technology has evolved from 17% in 2012 to 19,1% in 2021. +⋆ Mariel’s contribution was made while she was a Master’s student at UiT. +arXiv:2301.02532v1 [cs.CY] 6 Jan 2023 + +2 +E. Kozyri et al. +At university level in STEM, the percentage of female Bachelor, Master, and PhD +students is 20%, while the percentage of female professors is 15%. +Specifically for the Department of Computer Science at UiT The Arctic Uni- +versity of Norway, only 13% of students, 14% of PhD candidates and 21% of +faculty members are female. +Better Balance in Informatics (BBI), a program led by the CS department +at UiT and funded by the Research Council of Norway, aims to rectify this +imbalance and create a more diverse learning environment for Computer Science. +BBI is connected and builds upon an ecosystem of national and international +projects which address gender balance in CS acting on different levels: school +([13], [7]), university ([2], [6] [16]), industry ([17], [3], [5]), and the interplay of +these levels ([1]). +BBI aimed to identify some of the reasons that led to the current gender +dynamics in our CS department, and then propose measurements that could +address those reasons. Hearing directly from the CS students (Bachelor, Master) +seemed to be a sensible way for us to identify those reasons. So, BBI organized +structured discussion sessions, where we invited CS students (Bachelor, Master) +to share their thoughts about: +1. the reasons they picked CS for their studies, +2. their current experience with the CS studies, +3. their intention to pursue an academic career in CS, and +4. ways to make the CS community more diverse and inclusive. +The answers of the students illuminated points of intervention, which could +lead to a balanced flow of students into CS undergraduate program, a study +environment that embraces diversity, and a balanced flow of students into higher +levels of the CS academia. +This paper presents the methodology (§2) we employed to organize the dis- +cussion sessions, to collect responses, and to report the results. We then present +the specific questions we asked the students and the analysis of their answers +(§3). Finally, we list the recommendations (§4) submitted to the CS department +for achieving a gender-balanced environment, we discuss related work (§5), and +we conclude (§6) with reflections about the discussion sessions. +2 +Methodology +The end goal of the discussion sessions was to identify points of interventions +that could increase the gender balance among the incoming CS students, the +current CS students, and the CS graduates that are interested in entering the +CS academia. To identify those points, we were aiming for a high number of +participants in the discussion sessions: the more participants, the greater the +plurality of experiences, and thus, the higher the chances to find opportunities +for improvement. Deciding which questions to ask was crucial to ensure that +experiences from different aspects of the CS studies are captured and then an- +alyzed. But, we also had to create a trusting discussion environment for the + +Better Balance in Informatics: An Honest Discussion with Students +3 +students to honestly share those experiences with us. This section describes the +methodology we followed to prepare and organize the discussion sessions such +that all these targets are met. +2.1 +Outreach +Attracting the attention of students and persuading them to participate in the +discussion sessions was not trivial. Unless there is an immediate academic or +employment gain, motivating students to devote part of their busy schedules to a +university-led event is indeed challenging. Our strategy to address this challenge +comprised the following steps: +Hiring students as project assistants. We hired two talented and enthusiastic +female students as assistants for the project. They were our bridge to the student +community in our CS department. And this bridge was functioning in both ways. +Their thoughts, insights, and experience informed all aspects of the BBI project, +including the questions we asked during the discussions. At the same time, they +knew best how to reach their fellow-students and promote the agenda of BBI +(e.g., what advertisement means to employ and what to say in these means). +Website. The BBI website (https://uit.no/project/bbi) is the main official +space where the mission of BBI is communicated to the world. So, we redesigned +this website to include a clear and short motivation for the BBI mission, and +describe the upcoming BBI events, in particular the discussion sessions. +Advertisement. To reach a wider set of students and persuade them to partici- +pate in the BBI discussion sessions, we employed a variety of means. We posted +advertisements on the monitors of the Department, the social networks of the +Department, on Canvas, the UiT calendar, and the local student organization +forum, which is a Discord server that is maintained by the student organization +TD. The student assistants also gave 5-minutes talk about BBI and the dis- +cussion sessions to courses with high enrollment, they created and distributed +flyers, and they organized a stand with coffee and cookies, where students could +casually socialize and talk about BBI. In terms of registrations to the BBI dis- +cussion sessions, Canvas and TD seemed to have been the most effective, since +we observed a high correlation between the time a post about the BBI event was +created and the time students were registered. +Open to everyone. The invitation to participate in the BBI discussion sessions +was open to all students of the CS department, independently of their gender +(female, male, non-binary). This is because the gender imbalance is a problem +that owes to concern everyone—not only a part of the community. And because +any further actions that the Department will take to address the problem might +effect every student, there needs to be a wider understanding that these actions +are worthwhile. Leaving specific groups of students outside the discussion, would +not have increased this understanding. + +4 +E. Kozyri et al. +2.2 +Discussion Sessions +The discussion sessions were held at ´Ardna, UiT. ´Ardna is an iconic place in the +university, ideal for secluded discussions. Its central fire place and the surround- +ing wooden benches invites people to open up and discuss honestly. +In the BBI discussion sessions participated around 20 participants.4 Com- +paring to events organized in the past by BBI, this number of participants was +a very welcoming surprise. From those participants, around 50% were female or +non-binary students, and around 50% were male students. The vast majority +of the students participated physically, but there were some that participated +remotely. There were around three participants per discussion session. Each dis- +cussion session was moderated by two BBI members: one member was asking +the questions, and the other member was typing the answers (we used no video +or sound recording). At least one of the moderators was always one of the BBI +student assistants; having participants talking to their peers led to frank discus- +sions. To preserve anonymity, each participant was assigned a number, so the +recorded answers were associated with these numbers—not with the identity of +the student. For the discussion sessions, we gave the option to the student to +select either Norwegian or English as the speaking language. All participants, +apart from those that participated remotely, were offered a full meal and a free +cinema ticket. +2.3 +Selection of Questions +The selection of questions was inspired by questionnaires developed by other +gender-balance projects, such as EUGAIN [1], Prestige in UiT [2], and Balanse- +Hub [6]. However, the questions were tailored for the needs of the CS department +in UiT. And, in particular, the questions were intended to cover a wide range of +students’ experience: from the point they considered applying for CS, to their +current studies and their future plans. +2.4 +Reporting of Results +For most of the questions asked during the BBI discussion sessions, we compiled +the answers into graphs. Each graph depicts how answers are correlated with dif- +ferent genders. This information help us guide our selection of action items for +improving the gender balance. We protect the anonymity of the participants, so +we do not give specific numbers at graphs. Also, we do not have a separate cate- +gory for non-binary participants, because the number of non-binary participants +was not high enough to protect their anonymity. Instead, we group female and +non-binary participants together, and we explore the dynamics between majority +(males) and minorities (female, non-binary). +4 We do not give specific numbers to preserve the anonymity of the participants. + +Better Balance in Informatics: An Honest Discussion with Students +5 +Fig. 1. Reasons for choosing to study CS. Each column corresponds to a different rea- +son. The height of a column represents the number of participants that submitted the +corresponding reason. Dark blue represents female or non-binary participants (F/NB); +yellow represents male participants (M). +3 +Results +This section presents the questions we asked the participants and their answers +concerning: +1. the reasons they picked CS for their studies, +2. their current experience with the CS studies, +3. their intention to pursue an academic career in CS, and +4. ways to make the CS community more diverse and inclusive. +Correlating their answers with their gender, we identified action items that could +lead to a balanced flow of students into CS undergraduate program, a study +environment that embraces diversity, and a balanced flow of students into higher +levels of the CS academia. +3.1 +Intention to Study CS +To increase the balance in Computer Science, one first needs to increase the +balance in the new-coming students. So, when advertising CS to younger stu- +dents, one could also include aspects that attract minorities. We tried to identify +those aspects by asking the participants the reason they decided to study CS +in the first place. Figure 1 shows the different answers we received. The higher +the column, the more students gave that answer. The graph also shows how the +answers are split between the minority (F/NB) and majority (M). There is a +correlation between the gender and the reason for selecting CS studies. + +I started to study Cs because.. +Got interested Got interested Got interested CS combines +CS education CS has flexibleCS was 2nd +Like Tromso +through the +through +through +practice, math, creates more study program choice of study +girls-and-tech programming +computer +and problem +job +or interesting +day in UiT +courses at high +games +solving +opportunities +course +school +descriptions +F/NBM6 +E. Kozyri et al. +Action Items. Observing Figure 1, we can identify the reasons the minority +chose CS studies: the problem solving aspect of CS, the flexibility of the CS +studies, the job opportunities that CS graduates enjoy. To increase the diversity +of incoming students, we can then emphasize those reasons when advertising CS. +Also, as a possible means of advertisement Figure 1 indicates the UiT girls-and- +tech day. +Fig. 2. Ways for becoming familiar with CS. +Apart from the UiT girls-and-tech day, we wanted to understand what would +be other effective means of advertisement for attracting minorities to CS studies. +So, we asked the participants where did they hear about CS. Figure 2 plots the +answers, which are again correlated with the gender. +Action Items. Figure 2 indicates that one could use the highschool and the uni- +versity’s study catalog to better promote CS to minorities. Interestingly, friends +and relatives have a high impact on the decision of minorities to study CS. So, +one can make tech employees ambassadors of CS to young female and non-binary +members of their families. +In general, the vast majority of the participants, independently of their gen- +der, would have liked CS to have been introduced differently to them, as Figure +3 indicates. Participants indicated that CS should be introduced as something +that everyone can do and something that offers a plausible path to a regular job. +Action Item. When advertising to high-school students, we need to break +stereotypes on who can study CS. + +Where did you hear about Cs? +Highschool, Study catalog +Friends or +Event hosted +《Always》 +Gaming +Internet +career +relatives +by UiT +known +search +guidance +■F/NBMBetter Balance in Informatics: An Honest Discussion with Students +7 +Fig. 3. Independent of their gender, 81% of the participants said that they would have +liked to be introduced to CS in a different way. +3.2 +Your Experience in the Current CS Environment +At the very least, we want to sustain the diversity among the incoming students, +while these students progress to more senior years; we aim to decrease the number +of drop-outs, with an emphasis on the minority group. To achieve this, we need +to assess the student’s experience within the CS department and identify aspects +that can be improved to accommodate gender diversity. +We start by asking the participants whether the CS studies met their initial +expectations. Their answers are depicted in Figure 4. Almost all minority par- +ticipants gave a negative answer: they found their studies either more difficult +or more practical than expected. In particular, they found the learning curve of +programming to be particularly steep, something that might drive some of the +minority students to drop-out. We believe addressing this concern is important +for maintaining a diverse environment in the department. +Action Item. We propose the adoption of a smoother introduction to pro- +gramming, which will be appropriate for students with no prior programming +experience. +Returning to Figure 4, one also notices that, for most of the male participants, +their experience in the CS studies met or even exceeded their initial expectations. +So, this question highlights a striking difference between the minorities and the +male students in terms of how they view their studies. This difference might be +a cause of the current gender imbalance in the department. + +DOYOUWISHTHAT CS +WOULD'VE BEENINTRODUCED +TOYOUIN A DIFFERENT WAY +OREARLIER? +No +6% +Maybe +13% +Yes +81%8 +E. Kozyri et al. +Fig. 4. The first column corresponds to the answer that the CS studies met or exceeded +the expectations of the participant. The remaining four columns correspond to the +answer that the CS studies did not quite meet the expectations of the participant, +and they also represent different reasons why. The height of a column represents the +number of participants that submitted the corresponding answer. Dark blue represents +female or non-binary participants (F/NB); yellow represents male participants (M). +All the participants agreed, though, that the social environment built around +their studies exceeded their initial expectations. This is a great achievement of +the department that needs to be preserved for the future, too. +Participants were then explicitly asked whether they have thought to drop- +out of their study program. As Figure 5 shows, most of the participants answered +affirmatively. Notice that this is a concern across all genders, opposing the mis- +conception that the minorities are more likely to have such thoughts. Notice also +that even though most of the male students thought to drop-out of the program, +they still had an overall positive assessment of their study experience, according +to Figure 4. +As reasons for thinking to drop-out, the participants cited the difficulty of +some courses, the time-consuming assignments with overlapping deadlines, the +demanding task of writing thesis (a task for which they did not feel prepared), +and the complications that the COVID-19 pandemic brought. For a student to be +thinking to drop-out, it means that the student’s self esteem might be low at that +point. Figure 6 validates this claim, showing that most of the participants felt +”useless” or ”not-deserving” being in the CS program. Again, the answers do not +seem to be correlated with the gender. However there is an underlying difference: +many of the males had this feeling once, related to a specific assignment or for +short period of time, whereas the minority students had this feeling for a long +period of time (i.e, months or even years). + +Do your CS studies meet your initial expectations? if not, why? +Yes or exceeded: +Not quite: more +Not quite: more +Not quite: more +Not quite: lectures +more +theoretical than +difficult than +practical than +are not designed +coding/practical than +expected +expected +expected +carefully enough +expected +(programming) +■F/NB ■MBetter Balance in Informatics: An Honest Discussion with Students +9 +Fig. 5. The majority of the participants replied that they have thought of dropping +out of their CS study program. +Fig. 6. The majority of the participants replied that they have have felt “useless” or +“not-deserving to be here” during their CS study program. +When asked about the reasons they instead decided to stay in the program +and not drop out, the participants mentioned: +– the robust social network that they have built within and outside UiT, where +they could talk about their struggles, +– the senior student advisor Jan Fuglesteg, + +Have you ever thought of dropping out of your study program? +Yes +No +■F/NB +MHave you ever felt “"useless" or “"not-deserving being here" +(i.e., imposter syndrome) during your studies? +Yes +No +IF/NB +■M10 +E. Kozyri et al. +– their self-determination and discipline, +– taking time to relax. +Action Items. Given the stimulating power that the social groups exercised +on the students, we should further support actions and groups that promote +social networking in the department. Also, we will organize events where senior +students can offer tips and tricks from their experiences to the junior students, +where the main message will be “if we have made it, so can you”. +Fig. 7. More than half of the participants said that they have witnessed or heard of +sexual harassment incidents within our CS community. +Concentrating on minority students, one of the reasons they might feel un- +comfortable in an environment (and ultimately drop-out of the program) is when +they have experienced sexual harassment. So, we asked the participants whether +they have ever witnessed or heard of sexual harassment incidents within the CS +community. Figure 7 depicts their answers. More than half of the participants +answered positively. +Action Item. The “Yes” column in Figure 7 should not exist. So, we will pro- +pose to the department to devise videos and examples of unacceptable behavior, +so the student can recognize and dissociate from these phenomena. +The experience that a student gets from their CS program is a collection +of many educational aspects, such as lectures, colloquiums, and assignments. +For the CS program to be inclusive and diverse, all these aspects should pro- +mote inclusiveness and diversity. We asked the participants if they feel that the +educational aspects below promote only a particular way of thinking. + +Have you ever witnessed or heard of incidents of +sexual harassment within our cs community? +Yes +No +F/NB +MBetter Balance in Informatics: An Honest Discussion with Students +11 +Fig. 8. Almost all of the participants do not wish to have an academic career in CS. +– Lectures: Participants mentioned that examples employed in some lectures +are more appealing to male students. These examples usually involve games +or cars. +– Colloquiums:5 Participants, from all genders, mentioned that a colloquium +can quickly get a “boys-club” atmosphere if the TA is not attentive enough. +The participants also express the wish for more female TAs. +– Assignments: Some assignment are very focused on gaming or promote com- +petitiveness, which might be uncomfortable for some students. +Action Items. We will advise the faculty members to ensure that lectures and +assignments accommodate a variety of interests. The department should organize +a seminar for TAs in which they become sensitized on not allowing colloquiums +to become “boys-clubs” and accommodating different interests and needs. We +also need to brainstorm on how to hire more female TAs. +3.3 +Intention to Enter Higher Levels in CS Academia +We are striving to achieve balance at all levels of the CS academia, from students +to professors. At this section, we focus our attention to higher levels in CS +academia (from PhD candidates and above), and we want to understand the +intention of the current students to enter that level. According to Figure 8, the +vast majority, and in particular all female and non-binary participants, do not +wish to have an academic career in CS. Here are some reasons why: +5 A colloquium is associated with a course and it is often led by a TA, who answers +student’s questions about the course material and assignments. + +Do you wish to have an academic career in Cs? +Yes +No +F/NB +■M12 +E. Kozyri et al. +– The academic career seems difficult or exhausting. +– No interest in research or writing or teaching. +– Preference for work-life balance offered by the industry (pay, social, use tools, +practical). +– The description of an academic career is not clearly communicated. +Fig. 9. Most of the participants are either unsure about or do not know what the CS +PhD program. +In fact, as indicated by Figure 9, there is uncertainty between the students, +and in particular the minority students (i.e., female and non-binary), about +what a PhD program is—the first stepping stone towards building an academic +career. Given this uncertainty, it is expected that many students will not apply +to a PhD program, something that is affirmed by Figure 10. Notice, though, that +comparing Figures 8 and 10, the participants are less negative towards pursuing +a PhD program than ultimately following an academic career. This is because, +after obtaining a PhD degree, there is always the possibility to follow a career in +the industry. And some of the participants that replied “no” to the prospective +of applying to a PhD program now, they contemplate the possibility of applying +after working in the industry. +And if a participant does not want to apply to a PhD program, what are +their immediate plans after graduation? Figure 11 answers this question. Notice +that participants from the minority group explore a wider variety of options. +Also, for the participants that are not considering to apply to the CS PhD +program, Figure 12 gives the main reasons behind this disposition. Figure 12 + +Do you know what a PhD program is? +Yes +Maybe +No +■F/NB■MBetter Balance in Informatics: An Honest Discussion with Students +13 +Fig. 10. Most of the participants would not apply to a PhD program. +Fig. 11. Columns correspond to different options the participants consider to follow +after graduation. +informs how we can intervene and address some of these reasons, possibly leading +to more students applying to our PhD program. +Action Items. According to Figure 12, some participants said that the PhD +program “sounds too heavy”, and they described PhD students as being “alone” +and “depressed”. While this description might portray one aspect of the PhD + +Would you apply to a PhD program? +Yes +Maybe +No +■F/NB +■MWhat do you want to do when you graduate? +Work as +Work as +Work in +Work offshore Do a start-up Apply for PhD Do not know +developer +consultant +security, +intelligence, or +military. +F/NB +■M14 +E. Kozyri et al. +Fig. 12. Columns correspond to different reasons why the participants are not consid- +ering to apply to a PhD program. +experience, it is definitely not the entire truth. So, we are going to hold events +that clearly describe the characteristics of a PhD program, emphasizing the +positive aspects of being a PhD student. These events will also address the +uncertainty that was surfaced in Figure 9 about what is a PhD program. +The late deadline for applying to a PhD, which is not synchronized with the +job-search period of senior students, is another reason why current students do +not select a PhD program. To remedy this, we will advise the faculty members +of the CS department to consider moving the PhD application earlier in in the +academic year (i.e., fall semester). +Finally, given that many participants said that they might consider applying +to a PhD program in the future (i.e., after acquiring some experience in the +industry), we advocate to advertise new PhD positions explicitly to CS alumni. +For some of these alumni, these positions might seem attractive. +3.4 +The Gender Gap and Possible Measurements to Close it. +In previous sections, we attempted to understand the reasons why a gender +imbalance exists in the CS department, and concluded with action items that +could address those reasons. In this section, we explicitly discuss gender balance +with the students and record their opinions and proposals on the subject. +To begin with, the vast majority of the participants said that the current +gender-imbalance in the department is a problem. They actually see that gender- +balance has advantages: promotes innovation, enables plurality of perspectives, +and leads to a better study and work environment. Many of the participants said + +Why would you not apply to a PhD program? +Want a job in +Sounds too +Maybe in the +Do not want to +Happy with a + PhD application +industry +heavy +future +teach +bachelor +is too late +■F/NB■MBetter Balance in Informatics: An Honest Discussion with Students +15 +Fig. 13. Each row corresponds to a different measurement for improving gender balance +in CS. The length of each row represents the number of participants that agree with the +corresponding measurement. Dark blue represents female or non-binary participants +(F/NB); yellow represents male participants (M). +that there are no disadvantages with gender balance, although some expressed +the concern that gender-balance efforts, such as gender quotas, might “lead to +people being hired for the wrong reasons”. +The participants were then presented with different measurements that could +be employed to improve the gender balance within the department and asked to +say whether they are in favor or not of each presented measurement. Figure 13 +depicts their answers. Notice that measurements that blatantly favor minorities +in the education or in the career were among the least popular (for both minority +and male participants). We aim to focus on the most popular measurements. +Participants also proposed two additional measurements that did not appear +in our pre-selected list: +– Share success stories from minorities. +– Have a few preparatory weeks for programming before starting the first +semester in CS. +4 +BBI recommendations for the near future +Throughout this report we presented a variety of action items for improving +gender balance in our CS department. We have motivated these action items +using the findings from the discussions with the participants. We now summarize +those actions that BBI recommends for the near future. These actions aim to +achieve a balanced flow of students into CS studies, a balanced environment +within the CS department, and a balanced flow towards CS academia. Figure 14 + +Which of the following measurements are you in favor of? +Make the subject of CS courses easier +Use gender points +Educational events for minorities only +Student unions for minorities only +Career events for minorities only (e.g., visiting industry) +Focus on keeping minorities already in the program +Introduce more theoretical courses (e.g., complexity,. +Create more interdisciplinary programs (e.g., CS+Biology,.. +More advertisements aimed towards minorities +Social events for minorities only +Introduce more creative courses (e.g., HCl, computer graphics) +Deal with the issue at a younger age +Have more role models from minorities +■F/NB■M16 +E. Kozyri et al. +School +CS Studies +CS Academia +Balanced +Flow +Balanced +Environment +Balanced +Flow +Fig. 14. We propose action items for (i) a gender-balanced flow of students from the +school to CS studies, (ii) a gender-balanced student environment in our department, +and (iii) a gender-balanced flow of graduates from the CS studies to the CS Phd +program, and eventually CS academia. +gives a schematic representation of these three categories of actions, which are +listed below. +Action items for a balanced flow of students into CS. +– Student-ambassadors of all genders to present CS to high school and +middle high school students. +- Highlight problem solving aspect of CS, flexible and interesting program, +job opportunities. CS is something that everyone can do. +Action items for a balanced CS study environment. +– Organize social events where senior students offer tips and share experi- +ences and success stories with junior students. +– Have mandatory videos with examples of unaccepted behaviors (e.g, in- +appropriate jokes, stalking, etc). +– Advise faculty members to ensure that lectures and assignments accom- +modate a variety of interests. +– Advise faculty members to ensure that colloquiums are not transformed +into boys-clubs. +– Increase the number of female TAs. +– Explore the opportunity to have a few preparatory weeks for program- +ming before starting the first semester in CS. +Action items for a balanced flow of candidates into CS academia. +– Hold events that clearly describe the academic career and the PhD pro- +gram in CS. +– Advise faculty members to move PhD applications earlier at the aca- +demic year. + +Better Balance in Informatics: An Honest Discussion with Students +17 +5 +Related Work +The discussion sessions with the students helped us identify action items to +achieve (i) a balanced flow of students into CS studies, (ii) a balanced environ- +ment within the CS department, and (iii) a balanced flow towards CS academia +(i.e., PhD and beyond). This section discusses how prior work tackles these three +aspects separately. For an extensive overview of initiatives for improving gender +balance in CS, we refer the reader to Jaccheri et al. [14]. +Our discussion participants highlighted in Figure 13 that we need to “deal +with the issue [of gender balance] at a younger age”. A recent aggregated study [13] +collects 22 measures and strategies for CS educators in secondary education to +sustain the interest of female students in the CS classes. Our action items for a +balanced flow of students into CS are aligned with the proposed measurements +in [13] that aim to demolish stereotypes. Our observations are also aligned with +a study [9] concluding that: family and friends have high impact to the decision +of girls to study CS, courses for CS should be introduced earlier at school, one +should highlight the problem-solving aspect of CS and surface female role models. +A successful strategy for increasing the percentage of CS female undergraduate +students at CMU (close to 50% was the percentage of female students that en- +tered the CS program in 2018) is presented in [12]. Two points of this strategy are +that the curriculum does not have to change to become more “female-friendly”, +and that it is important to promote cultural changes within the institution (e.g., +create entry level courses for students with no prior programming experience, +increase the visibility of women, break stereotypes). These points address two +main reasons [23] for the low enrollment of female students in CS programs: +“bias in early socialization” and “anxiety towards technology”. A more recent +paper [22] refines those reasons into three categories: social (e.g., stereotypes), +educational (e.g., unattractive CS learning environment), and labor market (e.g, +unattractive jobs). Then the authors present ways to address those reasons, by +communicating different perspectives of CS and engaging female students to +various CS experiences. +Understanding the culture within the study environment of a CS department +is a prerequisite for decreasing the gender gap. CMU employed student inter- +views [11] to inform its strategy for better gender balance. Margolis et al. [18] +investigate how the interest of female students about their CS studies might de- +cline and eventually lead to drop-out. Rosenstein et al.[21] report that, within a +sample of 200 students, “57% were found to exhibit frequent feelings of the Im- +postor Phenomenon with a larger fraction of women (71%) experiencing frequent +feelings of the Imposter Phenomenon than men (52%)”. Miller et al. [19] focus +on students with minoritized identities of sexuality and/or gender (MIoSG) in +STEM, and concludes that these students are enduring a “dude culture” that +fosters hypermasculinity and suppresses discourses related to sexual orientations +other than heterosexuality. On the positive side, interviewing STEM students, +Rainey et al.[20] conclude that active teaching may improve the sense of belong- +ing for underrepresented students. Finally, Lagesen [15] interviews Malaysian + +18 +E. Kozyri et al. +female students, which form around 50% of the student body, to see how their +perception about CS differs from the western culture. +A more limited number of studies have been devoted to fostering a gender- +balanced flow of students towards PhD and beyond. For example, Moreno et +al. [10] interview doctoral CS students on the reasons that led them to apply to +a PhD program. The authors identified five mean reasons: academic career goal, +professional development, career change, employment opportunity and personal +fulfillment. Personal fulfillment was the most popular reason given. +6 +Conclusion +To understand how the gender balance in our CS department can be improved, +we organized discussion sessions among CS undergraduate students, who shared +their thoughts about: the reasons they picked CS for their studies, their current +experience with the CS studies, their intention to pursue an academic career +in CS, and ways to make the CS community more diverse and inclusive. From +their answers we identified action items for achieving a balanced flow of students +into CS undergraduate program, a study environment that embraces diversity, +and a balanced flow of students into higher levels of the CS academia. After +the completion of the discussion sessions, the students were able to submit their +feedback. We were pleased to see that they enjoyed the discussion and thought +that the questions we asked were important. The participants also appreciated +our effort to use neutral and not-offensive language for the questions and the +discussion that they triggered. +Acknowledgements +We would like to thank Lilli Mittner for recommending ´Ardna for holding the +discussion sessions and for giving inspiration for the discussion questions. Lynn +Nygaard gave us inspiration for these questions, too. We also thank Melina +Duarte for providing network support for BBI within UiT, and Ingeborg Owe- +sen for providing network support for BBI within BalanseHub. Finally, we are +grateful to the administration of the CS department and the members of BBI for +their help in organizing the discussion sessions. This work has been partially sup- +ported by the COST Action CA19122, from the European Network for Gender +Balance in Informatics, and by the NFR grant 321075 for BBI. +References +1. European network for gender balance in informatics. https://eugain.eu/ +2. Gender balance in research leadership. https://uit.no/research/prestige +3. Google 2022 diversity annual report 2022. https://about.google/belonging/ +diversity-annual-report/2022/ +4. ICT specialists in employment. https://ec.europa.eu/eurostat/ + +Better Balance in Informatics: An Honest Discussion with Students +19 +Fig. 15. A discussion session at ´Ardna, UiT. +5. Microsoft diversity & inclusion report. https://www.microsoft.com/en-us/ +diversity/default.aspx +6. Network support to promote gender balance in norwegian research. https://www. +forskningsradet.no/utlysninger/2021/balansehub/ +7. Booklet: Best practices from school to university. https://eugain.eu (2022) +8. Albusays, K., Bjorn, P., Dabbish, L., Ford, D., Murphy-Hill, E., Serebrenik, A., +Storey, M.A.: The diversity crisis in software development. IEEE Software 38(2), +19–25 (2021) +9. Alshahrani, A., Ross, I., Wood, M.I.: Using social cognitive career theory to under- +stand why students choose to study computer science. In: Proceedings of the 2018 +ACM Conference on International Computing Education Research. p. 205–214. +Association for Computing Machinery, New York, NY, USA (2018) +10. del Carmen Calatrava Moreno, M., Kollanus, S.: On the motivations to enroll in +doctoral studies in computer science — a comparison of phd program models. +In: 2013 12th International Conference on Information Technology Based Higher +Education and Training (ITHET). pp. 1–8 (2013) +11. Fisher, A., Margolis, J., Miller, F.: Undergraduate women in computer science: +Experience, motivation and culture. In: Proceedings of the Twenty-Eighth SIGCSE +Technical Symposium on Computer Science Education. p. 106–110. SIGCSE ’97, +Association for Computing Machinery, New York, NY, USA (1997) + +19V6H +Have more role models from +minorities20 +E. Kozyri et al. +12. Frieze, C., Quesenberry, J.L.: How computer science at CMU is attracting and +retaining women. Commun. ACM 62(2), 23–26 (jan 2019) +13. Happe, L., Buhnova, B., Koziolek, A., Wagner, I.: Effective measures to foster +girls’ interest in secondary computer science education. Education and Information +Technologies 26(3) (2021) +14. Jaccheri, L., Pereira, C., Fast, S.: Gender issues in computer science: Lessons learnt +and reflections for the future. In: 2020 22nd International Symposium on Symbolic +and Numeric Algorithms for Scientific Computing (SYNASC). pp. 9–16 (2020) +15. Lagesen, V.: A cyberfeminist utopia?: Perceptions of gender and computer science +among malaysian women computer science students and faculty. Science Technol- +ogy & Human Values 33, 5–27 (01 2008) +16. Letizia Jaccheri et. al.: Where are the female professors in STEM? (12 2022). +https://doi.org/10.36227/techrxiv.21760532.v1 +17. Lewin, A., Sathianathan, D., Lenhard, J., Britton, A.O., Warnock, E., Bavey, +N.: The state of diversity in european tech, https://eic.ec.europa.eu/system/ +files/2022-05/EIC%20report%20presentation.pdf +18. Margolis, J., Fisher, A., Miller, F.: The anatomy of interest: Women in undergrad- +uate computer science. Women’s Studies Quarterly 28(1/2), 104–127 (2000) +19. Miller, R.A., Vaccaro, A., Kimball, E.W., Forester, R.: “It’s dude culture”: Stu- +dents with minoritized identities of sexuality and/or gender navigating stem ma- +jors. Journal of Diversity in Higher Education 14, 340–352 (2021) +20. Rainey, K., Dancy, M., Mickelson, R., Stearns, E., Moller, S.: A descriptive study of +race and gender differences in how instructional style and perceived professor care +influence decisions to major in stem. International Journal of STEM Education +6(1) (2019) +21. 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Computer Science Education 20(4), 301–316 (2010) + diff --git a/f9E0T4oBgHgl3EQfpAES/content/tmp_files/load_file.txt b/f9E0T4oBgHgl3EQfpAES/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0fb2ef92a886c2a491e6b7275c614bd6d308969a --- /dev/null +++ b/f9E0T4oBgHgl3EQfpAES/content/tmp_files/load_file.txt @@ -0,0 +1,528 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf,len=527 +page_content='Better Balance in Informatics: An Honest Discussion with Students⋆ Elisavet Kozyri1, Mariel Evelyn Markussen Ellingsen2, Ragnhild Abel Grape1, and Letizia Jaccheri3 1 UiT The Arctic University of Norway 2 Woid AS 3 Norwegian University of Science and Technology Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In recent years, there has been considerable effort to pro- mote gender balance in the academic environment of Computer Science (CS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' However, there is still a gender gap at all CS academic levels: from students, to PhD candidates, to faculty members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This general trend is followed by the Department of Computer Science at UiT The Arctic Uni- versity of Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To combat this trend within the CS environment at UiT, we embarked on structured discussions with students of our depart- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' After analyzing the data collected from these discussions, we were able to identify action items that could mitigate the existing gender gap at our department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In particular, these discussions elucidated ways to achieve (i) a balanced flow of students into CS undergraduate program, (ii) a balanced CS study environment, and (iii) a balanced flow of grad- uates into higher levels of the CS academia (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', PhD program).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This paper presents the results of the discussions and the subsequent recom- mendations that we made to the administration of the department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We also provide a road-map that other institutions could follow to organize similar events as part of their gender-balance action plan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Keywords: Gender balance · computer science · diversity · inclusion · student study 1 Introduction Innovations in Computer Science shape the lives of everyone in our society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To create innovative solutions tailored to everyone, it is important that all groups of society are represented in the creation of these solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' However, this is still not the case in the field of Computer Science (CS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Having an awareness of the lack of representation and the different barriers people face in CS are fundamental in helping the field target those challenges and becoming more equitable and inclusive [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Statistics from Europe show that women are still highly underrepresented in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' According to Eurostat [4], the percentage of female specialists in Information and Communications Technology has evolved from 17% in 2012 to 19,1% in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' ⋆ Mariel’s contribution was made while she was a Master’s student at UiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='02532v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='CY] 6 Jan 2023 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' At university level in STEM, the percentage of female Bachelor, Master, and PhD students is 20%, while the percentage of female professors is 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Specifically for the Department of Computer Science at UiT The Arctic Uni- versity of Norway, only 13% of students, 14% of PhD candidates and 21% of faculty members are female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Better Balance in Informatics (BBI), a program led by the CS department at UiT and funded by the Research Council of Norway, aims to rectify this imbalance and create a more diverse learning environment for Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' BBI is connected and builds upon an ecosystem of national and international projects which address gender balance in CS acting on different levels: school ([13], [7]), university ([2], [6] [16]), industry ([17], [3], [5]), and the interplay of these levels ([1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' BBI aimed to identify some of the reasons that led to the current gender dynamics in our CS department, and then propose measurements that could address those reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Hearing directly from the CS students (Bachelor, Master) seemed to be a sensible way for us to identify those reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, BBI organized structured discussion sessions, where we invited CS students (Bachelor, Master) to share their thoughts about: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' the reasons they picked CS for their studies, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' their current experience with the CS studies, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' their intention to pursue an academic career in CS, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' ways to make the CS community more diverse and inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The answers of the students illuminated points of intervention, which could lead to a balanced flow of students into CS undergraduate program, a study environment that embraces diversity, and a balanced flow of students into higher levels of the CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This paper presents the methodology (§2) we employed to organize the dis- cussion sessions, to collect responses, and to report the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We then present the specific questions we asked the students and the analysis of their answers (§3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Finally, we list the recommendations (§4) submitted to the CS department for achieving a gender-balanced environment, we discuss related work (§5), and we conclude (§6) with reflections about the discussion sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 2 Methodology The end goal of the discussion sessions was to identify points of interventions that could increase the gender balance among the incoming CS students, the current CS students, and the CS graduates that are interested in entering the CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To identify those points, we were aiming for a high number of participants in the discussion sessions: the more participants, the greater the plurality of experiences, and thus, the higher the chances to find opportunities for improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Deciding which questions to ask was crucial to ensure that experiences from different aspects of the CS studies are captured and then an- alyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' But, we also had to create a trusting discussion environment for the Better Balance in Informatics: An Honest Discussion with Students 3 students to honestly share those experiences with us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This section describes the methodology we followed to prepare and organize the discussion sessions such that all these targets are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='1 Outreach Attracting the attention of students and persuading them to participate in the discussion sessions was not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Unless there is an immediate academic or employment gain, motivating students to devote part of their busy schedules to a university-led event is indeed challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Our strategy to address this challenge comprised the following steps: Hiring students as project assistants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We hired two talented and enthusiastic female students as assistants for the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' They were our bridge to the student community in our CS department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' And this bridge was functioning in both ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Their thoughts, insights, and experience informed all aspects of the BBI project, including the questions we asked during the discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' At the same time, they knew best how to reach their fellow-students and promote the agenda of BBI (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', what advertisement means to employ and what to say in these means).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The BBI website (https://uit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='no/project/bbi) is the main official space where the mission of BBI is communicated to the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, we redesigned this website to include a clear and short motivation for the BBI mission, and describe the upcoming BBI events, in particular the discussion sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Advertisement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To reach a wider set of students and persuade them to partici- pate in the BBI discussion sessions, we employed a variety of means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We posted advertisements on the monitors of the Department, the social networks of the Department, on Canvas, the UiT calendar, and the local student organization forum, which is a Discord server that is maintained by the student organization TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The student assistants also gave 5-minutes talk about BBI and the dis- cussion sessions to courses with high enrollment, they created and distributed flyers, and they organized a stand with coffee and cookies, where students could casually socialize and talk about BBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In terms of registrations to the BBI dis- cussion sessions, Canvas and TD seemed to have been the most effective, since we observed a high correlation between the time a post about the BBI event was created and the time students were registered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Open to everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The invitation to participate in the BBI discussion sessions was open to all students of the CS department, independently of their gender (female, male, non-binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This is because the gender imbalance is a problem that owes to concern everyone—not only a part of the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' And because any further actions that the Department will take to address the problem might effect every student, there needs to be a wider understanding that these actions are worthwhile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Leaving specific groups of students outside the discussion, would not have increased this understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='2 Discussion Sessions The discussion sessions were held at ´Ardna, UiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' ´Ardna is an iconic place in the university, ideal for secluded discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Its central fire place and the surround- ing wooden benches invites people to open up and discuss honestly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In the BBI discussion sessions participated around 20 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='4 Com- paring to events organized in the past by BBI, this number of participants was a very welcoming surprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' From those participants, around 50% were female or non-binary students, and around 50% were male students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The vast majority of the students participated physically, but there were some that participated remotely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' There were around three participants per discussion session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Each dis- cussion session was moderated by two BBI members: one member was asking the questions, and the other member was typing the answers (we used no video or sound recording).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' At least one of the moderators was always one of the BBI student assistants;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' having participants talking to their peers led to frank discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To preserve anonymity, each participant was assigned a number, so the recorded answers were associated with these numbers—not with the identity of the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' For the discussion sessions, we gave the option to the student to select either Norwegian or English as the speaking language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' All participants, apart from those that participated remotely, were offered a full meal and a free cinema ticket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='3 Selection of Questions The selection of questions was inspired by questionnaires developed by other gender-balance projects, such as EUGAIN [1], Prestige in UiT [2], and Balanse- Hub [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' However, the questions were tailored for the needs of the CS department in UiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' And, in particular, the questions were intended to cover a wide range of students’ experience: from the point they considered applying for CS, to their current studies and their future plans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='4 Reporting of Results For most of the questions asked during the BBI discussion sessions, we compiled the answers into graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Each graph depicts how answers are correlated with dif- ferent genders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This information help us guide our selection of action items for improving the gender balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We protect the anonymity of the participants, so we do not give specific numbers at graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Also, we do not have a separate cate- gory for non-binary participants, because the number of non-binary participants was not high enough to protect their anonymity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Instead, we group female and non-binary participants together, and we explore the dynamics between majority (males) and minorities (female, non-binary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 4 We do not give specific numbers to preserve the anonymity of the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Better Balance in Informatics: An Honest Discussion with Students 5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Reasons for choosing to study CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Each column corresponds to a different rea- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The height of a column represents the number of participants that submitted the corresponding reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Dark blue represents female or non-binary participants (F/NB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' yellow represents male participants (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 3 Results This section presents the questions we asked the participants and their answers concerning: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' the reasons they picked CS for their studies, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' their current experience with the CS studies, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' their intention to pursue an academic career in CS, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' ways to make the CS community more diverse and inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Correlating their answers with their gender, we identified action items that could lead to a balanced flow of students into CS undergraduate program, a study environment that embraces diversity, and a balanced flow of students into higher levels of the CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='1 Intention to Study CS To increase the balance in Computer Science, one first needs to increase the balance in the new-coming students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, when advertising CS to younger stu- dents, one could also include aspects that attract minorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We tried to identify those aspects by asking the participants the reason they decided to study CS in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 1 shows the different answers we received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The higher the column, the more students gave that answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The graph also shows how the answers are split between the minority (F/NB) and majority (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' There is a correlation between the gender and the reason for selecting CS studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' I started to study Cs because.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='. Got interested Got interested Got interested CS combines CS education CS has flexibleCS was 2nd Like Tromso through the through through practice, math, creates more study program choice of study girls-and-tech programming computer and problem job or interesting day in UiT courses at high games solving opportunities course school descriptions F/NBM6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Observing Figure 1, we can identify the reasons the minority chose CS studies: the problem solving aspect of CS, the flexibility of the CS studies, the job opportunities that CS graduates enjoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To increase the diversity of incoming students, we can then emphasize those reasons when advertising CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Also, as a possible means of advertisement Figure 1 indicates the UiT girls-and- tech day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Ways for becoming familiar with CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Apart from the UiT girls-and-tech day, we wanted to understand what would be other effective means of advertisement for attracting minorities to CS studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, we asked the participants where did they hear about CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 2 plots the answers, which are again correlated with the gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 2 indicates that one could use the highschool and the uni- versity’s study catalog to better promote CS to minorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Interestingly, friends and relatives have a high impact on the decision of minorities to study CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, one can make tech employees ambassadors of CS to young female and non-binary members of their families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In general, the vast majority of the participants, independently of their gen- der, would have liked CS to have been introduced differently to them, as Figure 3 indicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Participants indicated that CS should be introduced as something that everyone can do and something that offers a plausible path to a regular job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' When advertising to high-school students, we need to break stereotypes on who can study CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Where did you hear about Cs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Highschool, Study catalog Friends or Event hosted 《Always》 Gaming Internet career relatives by UiT known search guidance ■F/NBMBetter Balance in Informatics: An Honest Discussion with Students 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Independent of their gender, 81% of the participants said that they would have liked to be introduced to CS in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='2 Your Experience in the Current CS Environment At the very least, we want to sustain the diversity among the incoming students, while these students progress to more senior years;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' we aim to decrease the number of drop-outs, with an emphasis on the minority group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To achieve this, we need to assess the student’s experience within the CS department and identify aspects that can be improved to accommodate gender diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We start by asking the participants whether the CS studies met their initial expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Their answers are depicted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Almost all minority par- ticipants gave a negative answer: they found their studies either more difficult or more practical than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In particular, they found the learning curve of programming to be particularly steep, something that might drive some of the minority students to drop-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We believe addressing this concern is important for maintaining a diverse environment in the department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We propose the adoption of a smoother introduction to pro- gramming, which will be appropriate for students with no prior programming experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Returning to Figure 4, one also notices that, for most of the male participants, their experience in the CS studies met or even exceeded their initial expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, this question highlights a striking difference between the minorities and the male students in terms of how they view their studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This difference might be a cause of the current gender imbalance in the department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=" DOYOUWISHTHAT CS WOULD'VE BEENINTRODUCED TOYOUIN A DIFFERENT WAY OREARLIER?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' No 6% Maybe 13% Yes 81%8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The first column corresponds to the answer that the CS studies met or exceeded the expectations of the participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The remaining four columns correspond to the answer that the CS studies did not quite meet the expectations of the participant, and they also represent different reasons why.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The height of a column represents the number of participants that submitted the corresponding answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Dark blue represents female or non-binary participants (F/NB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' yellow represents male participants (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' All the participants agreed, though, that the social environment built around their studies exceeded their initial expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This is a great achievement of the department that needs to be preserved for the future, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Participants were then explicitly asked whether they have thought to drop- out of their study program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' As Figure 5 shows, most of the participants answered affirmatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Notice that this is a concern across all genders, opposing the mis- conception that the minorities are more likely to have such thoughts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Notice also that even though most of the male students thought to drop-out of the program, they still had an overall positive assessment of their study experience, according to Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' As reasons for thinking to drop-out, the participants cited the difficulty of some courses, the time-consuming assignments with overlapping deadlines, the demanding task of writing thesis (a task for which they did not feel prepared), and the complications that the COVID-19 pandemic brought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' For a student to be thinking to drop-out, it means that the student’s self esteem might be low at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 6 validates this claim, showing that most of the participants felt ”useless” or ”not-deserving” being in the CS program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Again, the answers do not seem to be correlated with the gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' However there is an underlying difference: many of the males had this feeling once, related to a specific assignment or for short period of time, whereas the minority students had this feeling for a long period of time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='e, months or even years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Do your CS studies meet your initial expectations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' if not, why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Yes or exceeded: Not quite: more Not quite: more Not quite: more Not quite: lectures more theoretical than difficult than practical than are not designed coding/practical than expected expected expected carefully enough expected (programming) ■F/NB ■MBetter Balance in Informatics: An Honest Discussion with Students 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The majority of the participants replied that they have thought of dropping out of their CS study program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The majority of the participants replied that they have have felt “useless” or “not-deserving to be here” during their CS study program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' When asked about the reasons they instead decided to stay in the program and not drop out, the participants mentioned: – the robust social network that they have built within and outside UiT, where they could talk about their struggles, – the senior student advisor Jan Fuglesteg, Have you ever thought of dropping out of your study program?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Yes No ■F/NB MHave you ever felt “"useless" or “"not-deserving being here" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', imposter syndrome) during your studies?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Yes No IF/NB ■M10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – their self-determination and discipline, – taking time to relax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Given the stimulating power that the social groups exercised on the students, we should further support actions and groups that promote social networking in the department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Also, we will organize events where senior students can offer tips and tricks from their experiences to the junior students, where the main message will be “if we have made it, so can you”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' More than half of the participants said that they have witnessed or heard of sexual harassment incidents within our CS community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Concentrating on minority students, one of the reasons they might feel un- comfortable in an environment (and ultimately drop-out of the program) is when they have experienced sexual harassment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, we asked the participants whether they have ever witnessed or heard of sexual harassment incidents within the CS community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 7 depicts their answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' More than half of the participants answered positively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The “Yes” column in Figure 7 should not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, we will pro- pose to the department to devise videos and examples of unacceptable behavior, so the student can recognize and dissociate from these phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The experience that a student gets from their CS program is a collection of many educational aspects, such as lectures, colloquiums, and assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' For the CS program to be inclusive and diverse, all these aspects should pro- mote inclusiveness and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We asked the participants if they feel that the educational aspects below promote only a particular way of thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Have you ever witnessed or heard of incidents of sexual harassment within our cs community?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Yes No F/NB MBetter Balance in Informatics: An Honest Discussion with Students 11 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Almost all of the participants do not wish to have an academic career in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Lectures: Participants mentioned that examples employed in some lectures are more appealing to male students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' These examples usually involve games or cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Colloquiums:5 Participants, from all genders, mentioned that a colloquium can quickly get a “boys-club” atmosphere if the TA is not attentive enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The participants also express the wish for more female TAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Assignments: Some assignment are very focused on gaming or promote com- petitiveness, which might be uncomfortable for some students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We will advise the faculty members to ensure that lectures and assignments accommodate a variety of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The department should organize a seminar for TAs in which they become sensitized on not allowing colloquiums to become “boys-clubs” and accommodating different interests and needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We also need to brainstorm on how to hire more female TAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='3 Intention to Enter Higher Levels in CS Academia We are striving to achieve balance at all levels of the CS academia, from students to professors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' At this section, we focus our attention to higher levels in CS academia (from PhD candidates and above), and we want to understand the intention of the current students to enter that level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' According to Figure 8, the vast majority, and in particular all female and non-binary participants, do not wish to have an academic career in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Here are some reasons why: 5 A colloquium is associated with a course and it is often led by a TA, who answers student’s questions about the course material and assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Do you wish to have an academic career in Cs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Yes No F/NB ■M12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – The academic career seems difficult or exhausting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – No interest in research or writing or teaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Preference for work-life balance offered by the industry (pay, social, use tools, practical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – The description of an academic career is not clearly communicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Most of the participants are either unsure about or do not know what the CS PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In fact, as indicated by Figure 9, there is uncertainty between the students, and in particular the minority students (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', female and non-binary), about what a PhD program is—the first stepping stone towards building an academic career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Given this uncertainty, it is expected that many students will not apply to a PhD program, something that is affirmed by Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Notice, though, that comparing Figures 8 and 10, the participants are less negative towards pursuing a PhD program than ultimately following an academic career.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This is because, after obtaining a PhD degree, there is always the possibility to follow a career in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' And some of the participants that replied “no” to the prospective of applying to a PhD program now, they contemplate the possibility of applying after working in the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' And if a participant does not want to apply to a PhD program, what are their immediate plans after graduation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 11 answers this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Notice that participants from the minority group explore a wider variety of options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Also, for the participants that are not considering to apply to the CS PhD program, Figure 12 gives the main reasons behind this disposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 12 Do you know what a PhD program is?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Yes Maybe No ■F/NB■MBetter Balance in Informatics: An Honest Discussion with Students 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Most of the participants would not apply to a PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Columns correspond to different options the participants consider to follow after graduation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' informs how we can intervene and address some of these reasons, possibly leading to more students applying to our PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action Items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' According to Figure 12, some participants said that the PhD program “sounds too heavy”, and they described PhD students as being “alone” and “depressed”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' While this description might portray one aspect of the PhD Would you apply to a PhD program?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Yes Maybe No ■F/NB ■MWhat do you want to do when you graduate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Work as Work as Work in Work offshore Do a start-up Apply for PhD Do not know developer consultant security, intelligence, or military.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' F/NB ■M14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Columns correspond to different reasons why the participants are not consid- ering to apply to a PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' experience, it is definitely not the entire truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' So, we are going to hold events that clearly describe the characteristics of a PhD program, emphasizing the positive aspects of being a PhD student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' These events will also address the uncertainty that was surfaced in Figure 9 about what is a PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The late deadline for applying to a PhD, which is not synchronized with the job-search period of senior students, is another reason why current students do not select a PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To remedy this, we will advise the faculty members of the CS department to consider moving the PhD application earlier in in the academic year (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', fall semester).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Finally, given that many participants said that they might consider applying to a PhD program in the future (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', after acquiring some experience in the industry), we advocate to advertise new PhD positions explicitly to CS alumni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' For some of these alumni, these positions might seem attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='4 The Gender Gap and Possible Measurements to Close it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In previous sections, we attempted to understand the reasons why a gender imbalance exists in the CS department, and concluded with action items that could address those reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' In this section, we explicitly discuss gender balance with the students and record their opinions and proposals on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' To begin with, the vast majority of the participants said that the current gender-imbalance in the department is a problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' They actually see that gender- balance has advantages: promotes innovation, enables plurality of perspectives, and leads to a better study and work environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Many of the participants said Why would you not apply to a PhD program?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Want a job in Sounds too Maybe in the Do not want to Happy with a PhD application industry heavy future teach bachelor is too late ■F/NB■MBetter Balance in Informatics: An Honest Discussion with Students 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Each row corresponds to a different measurement for improving gender balance in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The length of each row represents the number of participants that agree with the corresponding measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Dark blue represents female or non-binary participants (F/NB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' yellow represents male participants (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' that there are no disadvantages with gender balance, although some expressed the concern that gender-balance efforts, such as gender quotas, might “lead to people being hired for the wrong reasons”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The participants were then presented with different measurements that could be employed to improve the gender balance within the department and asked to say whether they are in favor or not of each presented measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 13 depicts their answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Notice that measurements that blatantly favor minorities in the education or in the career were among the least popular (for both minority and male participants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We aim to focus on the most popular measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Participants also proposed two additional measurements that did not appear in our pre-selected list: – Share success stories from minorities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Have a few preparatory weeks for programming before starting the first semester in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 4 BBI recommendations for the near future Throughout this report we presented a variety of action items for improving gender balance in our CS department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We have motivated these action items using the findings from the discussions with the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We now summarize those actions that BBI recommends for the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' These actions aim to achieve a balanced flow of students into CS studies, a balanced environment within the CS department, and a balanced flow towards CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Figure 14 Which of the following measurements are you in favor of?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Make the subject of CS courses easier Use gender points Educational events for minorities only Student unions for minorities only Career events for minorities only (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', visiting industry) Focus on keeping minorities already in the program Introduce more theoretical courses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', complexity,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Create more interdisciplinary programs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', CS+Biology,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='. More advertisements aimed towards minorities Social events for minorities only Introduce more creative courses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', HCl, computer graphics) Deal with the issue at a younger age Have more role models from minorities ■F/NB■M16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' School CS Studies CS Academia Balanced Flow Balanced Environment Balanced Flow Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We propose action items for (i) a gender-balanced flow of students from the school to CS studies, (ii) a gender-balanced student environment in our department, and (iii) a gender-balanced flow of graduates from the CS studies to the CS Phd program, and eventually CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' gives a schematic representation of these three categories of actions, which are listed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action items for a balanced flow of students into CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Student-ambassadors of all genders to present CS to high school and middle high school students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Highlight problem solving aspect of CS, flexible and interesting program, job opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' CS is something that everyone can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action items for a balanced CS study environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Organize social events where senior students offer tips and share experi- ences and success stories with junior students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Have mandatory videos with examples of unaccepted behaviors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g, in- appropriate jokes, stalking, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Advise faculty members to ensure that lectures and assignments accom- modate a variety of interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Advise faculty members to ensure that colloquiums are not transformed into boys-clubs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Increase the number of female TAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Explore the opportunity to have a few preparatory weeks for program- ming before starting the first semester in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Action items for a balanced flow of candidates into CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Hold events that clearly describe the academic career and the PhD pro- gram in CS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' – Advise faculty members to move PhD applications earlier at the aca- demic year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Better Balance in Informatics: An Honest Discussion with Students 17 5 Related Work The discussion sessions with the students helped us identify action items to achieve (i) a balanced flow of students into CS studies, (ii) a balanced environ- ment within the CS department, and (iii) a balanced flow towards CS academia (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', PhD and beyond).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This section discusses how prior work tackles these three aspects separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' For an extensive overview of initiatives for improving gender balance in CS, we refer the reader to Jaccheri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Our discussion participants highlighted in Figure 13 that we need to “deal with the issue [of gender balance] at a younger age”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' A recent aggregated study [13] collects 22 measures and strategies for CS educators in secondary education to sustain the interest of female students in the CS classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Our action items for a balanced flow of students into CS are aligned with the proposed measurements in [13] that aim to demolish stereotypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Our observations are also aligned with a study [9] concluding that: family and friends have high impact to the decision of girls to study CS, courses for CS should be introduced earlier at school, one should highlight the problem-solving aspect of CS and surface female role models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' A successful strategy for increasing the percentage of CS female undergraduate students at CMU (close to 50% was the percentage of female students that en- tered the CS program in 2018) is presented in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Two points of this strategy are that the curriculum does not have to change to become more “female-friendly”, and that it is important to promote cultural changes within the institution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', create entry level courses for students with no prior programming experience, increase the visibility of women, break stereotypes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' These points address two main reasons [23] for the low enrollment of female students in CS programs: “bias in early socialization” and “anxiety towards technology”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' A more recent paper [22] refines those reasons into three categories: social (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', stereotypes), educational (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=', unattractive CS learning environment), and labor market (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content='g, unattractive jobs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Then the authors present ways to address those reasons, by communicating different perspectives of CS and engaging female students to various CS experiences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Understanding the culture within the study environment of a CS department is a prerequisite for decreasing the gender gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' CMU employed student inter- views [11] to inform its strategy for better gender balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Margolis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' [18] investigate how the interest of female students about their CS studies might de- cline and eventually lead to drop-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Rosenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' [21] report that, within a sample of 200 students, “57% were found to exhibit frequent feelings of the Im- postor Phenomenon with a larger fraction of women (71%) experiencing frequent feelings of the Imposter Phenomenon than men (52%)”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Miller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' [19] focus on students with minoritized identities of sexuality and/or gender (MIoSG) in STEM, and concludes that these students are enduring a “dude culture” that fosters hypermasculinity and suppresses discourses related to sexual orientations other than heterosexuality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' On the positive side, interviewing STEM students, Rainey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' [20] conclude that active teaching may improve the sense of belong- ing for underrepresented students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Finally, Lagesen [15] interviews Malaysian 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Kozyri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' female students, which form around 50% of the student body, to see how their perception about CS differs from the western culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' A more limited number of studies have been devoted to fostering a gender- balanced flow of students towards PhD and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' For example, Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' [10] interview doctoral CS students on the reasons that led them to apply to a PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The authors identified five mean reasons: academic career goal, professional development, career change, employment opportunity and personal fulfillment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Personal fulfillment was the most popular reason given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' 6 Conclusion To understand how the gender balance in our CS department can be improved, we organized discussion sessions among CS undergraduate students, who shared their thoughts about: the reasons they picked CS for their studies, their current experience with the CS studies, their intention to pursue an academic career in CS, and ways to make the CS community more diverse and inclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' From their answers we identified action items for achieving a balanced flow of students into CS undergraduate program, a study environment that embraces diversity, and a balanced flow of students into higher levels of the CS academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' After the completion of the discussion sessions, the students were able to submit their feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We were pleased to see that they enjoyed the discussion and thought that the questions we asked were important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' The participants also appreciated our effort to use neutral and not-offensive language for the questions and the discussion that they triggered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Acknowledgements We would like to thank Lilli Mittner for recommending ´Ardna for holding the discussion sessions and for giving inspiration for the discussion questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Lynn Nygaard gave us inspiration for these questions, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' We also thank Melina Duarte for providing network support for BBI within UiT, and Ingeborg Owe- sen for providing network support for BBI within BalanseHub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' Finally, we are grateful to the administration of the CS department and the members of BBI for their help in organizing the discussion sessions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' This work has been partially sup- ported by the COST Action CA19122, from the European Network for Gender Balance in Informatics, and by the NFR grant 321075 for BBI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/f9E0T4oBgHgl3EQfpAES/content/2301.02532v1.pdf'} 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b/h9A0T4oBgHgl3EQfIP-u/content/tmp_files/2301.02073v1.pdf.txt @@ -0,0 +1,531 @@ +The earthquake network: the best time scale for +network construction +Nastaran Lotfia,∗ +a Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo, Caixa +Postal 668, 13560-970 S˜ao Carlos, SP, Brazil. +Abstract +Scientists mapped the seismic time series into networks by considering the geo- +graphical location of events as nodes and establishing links between the nodes +with different rules. Applying the successive defined laws to construct the net- +works of seismic data, a variety of features of earthquake networks are de- +tected (scale-free and small-world structures). +Network construction models +had changed in detail to optimize the performance of the verification of the +minimum geographical size defined for the node. In all the studies, people try +to use large data sets like years of data to ensure their results are good enough. +In this work, by proposing the temporal network construction and employing +the small-worldness property for data from Iran and California, we could achieve +the minimum time scale needed for the best results. We verified the importance +of this scale by analyzing two significant centrality measures (degree centrality +and PageRank) introduced in the concept of earthquake network. +Keywords: +Complex Networks, Temporal networks, earthquake networks +1. introduction +An earthquake is a sudden motion of a fault that releases an enormous +amount of energy and is considered a complex spatiotemporal phenomenon oc- +curring in the earth’s crust [18]. Transferring the stress of the movement of one +fault to the others results in triggering subsequent events [19, 9, 14]. Omori +∗n.lotfi@icmc.usp.br +Preprint submitted to Journal of Acta Geophysica Templates +January 6, 2023 +arXiv:2301.02073v1 [physics.geo-ph] 5 Jan 2023 + +law [26] and Gutenberg-Richter law [16] are empirical laws to characterize the +Temporal pattern of aftershocks, frequency, and magnitude, respectively. Be- +sides the visible properties, complex interaction exists in the internal of the +seismic system [8, 7, 15]. +While seismicity is assumed to be a complex phenomenon, the network ap- +proach offers a powerful tool for analyzing the dynamic structures of it [3, 4, 5, 6]. +Over the last decade, different models proposed to construct the earthquake net- +work [4, 20, 27, 25]. In the simple but basic model introduced by Abe-Suzuki [4], +the geographical region is divided into small square (cubic) cells, and seismic +events with time sequences get connected. Later, Lacasa et al. [20] proposed +a model to construct the network with a visibility graph. They converted the +time series into a graph by inheriting the properties of the series in its structure. +They explored periodicity, fractality, chaoticity, and non-linearity of the seismic +time series [21, 22, 13]. Rezaie et al. [27] introduced the hybrid model, which +inherits the bases of the Abe-Suzuki model mixed with a visibility graph. To +better capture the evolution of the earthquake network through time, a mul- +tiplex network was employed [25]. Analyzing the seismic data with a network +approach through different models helped reveal many features of the seismic +activity just by knowing the basic information of magnitude, time of occurrence, +and the location of seismic events [7, 3, 23, 2, 24]. It had been verified that the +the earthquake networks that constructed from the seismic data taken from Cal- +ifornia and Japan [3, 5, 4], Iran [23, 24], Chile [1] , Greece [11], and Italy [27] +are scale-free and small-world. +Most recent works focused on improving the proposed models to capture the +best minimum resolution of the cell size needed for network construction. It +means the cell size should be smaller than the specified limit to be trustable. +The main question is how we ensure that the time window, in the scale of +dates, months, or years, is large enough for constructing the network. In all +the studies done till now, scientists considered the time on such a big scale of +years. And the concept of the minimum necessary time window for achieving +the best results are missed. In this work, we employ the definition of temporal +2 + +network construction and capture the lowest time window essential for network +construction. +We found that depending on the region of consideration, the +value of the time window threshold would change. We verified the trustiness +of this time window size by analyzing two important centrality parameters, +degree centrality, and PageRank. +If the time window is small, we miss the +information in centrality, and if it is bigger than the threshold, we do not gain +extra knowledge than in the threshold time region. +The rest of the paper is organized as follows. +In section 2, we provide +information about the data sets we employ, and Section 3 is devoted to our +results. +2. Database +We applied our model for the latest four years of data, 01 Jan 2018 to 31 Dec +2021, for Iran in the range of 24N − 44N latitude and 40E–62E longitude with +14062 total events obtained from Iranian Seismological Center1, and California +in the range of 32N–42N latitude and 114W–124W longitude with 7575 total +events gained from the Northern California Earthquake Catalog2. In both of +the considered data sets, we examined only events with a magnitude larger than +2.5. +3. Results +Through different models introduced for earthquake network construction, +we used the simple model introduced by Abe-Suzuki [4]. Dividing the geograph- +ical region into small square cells and having seismic events data ordered by the +occurrence time, each square is regarded as one node if an earthquake with any +magnitude occurred, and two nodes with consecutive events are connected. +We also divided the seismic data of four years length into small time windows +in the following way; In the first step, we construct the Abe-Suzuki network for +1http://irsc.ut.ac.ir +2http://www.usgs.gov/ +3 + +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(a) +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(b) +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(c) +100 +101 +102 +k +10−3 +10−2 +10−1 +100 +p(k) +100 +101 +102 +k +10−3 +10−2 +10−1 +100 +100 +101 +102 +k +10−3 +10−2 +10−1 +100 +(d) +(e) +(f) +Figure 1: (a)-(c) The schematic representation of temporal earthquake networks of Iran for +three different time steps with time windows of (a) one month, (b) 10 months, and (c) 46 +months with filtered data magnitudes > 4.0, (d) to (f) are the degree distribution of the +networks for the three defined networks respectively for data with magnitudes > 2.5). +4 + +35◦N +40◦N +120◦W +(a) +35◦N +40◦N +120◦W +(b) +35◦N +40◦N +120◦W +(c) +100 +101 +102 +k +10−3 +10−2 +10−1 +100 +p(k) +100 +101 +102 +k +10−3 +10−2 +10−1 +100 +100 +101 +102 +k +10−3 +10−2 +10−1 +100 +(d) +(e) +(f) +Figure 2: (a)-(c) The schematic representation of temporal earthquake networks of California +for three different time steps with time windows of (a) 10 months, (b) 19 months, and (c) +46 months with filtered data magnitudes > 3.0, (d) to (f) are the degree distribution of the +networks for the three defined networks respectively for data with magnitudes > 2.5). +0 +10 +20 +30 +40 +50 +t(month) +102 +2 × 102 +Sw +0 +10 +20 +30 +40 +50 +t(month) +101 +102 +Sw +Iran +California +Figure 3: The variation of small-worldness in the scale of time windows computed for earth- +quake networks of Iran and California. +5 + +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(a) +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(b) +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(c) +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(d) +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(e) +25◦N +30◦N +35◦N +40◦N +50◦E +60◦E +(f) +Figure 4: Degree centrality and PageRank for earthquake network of Iran for time windows +of (a), (d) one month, (b), (e) 10 months, and (c), (f) 46 months respectively. +6 + +35◦N +40◦N +120◦W +(a) +35◦N +40◦N +120◦W +(b) +35◦N +40◦N +120◦W +(c) +35◦N +40◦N +120◦W +(d) +35◦N +40◦N +120◦W +(e) +35◦N +40◦N +120◦W +(f) +Figure 5: Degree centrality and PageRank for earthquake network of California for time +windows of (a), (d) 10 month, (b), (e) 19 months, and (c), (f) 46 months respectively. +7 + +the data for the length of one month and study the characteristics of interest. +Then, we added the data from the second month to the previous one and re- +constructed the network. The process of adding data by time windows of the +size of one month continues until the whole 48 months of data are covered. +The schematic representation of the temporal network construction is plotted +in Fig. 1 (a)-(c) for Iran and Fig. 2 (a)-(c) for California. Fig. 1(a) and 2(a) +belong to the data of the length of one month for Iran and ten months for Cal- +ifornia. As the number of events is low, having a sparse network is predictable. +The second Fig. 1(b) and 2(b) is in the middle time when the network is not +sparse as the first month and is not too connected as the last, and the Fig. 1(c) +and 2(c) represent the networks for the whole four years data which have a very +dense connection. +The second step would be building the adjacency matrix A for facilitating +the analysis; aij = 1 if nodes i and j are connected, and 0 otherwise. In the +network definitions, the degree of the node is the number of connections a node +could have and is calculated from the adjacency matrix ki = � +j aij. The degree +distribution of the earthquake network of different regions is power law [3, 5, 23]. +To check the validity of this characteristic, for each of the above mentioned +networks (Fig. 1 and 2), we plot the degree distribution Fig. 1(d) and 2(d). +One could see that no matter the time length, we would have approximately +the power-law distribution. +The other famous characteristic of earthquakes is being small-world [4, 23]. +In a small-world network with N nodes and M links, the value of the shortest +path is similar to the random network with the same number of nodes and links, +while the clustering coefficient has a higher value. The clustering coefficient of a +node i is the fraction of connection existing among its nearest neighbor nodes to +the maximum number of possible links among them. The clustering coefficient +of the network would be the average clustering of all nodes: +Ci = +1 +ki(ki − 1) +� +j,k +aijajkaki , C = 1 +N +N +� +i=1 +Ci +(1) +where N is the total number of nodes in the network. +In other words, the +8 + +clustering coefficient is the probability of the tendency of the nodes in the graph +to cluster together and has a value 0 ≤ C ≤ 1. On the other hand, the shortest +path is the minimum path length needed to traverse to get from one node to +the other. The average over all nodes would result in the shortest path of the +network: +L = +1 +N(N − 1) +� +i,j=1,N;i̸=j +dij +(2) +in which dij is the minimum length of the path between two nodes of i and j. +By having the clustering coefficient and shortest path of the network, Humphries +et al. [17] introduced a small-worldness metric defined with the averaged clus- +tering coefficient and path length relative to these metrics for random networks. +This metric helps to provide an overview of connectivity in the entire network: +Sw = C/Crand +L/Lrand +(3) +Crand and Lrand are the values obtained for random networks by randomizing +the connections of each earthquake network by keeping the same number of +nodes and links. +The variation of Sw by time (in the scale of the length of the month) is +shown in Fig. 3. One could see that this value is small for the first months +of consideration. +It starts to increase until a threshold and gets stationery +later. This behavior could emphasize that until a specific time window, the +variation of the parameters is high. The fluctuations disappear while a person +considers a large enough time window, and the system gets stationary. The +geographical region under consideration and frequency of the seismic event could +result in observing different values. This value for Iran’s data is approximately +ten months, while for California it is around 19 months. +To clarify the importance of having the minimum time window, we calculate +two of the most important centralities in the concept of earthquake networks and +compare them in three different time windows. Looking through the literature, +one could find different parameters to calculate the centrality of nodes in the +seismic networks. The simplest and most common centrality that uses the local +9 + +structure around the nodes is the degree centrality. In Fig. 4 and 5 (a)-(c), +we plotted the degree centrality for three different time scales as the following: +4(a) and 5(a) are for the time window of length 1 month for Iran and 10 months +for California. Near the time window of the threshold, we selected the network +with the length of 10 months of data for Iran (Fig. 4(b)) and 19 months for +California (Fig.5(b)). And Fig. 4(c) and 5(c) belong to the largest time window +(48 months). +The second famous centrality in the concept of earthquake network is PageR- +ank [12, 28]. PageRank is an algorithm used to assess the ranks of nodes in a +network based on their connections’ levels used in the Google search engine for +ranking web pages for the first time [10]. PageRank explained through the ran- +dom walk. The random walker starts from one node and selects the next one +randomly. In this definition, PageRank of node i is the asymptotic probability +that the walker meets the node. One could infer that the possibility of reaching +one important node is higher than the unimportant ones. This centrality is an +iterative procedure in which the PageRank of nodes depends on all its neighbors’ +PageRanks. The following equation describes such a random walking procedure: +PRi = d +N + (1 − d) +� +j∈Bi +PRj +Kout +j +(4) +in which PRi is the PageRank of node i, Bi is the set of nearest neighbors of +node i, and kout is the out-degree of each node. d is a fixed value (0.15) defined +as the probability of jumping to any vertex. Fig. 4 and 5 (d)-(f) are representing +the PageRank of the networks for the three different time windows. Taking into +account both above-introduced centralities, one could see that in the small-time +windows, we could not find enough information about the central locations of +the regions as it should. If we increase the length of the time window up to +the threshold, the results capture the same central regions as the largest time +window. +10 + +4. Conclusion +Recently, different models proposed to study the earthquake phenomena +to explore the features of this harmful disaster. +Although this phenomenon +is very complex from a fault and inside earth interactions point of view, it is +possible to study it with the complex network with the minimum information: +geographical location, time, and magnitude. Among the most famous models +proposed, Abe-Suzuki and visibility models, scientists were trying to improve +the model’s performances. The main idea that got most of the attention from +those studying was how they could introduce the best minimum geographical +cell size. +Here, we proposed the temporal earthquake network construction for cap- +turing another essential factor of network analysis, the best time window size. +We start with constructing an earthquake network in windows of the month +length and adding data with a length of one month in each step. We used the +most straightforward model introduced by Abe-Suzuki to build our networks. +For each constructed network, the small-worldness is evaluated. Studying how +this parameter changes by increasing the time window, we could verify the min- +imum length of time window needed. This value is smaller than its value in the +threshold time window and gets stationary by enlarging the time lengths. The +time threshold differs for disparate geographical regions as the construction of +the earth is different. One point of these differences appears in the frequency +of the events on the same time scale. Then, it is a delicate factor to study +the minimum and efficient time window size for different geographical regions +before the rest of the analysis to ensure obtaining the best results. 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[2019], ‘Pagerank: An +alarming index of probable earthquake occurrence’, Chaos 29(6), 063114. +14 + +Conflict of interests +Dear Prof. lakshmi somasundaram, +The authors declare that they have no known competing financial interests or +personal relationships that could have appeared to influence the work reported in this +paper. +Nastaran Lotfi, +On behalf of the authors, +Instituto de Ciências Matemáticas e de Computação (ICMC), +Universidade de São Paulo (USP), +São Carlos, SP, Brasil. + diff --git a/h9A0T4oBgHgl3EQfIP-u/content/tmp_files/load_file.txt b/h9A0T4oBgHgl3EQfIP-u/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a491574b33b5d1d7e78a1f5df64569c55cffb55 --- /dev/null +++ b/h9A0T4oBgHgl3EQfIP-u/content/tmp_files/load_file.txt @@ -0,0 +1,303 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf,len=302 +page_content='The earthquake network: the best time scale for network construction Nastaran Lotfia,∗ a Instituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo, Caixa Postal 668, 13560-970 S˜ao Carlos, SP, Brazil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Abstract Scientists mapped the seismic time series into networks by considering the geo- graphical location of events as nodes and establishing links between the nodes with different rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Applying the successive defined laws to construct the net- works of seismic data, a variety of features of earthquake networks are de- tected (scale-free and small-world structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Network construction models had changed in detail to optimize the performance of the verification of the minimum geographical size defined for the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In all the studies, people try to use large data sets like years of data to ensure their results are good enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In this work, by proposing the temporal network construction and employing the small-worldness property for data from Iran and California, we could achieve the minimum time scale needed for the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' We verified the importance of this scale by analyzing two significant centrality measures (degree centrality and PageRank) introduced in the concept of earthquake network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Keywords: Complex Networks, Temporal networks, earthquake networks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' introduction An earthquake is a sudden motion of a fault that releases an enormous amount of energy and is considered a complex spatiotemporal phenomenon oc- curring in the earth’s crust [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Transferring the stress of the movement of one fault to the others results in triggering subsequent events [19, 9, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Omori ∗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='lotfi@icmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='br Preprint submitted to Journal of Acta Geophysica Templates January 6, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='02073v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='geo-ph] 5 Jan 2023 law [26] and Gutenberg-Richter law [16] are empirical laws to characterize the Temporal pattern of aftershocks, frequency, and magnitude, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Be- sides the visible properties, complex interaction exists in the internal of the seismic system [8, 7, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' While seismicity is assumed to be a complex phenomenon, the network ap- proach offers a powerful tool for analyzing the dynamic structures of it [3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Over the last decade, different models proposed to construct the earthquake net- work [4, 20, 27, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In the simple but basic model introduced by Abe-Suzuki [4], the geographical region is divided into small square (cubic) cells, and seismic events with time sequences get connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Later, Lacasa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' [20] proposed a model to construct the network with a visibility graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' They converted the time series into a graph by inheriting the properties of the series in its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' They explored periodicity, fractality, chaoticity, and non-linearity of the seismic time series [21, 22, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Rezaie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' [27] introduced the hybrid model, which inherits the bases of the Abe-Suzuki model mixed with a visibility graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' To better capture the evolution of the earthquake network through time, a mul- tiplex network was employed [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Analyzing the seismic data with a network approach through different models helped reveal many features of the seismic activity just by knowing the basic information of magnitude, time of occurrence, and the location of seismic events [7, 3, 23, 2, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' It had been verified that the the earthquake networks that constructed from the seismic data taken from Cal- ifornia and Japan [3, 5, 4], Iran [23, 24], Chile [1] , Greece [11], and Italy [27] are scale-free and small-world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Most recent works focused on improving the proposed models to capture the best minimum resolution of the cell size needed for network construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' It means the cell size should be smaller than the specified limit to be trustable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The main question is how we ensure that the time window, in the scale of dates, months, or years, is large enough for constructing the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In all the studies done till now, scientists considered the time on such a big scale of years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' And the concept of the minimum necessary time window for achieving the best results are missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In this work, we employ the definition of temporal 2 network construction and capture the lowest time window essential for network construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' We found that depending on the region of consideration, the value of the time window threshold would change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' We verified the trustiness of this time window size by analyzing two important centrality parameters, degree centrality, and PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' If the time window is small, we miss the information in centrality, and if it is bigger than the threshold, we do not gain extra knowledge than in the threshold time region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In section 2, we provide information about the data sets we employ, and Section 3 is devoted to our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Database We applied our model for the latest four years of data, 01 Jan 2018 to 31 Dec 2021, for Iran in the range of 24N − 44N latitude and 40E–62E longitude with 14062 total events obtained from Iranian Seismological Center1, and California in the range of 32N–42N latitude and 114W–124W longitude with 7575 total events gained from the Northern California Earthquake Catalog2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In both of the considered data sets, we examined only events with a magnitude larger than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Results Through different models introduced for earthquake network construction, we used the simple model introduced by Abe-Suzuki [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Dividing the geograph- ical region into small square cells and having seismic events data ordered by the occurrence time, each square is regarded as one node if an earthquake with any magnitude occurred, and two nodes with consecutive events are connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' We also divided the seismic data of four years length into small time windows in the following way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In the first step, we construct the Abe-Suzuki network for 1http://irsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='ir 2http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='usgs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='gov/ 3 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (a) 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (b) 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (c) 100 101 102 k 10−3 10−2 10−1 100 p(k) 100 101 102 k 10−3 10−2 10−1 100 100 101 102 k 10−3 10−2 10−1 100 (d) (e) (f) Figure 1: (a)-(c) The schematic representation of temporal earthquake networks of Iran for three different time steps with time windows of (a) one month, (b) 10 months, and (c) 46 months with filtered data magnitudes > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='0, (d) to (f) are the degree distribution of the networks for the three defined networks respectively for data with magnitudes > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 4 35◦N 40◦N 120◦W (a) 35◦N 40◦N 120◦W (b) 35◦N 40◦N 120◦W (c) 100 101 102 k 10−3 10−2 10−1 100 p(k) 100 101 102 k 10−3 10−2 10−1 100 100 101 102 k 10−3 10−2 10−1 100 (d) (e) (f) Figure 2: (a)-(c) The schematic representation of temporal earthquake networks of California for three different time steps with time windows of (a) 10 months, (b) 19 months, and (c) 46 months with filtered data magnitudes > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='0, (d) to (f) are the degree distribution of the networks for the three defined networks respectively for data with magnitudes > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 0 10 20 30 40 50 t(month) 102 2 × 102 Sw 0 10 20 30 40 50 t(month) 101 102 Sw Iran California Figure 3: The variation of small-worldness in the scale of time windows computed for earth- quake networks of Iran and California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 5 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (a) 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (b) 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (c) 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (d) 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (e) 25◦N 30◦N 35◦N 40◦N 50◦E 60◦E (f) Figure 4: Degree centrality and PageRank for earthquake network of Iran for time windows of (a), (d) one month, (b), (e) 10 months, and (c), (f) 46 months respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 6 35◦N 40◦N 120◦W (a) 35◦N 40◦N 120◦W (b) 35◦N 40◦N 120◦W (c) 35◦N 40◦N 120◦W (d) 35◦N 40◦N 120◦W (e) 35◦N 40◦N 120◦W (f) Figure 5: Degree centrality and PageRank for earthquake network of California for time windows of (a), (d) 10 month, (b), (e) 19 months, and (c), (f) 46 months respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 7 the data for the length of one month and study the characteristics of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Then, we added the data from the second month to the previous one and re- constructed the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The process of adding data by time windows of the size of one month continues until the whole 48 months of data are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The schematic representation of the temporal network construction is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 1 (a)-(c) for Iran and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 2 (a)-(c) for California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 1(a) and 2(a) belong to the data of the length of one month for Iran and ten months for Cal- ifornia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' As the number of events is low, having a sparse network is predictable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The second Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 1(b) and 2(b) is in the middle time when the network is not sparse as the first month and is not too connected as the last, and the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 1(c) and 2(c) represent the networks for the whole four years data which have a very dense connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The second step would be building the adjacency matrix A for facilitating the analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' aij = 1 if nodes i and j are connected, and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In the network definitions, the degree of the node is the number of connections a node could have and is calculated from the adjacency matrix ki = � j aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The degree distribution of the earthquake network of different regions is power law [3, 5, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' To check the validity of this characteristic, for each of the above mentioned networks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 1 and 2), we plot the degree distribution Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 1(d) and 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' One could see that no matter the time length, we would have approximately the power-law distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The other famous characteristic of earthquakes is being small-world [4, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In a small-world network with N nodes and M links, the value of the shortest path is similar to the random network with the same number of nodes and links, while the clustering coefficient has a higher value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The clustering coefficient of a node i is the fraction of connection existing among its nearest neighbor nodes to the maximum number of possible links among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The clustering coefficient of the network would be the average clustering of all nodes: Ci = 1 ki(ki − 1) � j,k aijajkaki , C = 1 N N � i=1 Ci (1) where N is the total number of nodes in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In other words, the 8 clustering coefficient is the probability of the tendency of the nodes in the graph to cluster together and has a value 0 ≤ C ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' On the other hand, the shortest path is the minimum path length needed to traverse to get from one node to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The average over all nodes would result in the shortest path of the network: L = 1 N(N − 1) � i,j=1,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='i̸=j dij (2) in which dij is the minimum length of the path between two nodes of i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' By having the clustering coefficient and shortest path of the network, Humphries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' [17] introduced a small-worldness metric defined with the averaged clus- tering coefficient and path length relative to these metrics for random networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' This metric helps to provide an overview of connectivity in the entire network: Sw = C/Crand L/Lrand (3) Crand and Lrand are the values obtained for random networks by randomizing the connections of each earthquake network by keeping the same number of nodes and links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The variation of Sw by time (in the scale of the length of the month) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' One could see that this value is small for the first months of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' It starts to increase until a threshold and gets stationery later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' This behavior could emphasize that until a specific time window, the variation of the parameters is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The fluctuations disappear while a person considers a large enough time window, and the system gets stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The geographical region under consideration and frequency of the seismic event could result in observing different values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' This value for Iran’s data is approximately ten months, while for California it is around 19 months.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' To clarify the importance of having the minimum time window, we calculate two of the most important centralities in the concept of earthquake networks and compare them in three different time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Looking through the literature, one could find different parameters to calculate the centrality of nodes in the seismic networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The simplest and most common centrality that uses the local 9 structure around the nodes is the degree centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 4 and 5 (a)-(c), we plotted the degree centrality for three different time scales as the following: 4(a) and 5(a) are for the time window of length 1 month for Iran and 10 months for California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Near the time window of the threshold, we selected the network with the length of 10 months of data for Iran (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 4(b)) and 19 months for California (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' And Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 4(c) and 5(c) belong to the largest time window (48 months).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The second famous centrality in the concept of earthquake network is PageR- ank [12, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' PageRank is an algorithm used to assess the ranks of nodes in a network based on their connections’ levels used in the Google search engine for ranking web pages for the first time [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' PageRank explained through the ran- dom walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The random walker starts from one node and selects the next one randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' In this definition, PageRank of node i is the asymptotic probability that the walker meets the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' One could infer that the possibility of reaching one important node is higher than the unimportant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' This centrality is an iterative procedure in which the PageRank of nodes depends on all its neighbors’ PageRanks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The following equation describes such a random walking procedure: PRi = d N + (1 − d) � j∈Bi PRj Kout j (4) in which PRi is the PageRank of node i, Bi is the set of nearest neighbors of node i, and kout is the out-degree of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' d is a fixed value (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content='15) defined as the probability of jumping to any vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 4 and 5 (d)-(f) are representing the PageRank of the networks for the three different time windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Taking into account both above-introduced centralities, one could see that in the small-time windows, we could not find enough information about the central locations of the regions as it should.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' If we increase the length of the time window up to the threshold, the results capture the same central regions as the largest time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Conclusion Recently, different models proposed to study the earthquake phenomena to explore the features of this harmful disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Although this phenomenon is very complex from a fault and inside earth interactions point of view, it is possible to study it with the complex network with the minimum information: geographical location, time, and magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Among the most famous models proposed, Abe-Suzuki and visibility models, scientists were trying to improve the model’s performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The main idea that got most of the attention from those studying was how they could introduce the best minimum geographical cell size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Here, we proposed the temporal earthquake network construction for cap- turing another essential factor of network analysis, the best time window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' We start with constructing an earthquake network in windows of the month length and adding data with a length of one month in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' We used the most straightforward model introduced by Abe-Suzuki to build our networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' For each constructed network, the small-worldness is evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Studying how this parameter changes by increasing the time window, we could verify the min- imum length of time window needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' This value is smaller than its value in the threshold time window and gets stationary by enlarging the time lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' The time threshold differs for disparate geographical regions as the construction of the earth is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' One point of these differences appears in the frequency of the events on the same time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Then, it is a delicate factor to study the minimum and efficient time window size for different geographical regions before the rest of the analysis to ensure obtaining the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' By consid- ering two famous centralities measures in the concept of earthquake networks (degree centrality, and PageRank), we show that if this size is smaller than the threshold, we will miss the information we should have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' If the time window is too large, it doesn’t provide extra information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Acknowledgments N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Lotfi is thankful to the FAPESP (grant with number 2020/08359-1) for the support given to this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' References [1] Abe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=', Past´en, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=', Mu˜noz, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' and Suzuki, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' [2011], ‘Universalities of earthquake-network characteristics’, Chinese Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Bull 56(34), 3697–3701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' [2] Abe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=', Past´en, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' and Suzuki, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' [2011], ‘Finite data-size scaling of clus- tering in earthquake networks’, Physica A 390(7), 1343–1349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' 14 Conflict of interests Dear Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' lakshmi somasundaram, The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} +page_content=' Nastaran Lotfi, On behalf of the authors, Instituto de Ciências Matemáticas e de Computação (ICMC), Universidade de São Paulo (USP), São Carlos, SP, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/h9A0T4oBgHgl3EQfIP-u/content/2301.02073v1.pdf'} diff --git a/jNFPT4oBgHgl3EQf0zUa/vector_store/index.faiss b/jNFPT4oBgHgl3EQf0zUa/vector_store/index.faiss new file 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/dev/null +++ b/k9AzT4oBgHgl3EQfNfvd/content/tmp_files/2301.01151v1.pdf.txt @@ -0,0 +1,1380 @@ +MNRAS 000, 1–12 (2022) +Preprint 4 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Constraining the Minimum Halo Mass that Supports Water Formation in +a CCSN Remnant +Christopher T. D. Jessop,1★ +1Institute of Cosmology and Gravitation, 1-8 Burnaby Rd, Portsmouth, PO1 3FX, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We present a simulation probing the formation of water in the remnant of low-mass Population III supernovae in a cosmological +minihalo, and provide a tentative lower mass limit on host minihaloes that can recollapse on a short enough timescale and +efficiently mix metals at high densities. We start from cosmological initial conditions and end the simulation when the central +density undergoes catastrophic recollapse, whereby the water abundance is reported. During the Population III stars lifetime, +the minihalo (M= 5 × 105 M⊙) becomes blown out, and consequently the faint supernova explosion (ESN = 5 × 1050 ergs) is +completely unconfined to the virial radius of the minihalo. At the end of the simulation there is no significant water formation +anywhere throughout the remnant, and the central recollapsing region is inefficient at incorporating the first metals into itself, +remaining at low metallicity. The majority of metals are ejected from the core via bipolar outflow into the void and reach a peak +metallicity of Z ∼ 10−6 Z⊙ at very low densities. The mass of the minihalo is low enough such that the recollapse timescale +is unreasonable for this configuration to be the primary avenue of water formation in the early universe. We also provide a +comparison with a regular CCSN (ESN = 1051 ergs) and find the same effect, but amplified. As such, we can suggest that the +minimum minihalo mass required for a confined explosion, and therefore the possibility of water formation is at least 106 M⊙ +and the chemo-thermal evolution of a supernova remnant is more sensitive to the mass of the host minihalo than the mass of the +Population III star residing within it. +Key words: hydrodynamics – stars: Population III – astrochemistry – HII regions – stars: low-mass +1 INTRODUCTION +The emergence of life in the universe is one of the most fundamental +questions that arises to some degree within most areas and disciplines +of science. A question of such complexity and depth cannot possibly +be answered in a single paper, however, beginning to understand +the cosmological conditions that are sufficient or even promote the +inception of life is just one way we can begin to partly understand +and answer this question. +Water is an essential ingredient for life as we know it (Kasting +2012), which is why an understanding of early-universe chemistry +with an emphasis on numerous life-giving molecules, in particular +water, is a crucial first step to take. There is currently very little in +the way of hydrodynamical simulations or first principle calculations +when it comes to modelling early universe water formation. +Cosmological simulations on the collapse of primordial molecular +clouds (Gnedin & Ostriker 1997) suggest that the first generation of +stars, known as Population III stars (Pop III henceforth) may have +had extremely large masses, anywhere within the range of 10 M⊙ to +> 1000 M⊙. This first generation of stars are characterised by their +extremely low metallicity, of at most 10−6 𝑍⊙ due to the availability +of predominantly hydrogen and helium at their formation. Formation +pathways for Pop III stars can be categorised into high and low +★ E-mail: christopher.jessop@port.ac.uk (KTS) +mass regimes. Firstly, metal-free molecular gas clouds cool very +inefficiently and may result in Jeans-mass fragments on the order of +1000 M⊙ (Barkana & Loeb 2001; Bromm & Larson 2004; Bromm +et al. 2009). This leads to the possibility of either the formation +of a free-growing dense core due to unopposed accretion from its +host halo (Abel et al. 2002; Yoshida et al. 2006) or the formation +of a secondary core (Turk et al. 2009). Additionally, there exists a +prediction for lower-mass Pop III star formation (< 10 M⊙) under +photodissociative feedback in systems where a minihalo forms at later +redshifts, whereby the accretion rate of emerging stellar systems is +seen to be a lot lower than stellar systems within minihaloes that have +formed at earlier redshifts (Stacy & Bromm 2014). +Big Bang Nucleosynthesis allows only the production of the light- +est elements, namely isotopes of hydrogen and helium, with trace +amounts of lithium (Copi et al. 1995). These elements underwent +gravitational collapse and formed the first cosmological minihaloes, +inside of which Pop III star formation occurred to mark the end of the +cosmological dark ages. These stars are thought to be the first great +nucleosyntethic engines in the universe, expelling vast quantities of +heavier elements outwards and enriching both their own host mini- +halo, and other surrounding minihaloes. Depending on the mass of a +Pop III star at the end of its lifetime, it is signified by a supernova that +takes one of two types. Pop III stars in the mass range of 10 − 25 M⊙ +explode as Type-II Core-Collapse Supernovæ (CCSN) with a corre- +sponding energy ESN = 0.5 × 1051 ergs and leave behind a compact +© 2022 The Authors +arXiv:2301.01151v1 [astro-ph.GA] 3 Jan 2023 + +2 +C. T. D. Jessop +remnant. At the higher end of the Pop III mass spectrum, stars within +the mass range of 140 − 260 M⊙ end their lives as Pair-Instability +Supernovæ (PISN) releasing an order of magnitude greater energy, +and is completely disrupted leaving behind no remnant (Heger et al. +2003). The most massive stars at the top of this range, i.e. 250 M⊙ +and above, undergo complete photodisintegration. This reaction ab- +sorbs excess energy before runaway collapse leads to a hypernova, +before collapsing as a black hole (Fryer et al. 2001). Both CCSN +and PISN forge the heavier elements crucial for complex molecule +formation in their core, such as oxygen. Chemical enrichment can be +categorised into 3 distinct methods depending on the mechanisms at +play. Primarily, metals can be incorporated into massive haloes via a +hierarchical structure formation process. Greif et al. (2007) explore +this scenario where they explode a 200 M⊙ PISN, which expels an +interior, metal enriched, bubble that expands adiabatically into the +void and inter-galactic medium (IGM) through cavities created by +the supernova shock. It is concluded that a dark matter halo with a +virialised mass > 108 M⊙ is required to recollect the shocked gas +and allow mixing to occur. The second method of metal enrichment +of a halo can be referred to as the IE mechanism. This mechanism +refers to the way enriched material falls back into the host halo after +a period of recovery, and is applicable to describe the sequence of +metal enrichment only when the supernova progenitor is on the order +of tens of solar masses. The reason for this is to ensure that the host +halo has not been blown out as described previously, and so that +the HII (Singly ionised Hydrogen) region is relatively confined to +well within the virial radius. It is reported that a moderately massed +progenitor, and by extension, a moderate energy supernova, initially +is only able to photoionise a small region within its host halo, i.e. a +trapped HII region. Following the instability created due to the su- +pernova explosion, ejection material begins to fall back to the central +region, reaching a steady accretion rate after approximately 5 Myr, +and less than half of the ejecta is able to escape the virial radius. +(Ritter et al. 2012, 2016). A relatively new concept for metal enrich- +ment was explored to explain the observations of the most metal-poor +stars in a study that looked to describe the chemical abundance pat- +terns of Carbon Enhanced Metal Poor (CEMP) stars, in particular, +one with an Iron-to-Hydrogen ratio, [Fe/H] = −5.54 (Norris et al. +2013). Minihaloes that lie external to a Pop III stars host minihalo +can undergo external enrichment (EE), whereby enriched supernova +material escapes the host minihalo and impacts these nearby mini- +haloes, potentially paving a way for the next generation of Population +II (Pop II) star formation as long as these externally located mini- +haloes have not yet formed stars of their own. The pioneering study +to look specifically at the EE mechanism, concludes that turbulence +from the virialisation of the impacted minihalo is able to incorpo- +rate enriched material into a 0.01 pc radius to a uniform metallicty +Z ∼ 2 × 10−5 Z⊙ (Smith et al. 2015). +The possibility of early-universe water formation has not been vig- +orously tested until relatively recently. Bialy et al. (2015) demonstrate +that in the low-metallicity environments (𝑍 = 10−3 𝑍⊙) typical of the +metal enrichment epoch, significant quantities of water vapour could +have been present, using idealised one-dimensional calculations in a +system of partially shielded gas. Water formation in the present day +is dominated by reactions that occur on the surface of dust grains, for +example through the hydrogenation of oxygen (van Dishoeck 2014). +Due to the lack of metals that constitute dust grains in the early uni- +verse, water formation on dust is not a reasonable avenue to explore +at these higher redshifts, and we therefore must look at alternative +formation pathways. At temperatures ≥ 300 K, water is able to form +via gas-phase neutral-neutral reactions. Hydroxyl (OH) is initially +formed when oxygen reacts with H2, which in turn reacts with H2 +forming water. +O + H2 → OH + H +(1) +OH + H2 → H2O + H +(2) +The conditions for this formation pathway are thought to be typical +of shocks, where most of the oxygen can be driven into water where +the gas has heated to higher temperatures (Draine et al. 1983). To +summarise what is required within a region for water formation to +occur in the primordial universe, there needs to be a region of warm, +dense gas at a relatively high metallicity (there needs to be sufficiently +enough oxygen) +2 METHOD +We run our simulations using the Enzo v2.5 simulation code (O’shea +et al. 2005; Bryan et al. 2014). Enzo is an open-source, Eulerian, +adaptive mesh refinement (AMR) code that solves N-body dark mat- +ter dynamics. The equations of hydrodynamics are solved using a +second-order piecewise parabolic method (PPM), fully detailed in +Colella & Woodward (1984). Conventionally, Enzo uses a fixed +value for the adiabatic index 𝛾 = 5/3 if the selected hydro method +is PPM. However, at very low metallicities 𝑍⊙ < 10−3 and at densi- +ties nH ≥ 108 cm−3, the adiabatic index approaches 𝛾 = 7/5 as the +gas becomes fully molecular. Therefore, the hydrodynamics solver is +extended to consider a variable adiabatic index. +2.1 Chemistry - Heating and Cooling +Enzo natively is limited to the primordial 12-species model: e−, H, +H+, He, He+, He++, H2, H+ +2, H−, D, D+, and HD. These species can +be solved either within Enzo’s internal chemistry solver or with the +external chemistry and radiative cooling package Grackle, which +can be used to advect and evolve non-equilibrium primordial chem- +istry (Smith et al. 2017). Metal cooling rates are interpolated from +tables using the CLOUDY (Smith et al. 2008) cooling photoioni- +sation code, which are four-dimensional tables that interpolate over +density, metallicity, electron fraction, and temperature. Any further +chemistry, such as molecular chemistry will require a significantly +expanded chemical network. A modified Grackle scheme has been +implemented (Chiaki & Wise 2018) and extended to include 49 re- +actions for the primordial species listed above, in addition to HeH+, +D−, and HD+ to create a 15-species primordial model. This model +includes various processes such as collisional ionisation and recom- +bination of H2, in addition to its formation and destruction pathways +from three-body reactions. Primordial heating and cooling processes +are included, such as gas heating. In the temperature regime T > 8000 +K, inverse Compton cooling, brehmsstrahlung, H and He species +transitional line cooling, and ionisation/recombination processes are +considered. For temperatures under 10000 K, the contribution of +molecular cooling from H2 and HD are calculated. The hydrogen +mass fraction remains constant at 𝑋𝐻 = 0.76, whereas the helium +fraction varies in the region surrounding the SN ejecta where it be- +comes enhanced. The effect of metals when first introduced from Pop +III stars have a profound impact on the evolution of a halo (Bromm +& Loeb 2003). Therefore to follow these effects in detail, the ex- +tra species and reactions are taken from Omukai et al. (2005) and +added to the extended version of Grackle. The additional species +MNRAS 000, 1–12 (2022) + +Forming the First Water +3 +being considered are: C, C+, O, O+, CH, CH+ +2, CO, OH, H2O, O2, +CO+, O+ +2, OH+, H2O+, H3O+, CO2, Si, SiO, and SiO2 for a total of +40 non-primordial reactions. The cooling due to fine-structure level +transitions for C, C+, and O are included, in addition to the rotational +transitions of H2O, OH, and CO by interpolating over pre-computed +tables. +2.2 Dust Treatment +Similarly to the metal species, the chemical network has been +extended to eight species of dust: both metallic silicon (Si) and +metallic iron (Fe), forstertite (Mg2SiO4), magnesia (MgO), enstatite +(MgSiO3), silica (SiO2), amorphous carbon (C), and troilite (FeS). +Smith et al. (2015) demonstrate the significance that dust has on the +thermal properties of a gas cloud. Dust can very effectively enhance +gas cooling which in turn initiates fragmentation. The thermal effects +of dust on a gas cloud can be discretised in four ways: +(i) Formation of H2 on dust grains (crucial to consider for water forma- +tion) +(ii) Gas cooling via thermal emission from dust grains +(iii) Accretion of metals in the gas-phase onto dust grain surfaces +(iv) Creates a continuum opacity +A detailed description of the formulation for each rate in 2.2 is +provided in Chiaki et al. (2014). Each grain species has its rates +tabulated in its own table, and these rates are interpolated from the +gas temperature T, density 𝜌, density of a grain species 𝑖(𝜌𝑖), and +the density of the species corresponding key element 𝑋(𝜌𝑋) at each +timestep for a fluid cell. The continuum opacity is approximated as +𝜏cont = (𝜅p𝜌 + � +𝑖 𝜅𝑖𝜌𝑖)𝑙jeans, where 𝜅p is the Planck mean opacity +of primordial gas, 𝜅𝑖 is the mean opacity of a given grain species 𝑖, +𝜌𝑖 is the mass density of again, a grain species 𝑖, and the 𝑙jeans term +is the Jeans length, which is used as a shielding length. A diffusion +approximation that effects the continuum cooling rate processes such +as dust thermal emission and collisionally induced excitation are +reduced by a factor 𝛽cont = min{1, 𝜏−2 +cont}. +2.3 Simulation Setup +We initialise the primary isolated minihalo runs using the MU- +SIC initial conditions generator, an algorithm that generates multi- +scale initial conditions with multiple levels of refinement for cos- +mological simulations (Hahn & Abel 2011). We initialise a 0.25 +Mpc/h comoving box using the Planck 2016 cosmological parame- +ters [Ω𝑚 = 0.3089, Ω𝜆 = 0.6911, Ω𝑏 = 0.04860, 𝐻0 = 67.74, 𝜎8 = +0.8159, 𝑛𝑠 = 0.9667] (Ade et al. 2016). For our exploratory run we +use a resolution of 2563 corresponding to a dark matter mass res- +olution of 67.27 M⊙, with a refinement level 𝑙 = 1 from 𝑧 = 200 +to 𝑧 = 15 to find a minihalo with a total mass of 106 M⊙ using the +ROCKSTAR halo finder (Behroozi et al. 2012). Simulations are then +set up using the same random seed generators and initial conditions +as the exploratory run, centred around the 106 M⊙ minihalo, and +placing a nested grid with a resolution of 1024 zones. This target +minihalo at SF will be approximately 105 M⊙ and be classed as a +low-mass minihalo. The simulation is started from 𝑧 = 200 with the +added chemistry, radiative feedback and star formation mechanisms +described above, with data being output at every time step, initially +in code units of 5.0. Adaptive mesh refinement of cells occurs by a +factor of 2 when the following criteria are met: +(i) Gas mass is > 4 × ¯𝑀baryon × 2−𝑙×0.2, where ¯𝑀baryon is the +average gas mass in a cell and l is the refinement level +(ii) The dark matter in a cell is > 4 × 𝑀initial, where 𝑀initial is the +initial mass of a cell +(iii) Local Jeans length is < 16 cells wide +The reason for employing such criteria is as follows. Firstly, we +have a negative exponent in the baryon density function allowing a +greater rate of refinement with increasing levels of adaptive mesh re- +finement, ensuring super-Lagrangian behaviour (O’Shea & Norman +2007). The local Jeans length cell width constraint ensures that the +Truelove criterion, which states that Jeans length has to be resolved +by a minimum of 4 cells on each axis, is satisfied (Truelove et al. +1997). This ensures that there is no occurrence of artificial fragmen- +tation. A refinement level 𝑙 = 15 is set which is reached before the +formation of the first Pop III star in our simulation, and eventually +gives the required resolution to adequately show metal mixing and +halo enrichment from SNe and consequent water formation. Analysis +of all data and all images created from the raw data is done through +the YT project, a community developed, open source astrophysical +analysis and visualization toolkit developed in Python (Turk et al. +2010). +2.4 Star Formation +We model star formation and the resulting feedback of Pop III stars +using the Cen & Ostriker (1992) algorithm extension (Wise & Abel +2008) built into Enzo. This extension inserts a star particle into a +grid cell when the following criteria are met: +(i) An overdensity ≥ 106 cm−3 +(ii) A converging gas flow/velocity field, i.e. (∇ · vgas < 0) +(iii) A molecular hydrogen (H2) fraction > 5 × 10−4 +(iv) The metallicity is less than some critical metallicity value +(𝑍 < 6 × 10−8) +(v) The cooling time is less than the dynamical time (Tcool < 𝑡dyn) +These conditions are thought to be typical of metal-free clouds +undergoing collapse approximately 10 Myr before the birth of a +Pop III star (Abel et al. 2002). The criteria regarding the molecular +hydrogen fraction are required to limit star formation to molecular +hydrogen clouds with large rates of H2 formation relative to H2 +radiative dissociation. Once the conditions are met, a star particle +will then form 10 Myr later, whereby the mass of the star is uniformly +removed from a sphere containing twice the stellar mass, centred on +the star particle. The Pop III star mass is sampled from the IMF +𝑓 (log𝑀Pop III) = 𝑀−1exp +� +− +� 𝑀char +𝑀Pop III +�1.6� +, +(3) +where Mchar = 20 𝑀⊙, which is the characteristic mass of Pop III +stars. The radiation field evolution from point sources, as in the case +of Pop III stars, can be accurately calculated using adaptive ray tracing +(Abel et al. 2007), utilising the MORAY radiation field solver (Wise +& Abel 2011). Each individual star particle is treated as a point source +of ionising radiation, where the photons in each ray are equal to one +another. Furthermore, the sum of the photons in the initial rays are +equal to the stellar luminosity. On incidence with a higher resolution +AMR grid, the primary ray is split into 4 daughter rays if and when +the associated solid angle 𝜃 > 20% of the area of the cell. At the end +of the stars lifetime, Tlife, the supernova energy, ESN = 0.5 × 1051 +ergs is uniformly injected into a sphere with a radius of 10 pc. The +metals corresponding to that of a progenitor are likewise assumed +to be uniformly mixed within the ejecta. At a radius of 7.5 pc, a +contact discontinuity (CD) is placed which contains all metals from +MNRAS 000, 1–12 (2022) + +4 +C. T. D. Jessop +5 +10 +15 +20 +25 +30 +35 +Atomic Number [A] +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +Metal Ejecta [M ] +C +O +MgSi +Fe +Al +S +CCSN Abundance Patterns (13 M ) +Figure 1. Metal ejecta in solar masses of the major elements for the 13 M⊙ +faint-SN model used in this simulation. +the explosion. Only when the shock undergoes Rayleigh-Taylor (RT) +instabilities can the metals break through the CD. +2.5 Supernovæ Yields +The metal and dust abundances, as well as the dust size distribution +are derived from Umeda & Nomoto (2002); Nozawa et al. (2007) +for our Pop III progenitor. The model considers SN yields for the +following core-collapse masses: 13, 20, 25, and 30 M⊙. In addition to +the pair-instability masses: 170 and 200 M⊙. For our purposes, only +the 13 M⊙ model is selected and explodes with energy ESN = 0.5 × +1051 ergs and outputs Mmet = 0.097 M⊙ of metals. This represents +a ‘subdued’ SNe, where fallback onto the core is significant and +dampens the effect of the explosion. Pop III metal abundance differ +somewhat to the present-time ISM, specifically they show an alpha- +species enhancement relative to that of the solar abundance ratio of +[𝛼/Fe] ≃ 0.4. Note that in the case of a 13 M⊙ CCSN, magnesium +([Mg/Fe] = −0.23) and oxygen ([O/Fe] = −0.15) abundances are +relatively depleted with respect to the solar ratio, whilst sulphur +([S/Fe] = 0.17) and silicon ([Si/Fe] = 0.11) are enhanced. +3 RESULTS +The entire sequence of minihalo collapse, Pop III star formation and +HII region generation, supernova explosion feedback, and finally re- +collapse are followed in detail. Fig. 2 display slice plots of density, +temperature, metallicity, and HII abundance centred on the coor- +dinates of SF. Each row of the figure corresponds to a significant +period in the evolution of the system. Firstly, the state of the mini- +halo immediately prior to SF, after virialisation and gas collapse via +HII cooling has created conditions congruent to SF (Fig. 3 Top left), +shortly followed by SF and evolution itself (Fig. 3 Top right). The +creation of the HII region resultant of the Pop III star is followed, +which includes breakout regions that extend past 1 kpc, in addition to +a D-type ionisation front (I-front). After the lifetime of the star Tlife, +has elapsed the CCSN occurs, whereby the first source of metals and +dust are produced and expelled into the immediate environment (Fig. +3 Middle left). +3.1 Star Formation and Feedback +The top row of Fig. 2 displays the state of the minihalo immediately +prior to SF. Gas has accreted onto the minihalo and collapsed to +densities around nH = 1 cm−3, at which point the cooling process is +dominated by H2 through rotational and vibrational transitions (ro- +vibrational lines) until the temperature decreases to 200 K, and the +density increases to nH ∼ 103 cm−3. At this point, the ro-vibrational +levels of H2 are fully populated at their equilibrium levels and an +apparent hydrostatic equilibrium is reached. Additionally, the cool- +ing rate becomes independent of density. Early simulations for the +collapse of primordial minihaloes (Gnedin & Ostriker 1997) sug- +gest that neutral, metal-free pristine gas cools extremely inefficiently, +and therefore result in Jeans-mass scale fragments that could reach +> 1000M⊙ (Omukai 2000; Barkana & Loeb 2001; Bromm & Lar- +son 2004; Bromm et al. 2009). In this scenario, the fragment results +in a Pop III star formed with a mass 𝑀PopIII = 13 M⊙ and corre- +sponding lifetime Tlife = 12 Myr that forms at redshift 𝑧 = 19.02. +The Pop III star mass is sampled from the IMF given by eq. 3 in +this instance. Immediately prior to SF, the mass of the minihalo is +𝑀halo = 4.8×105 M⊙. During its main-sequence lifetime (MSL), the +star particle isotropically emits ionising and H2 dissociating Lyman- +Werner photons (LW) with rates of 𝑄(H) = 1.09 × 1048 s−1 and +𝑄(LW) = 1.54 × 1048 s−1 respectively (Schaerer 2002). The ionisa- +tion rates are given by +𝑄(H) = 1043.61+4.9𝑥−0.83𝑥2, +(4) +𝑄(H0) = 1042.51+5.69𝑥−1.01𝑥2, +(5) +𝑄(He) = 1026.71+18.14𝑥−3.58𝑥2, +(6) +𝑄(LW) = 1044.03+4.59𝑥−0.77𝑥2, +(7) +where 𝑥 = log10(Mstar) and 𝑥2 = 𝑥2. The effects of the progenitor +star are visible in both Fig. 2 and 3. Specifically, Fig. 3 shows the +formation of the low density, diffuse region surrounding the star, +where the temperature increases to T> 105 K, at a minimum den- +sity of n𝐻 ∼ 10−3 cm−1. Within this region, hydrogen has become +fully ionised, and this becomes clearly visible in Fig. 2, as it presents +itself as the ‘butterfly’ structure. The compact D-type I-front is vis- +ible within the density slice plot extending to approximately 100 +pc in each direction although the structure remains inhomogeneous, +whereas it remains somewhat hidden by the breakout regions in the +temperature slice plot. Along the ‘principal’ axis (main horizontal +structure when looking at Fig. 2), the density remains relatively high +(n𝐻 ∼ 10−1 cm−1) throughout as the edges of the minihalo meets +the filamentary structure, compared to the regions above and be- +low the minihalo that borders the void (n𝐻 ≤ 10−2 cm−1). As the +main structure of the minihalo meets the voids, Lyman continuum +leakage (LyC leakage) extends the HII region, as LyC photons are +able to escape through holes of low column density channels per the +ionisation-bound LyC leakage mechanism, however in the direction +of the halo-filament intersection, the I-front is bound within the halo. +Therefore, in this instance a mostly confined HII region has been +produced. A secondary clump within the host minihalo located in +close proximity to the SF region is discernible in the first 3 rows of +Fig. 2. This clump is only marginally disrupted by the Pop III star +but overall retains its high density throughout the stars MSL. As the +Pop III star resides at the lower-end of the Pop III mass range, a +MNRAS 000, 1–12 (2022) + +Forming the First Water +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Density (cm +3) +101 +102 +103 +104 +105 +Temperature (K) +10 +16 +10 +13 +10 +10 +10 +7 +10 +4 +10 +1 +Metallicity (Z ) +10 +6 +10 +4 +10 +2 +100 +102 +y(HII) +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Density (cm +3) +101 +102 +103 +104 +105 +Temperature (K) +10 +16 +10 +13 +10 +10 +10 +7 +10 +4 +10 +1 +Metallicity (Z ) +10 +6 +10 +4 +10 +2 +100 +102 +y(HII) +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Density (cm +3) +101 +102 +103 +104 +105 +Temperature (K) +10 +16 +10 +13 +10 +10 +10 +7 +10 +4 +10 +1 +Metallicity (Z ) +10 +6 +10 +4 +10 +2 +100 +102 +y(HII) +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Density (cm +3) +101 +102 +103 +104 +105 +Temperature (K) +10 +16 +10 +13 +10 +10 +10 +7 +10 +4 +10 +1 +Metallicity (Z ) +10 +6 +10 +4 +10 +2 +100 +102 +y(HII) +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Density (cm +3) +101 +102 +103 +104 +105 +Temperature (K) +10 +16 +10 +13 +10 +10 +10 +7 +10 +4 +10 +1 +Metallicity (Z ) +10 +6 +10 +4 +10 +2 +100 +102 +y(HII) +Figure 2. From left to right, slices of density, temperature, metallicity, and HII abundance centred on the coordinates of SF and SNe with a side length of 2 kpc +comoving. From top to bottom, immediately prior to SF (𝑧 = 19.10), SF (𝑧 = 18.53), 1 Myr post-SNe (𝑧 = 18.17), 18 Myr post-SNe (𝑧 = 17.41), and 112 Myr +post-SNe (𝑧 = 13.34). +MNRAS 000, 1–12 (2022) + +中726 +C. T. D. Jessop +10-2 +10-1 +100 +101 +102 +Density (cm−3) +101 +102 +103 +Temperature (K) +Immediately prior to SF (z = 20.21) +10-1 +100 +101 +102 +103 +104 +105 +Cell Mass (M ⊙ ) +10-3 +10-2 +10-1 +100 +101 +102 +103 +Density (cm−3) +101 +102 +103 +104 +105 +Temperature (K) +Immediately prior to SNe (z = 18.53) +10-4 +10-2 +100 +102 +104 +Cell Mass (M ⊙ ) +10-3 +10-2 +10-1 +100 +101 +102 +Density (cm−3) +101 +102 +103 +104 +105 +106 +107 +Temperature (K) +Post SNe (z = 18.17) +10-3 +10-1 +101 +103 +105 +Cell Mass (M ⊙ ) +10-4 +10-3 +10-2 +10-1 +100 +101 +102 +Density (cm−3) +101 +102 +103 +104 +105 +Temperature (K) +18 Myr Post SNe (z = 17.14) +10-2 +100 +102 +104 +Cell Mass (M ⊙ ) +10-4 +10-3 +10-2 +10-1 +100 +101 +Density (cm−3) +101 +102 +103 +104 +Temperature (K) +112 Myr Post SNe (z = 13.34) +10-2 +10-1 +100 +101 +102 +103 +104 +Cell Mass (M ⊙ ) +10-3 +10-1 +101 +103 +105 +107 +Density (cm−3) +101 +102 +103 +104 +Temperature (K) +192 Myr Post SNe (z = 11.35) +10-5 +10-3 +10-1 +101 +103 +105 +107 +Cell Mass (M ⊙ ) +Figure 3. Two-dimensional phase plots displaying temperature and density at the same epochs as Fig. 2. +MNRAS 000, 1–12 (2022) + +Forming the First Water +7 +weaker I-front is driven through the minihalo that is unable to revert +to R-type for most of the MSL. The bound component of the HII +region evolves as follows. If it is assumed that the Franco density +column is static, 𝑤 = 1 and follows the power-law 𝑟−𝑤 (Franco et al. +1990), then the radius of the I-front evolves to good approximation +as +𝑅(𝑡) = 𝑅𝑠 +� +1 + W0 +� +−exp +� +− +𝑟𝑐𝑡 +𝑅𝑠𝑡rec,core +��� +(8) +where 𝑅𝑠 = L/K is the Strömgren radius (Strömgren 1939) and +W0(𝑥) is the principal branch of the Lambert function W. Addition- +ally, 𝐿 ≡ �𝑁𝛾/(4𝜋𝑛𝑐𝑟𝑐) and 𝐾 ≡ 𝑛𝑐𝑟𝑐𝐶𝛼𝛽, where 𝐶 is the clumping +factor. Whalen et al. (2004) demonstrate the solution of W0(𝑥) for a +bound I-front. The mass of the minihalo allows for an HII region to +form in this manner, consistent with findings of Whalen et al. (2008). +If the minihalo mass was greater, then outward radiative pressure +would not be able to compete with the downward baryonic pressure +from the larger potential well and the resultant HII region would +have a negligible radius (too small to be resolved). This minihalo +represents an intermediate case between a trapped and unconfined +HII region, where it remains confined for the majority of the MSL, +breaking through the neutral shell as an R-type front only at the +end of the MSL. The pressure within the HII region evacuates the +central region driving the temperature past T= 20, 000 K whilst si- +multaneously reducing the density to an average of n𝐻 ∼ 0.1 cm−1. +The density reduction is enhanced at the north and south poles of +the minihalo where the structure borders on the void. As the I-front +transitions to a R-type front at these locations, the evolution of the +I-front may reduce to a simple unbound solution +𝑟𝐼 = 𝑟0(1 + 2𝑡/𝑡rec,core)1/2, +(9) +where Trec,core is the recombination time in the core if it is assumed +that the the photon output per unit time, �𝑁𝛾 = 16𝜋𝑟3 +0𝑛2 +0𝐶𝛼𝛽/3 and +𝑟(0) = 𝑟0, n0 is the gas number density at the characteristic radius, 𝑟0 +(Mellema et al. 2006). Additionally, this holds only for static Franco +density fields, i.e. 𝑤 = 2, but provides a good approximation. +The MSL of the Pop III star ends after Tlife has elapsed, whereby +the SN explosion occurs after removal of the star particle, carrying +with it enriched material that propagates throughout the diffuse HII +region. The energy and metal contents of the SN are deposited into +a region 10 pc in radius centred on the coordinates of the star parti- +cle. Approximately 4.5 Myr after the SN explosion, the shock front +impacts the secondary clump and violently disrupts it, reducing its +density. The clump falls into the recollapsing region of the SN rem- +nant after a period of brief recovery ∼ 54 Myr after the explosion. +The SN ejecta exhibits distinct bipolar outflow from the central re- +gion, as it is forced through regions of lower density characterised +by HII breakout that is observed at the north and south poles of Fig. +2. Finally after 190 Myr, the halo begins to recover and initiate the +process of recollapse. The selected host minihalo represents a unique +scenario where the binding energy of the halo and the energy of the +SNe are finely balanced. The minihalo of mass Mhalo = 4.8 × 105 +M⊙ at 𝑧 = 19.10 yields a gravitational binding energy +𝐸bind = −3 +5 +𝐺M2 +halo +𝑅vir += 7.45 × 1050, ergs +(10) +where 𝐺 is the gravitational constant and Rvir is the virial radius +given as approximately 145 pc. The SNe energy 𝐸SN = 5 × 1050 +ergs. Therefore in this instance 𝐸bind ≲ 𝐸SN, although they can be +approximated as the same with regards to the method of enrichment. +In theory, this presents an opportunity for both EE and/or internal +enrichment (IE) being realised, or at the very least increasing the +timescale for recollapse to occur significantly, when compared to +the same configuration with a higher mass halo (𝐸bind ≫ 𝐸SN) or +a more energetic SNe (𝐸bind ≪ 𝐸SN). The first case would be a +strong candidate for complete IE, whereas the second for complete +EE rather than a mixture of the two. +3.2 Long-term Remnant Evolution +The unconfined nature of the HII region gives rise to the visually +distinct bipolar outflow of metal, forced through the poles where +the minihalo borders the void and extending further than 2 kpc/h +after 112 Myr. The consequences of such a blowout are extremely +significant for the subsequent chemical enrichment of the remnant. +The metallicity within the central 100 pc region is radially reduced +almost uniformly from 𝑍 = 10−15 𝑍⊙ right at the coordinates of the +dead star, as the over dense region lying in close proximity falls into +this region. In fact, the bulk of the metal ejecta is transported out of the +minihalo within the bipolar outflow into the void. Evident in Fig. 3.2, +the highest average metallicity at this time is located ∼ 600 pc away +in the extrema of the ejecta, and reaches 𝑍 > 10−6 𝑍⊙. Although the +metallicity is relatively high within these regions, the density is so +low that the primary reactions governing water formation and other +heavy molecule formation are effectively stifled. For any significant +quantities of water to form, a recollapsing region exceeding nH = 103 +cm−3 and enriched to a minimum metallicity (𝑍 ∼ 5 × 10−5𝑍⊙ for a +fiducial value) is required to initiate runaway recollapse. +Immediately prior to the recollapse of multiple small regions +throughout the remnant, the distribution of metals within this region +are visible in Fig. 5. Perhaps the most obvious feature within the +top series of slice plots is the distribution of higher water abundance +(although marginal) at the same position as the relatively higher +metallicities. Specifically, the darker orange/red of water abundance +matches the position of the brighter spots in the adjacent metallicity +plots, whereas the centre has effectively 0 water abundance denoted +by the dark blue, and is in the same location as the darker region +of metallicity. The region of high density that dominates the top left +plot is the consequence of the high density region that lay in close +proximity to the blast, falling into the central region 4.5 Myr after +the explosion. 192 Myr post SNe, the clump reaches a faux equi- +librium with the surrounding remnant, remaining in essentially the +same state from fall-in until the onset of recollapse. The series of +projection plots in the lower half of Fig. 5 gives a greater representa- +tion of the distribution of the dense knots that permeate the remnant. +Forming a filamentary structure with the aforementioned clump at +the centre extends horizontally across the figure, the collapsing re- +gions remain at low metallicity. The incorporation of metals into the +surrounding pristine minihalo structure via mixing has been studied +by Cen & Riquelme (2008), and in some instances 90 % of high +density gas inside the 106 M⊙ minihalo is enriched to only 3 % of +the surrounding metallicity by 𝑧 = 6 suggesting that metal-free Pop +III SF may be possible down to lower redshifts. In our minihalo, the +mass at SF is an order of magnitude lower and therefore the apparent +inability of metal-enriched material to mix with pristine gas may be +amplified, as evident in Fig. 5. +3.3 Water Formation in a Low-Mass Minihalo +In our simulation, there is negligible water formation in the remnant +of the SN after significant time (192 Myr) has passed. We define a +significant amount of time as the dynamical time when recollapse +would be expected in a halo of mass M ∼ 1 × 106 M⊙ which is +MNRAS 000, 1–12 (2022) + +8 +C. T. D. Jessop +100 +101 +102 +103 +Radius (pc) +10-17 +10-15 +10-13 +10-11 +10-9 +10-7 +10-5 +Zmet +100 +101 +102 +103 +Radius (pc) +10-16 +10-14 +10-12 +10-10 +10-8 +10-6 +10-4 +Zmet +Figure 4. Spherically averaged radial profile of the SN remnant in both simulations 112 Myr post-SNe displaying the metallicity weighted by cell mass as a +function of radius, centred on the coordinates of the SNe. Top: Main faint-CCSN simulation. Bottom: Normal CCSN simulation. The increase of metallicity +as the radius grows demonstrates the bulk of the metals being transported out of the central region. Note the CCSN plot that demonstrates a more consistent +distribution of metals due to the greater metal ejecta mass. +MNRAS 000, 1–12 (2022) + +Forming the First Water +9 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Density (cm +3) +101 +102 +103 +104 +105 +Temperature (K) +10 +20 +10 +16 +10 +12 +10 +8 +10 +4 +100 +Metallicity (Z ) +10 +19 +10 +18 +10 +17 +10 +16 +y(H2O) +10 +3 +10 +2 +10 +1 +100 +Density (cm +3) +101 +102 +103 +Temperature (K) +10 +16 +10 +14 +10 +12 +10 +10 +10 +8 +10 +6 +Metallicity (Z ) +10 +17 +y(H2O) +Figure 5. Top: Slices of density, temperature, metallicity, and water abundance centred on the coordinates of SF 192 Myr post-SNe. Bottom: Density weighted +projections centred on the same coordinates at the same epoch. Each figure has a side length of 4 kpc comoving. +approximately 80 Myr. There are regions within the remnant region +that have an elevated water abundance with respect to the background +value by a few orders of magnitude, indicating that the remnant is not +completely devoid of water. These values however, are at a maximum +𝑦_H2O ∼ 1015 and exceedingly low. The metallicity in these "high" +water abundance regions has a minimum value of Z = 10−9 Z⊙, +below this metallicity value the water abundance does not increase +further than its background initialisation values. Furthermore, the +high metallicity regions exist at significant distance from the central +overdense region. This suggests that a significant fraction of metal +would need to halt further outward expansion and fall back onto the +core before the potential for mixing could occur, let alone effective +mixing of metal with primordial gas. +A simulation was initialised with identical initial conditions to en- +sure the characteristics of the minihalo were unchanged. An identical +mass (13 M⊙) Pop III star was inserted at the same time described +above, only in this instance the SN model was altered to represent a +‘nominal’ CCSN, i.e. the energy was ESN = 1051 ergs and the SN +model outputs almost 10 times the metals at Mmet = 0.784 M⊙ as +the faint-SNe. The progenitor star evolves in an identical fashion and +so the initial conditions for the explosion to occur in are consistent +between the two simulations. After the SNe occurs, the dense clump +lying in close proximity is impacted by the shock wave and is partially +disrupted. The clump falls into the central region in a similar fashion +after a period of brief recovery. After 120 Myr, the same filamentary +structure that runs horizontally connecting regions of high density +becomes visible, with the pristine clump of gas forming the centre. +Fig. 6 displays the state of the remnant and clearly shows this struc- +ture forming. Although taken at a different epoch, comparing with +Fig. 5 the similarities are obvious. The bulk of the metals are trans- +ported away from the central region, and marginal water formation +occurs within the outer extremities of the remnant where the densities +are the lowest. In addition, the collapsing dense substructures that +make up the horizontal filament have the lowest water abundance +values throughout the remnant at y_H2O < 10−18, over two orders +of magnitude less than the lower density regions. +4 DISCUSSION AND CONCLUSIONS +Here we present the first results in a series where we attempt to find +the ideal conditions for water formation in the remnant of Pop III +supernovae. For the first time, we solve all the relevant metal chem- +istry that governs complex molecule formation and include a detailed +prescription for dust chemistry, in addition to mass dependant Pop +III SN yields in a self-consistent manner. In the target minihalo (105 +M⊙) initialised from cosmological initial conditions, pristine gas col- +lapses via H2 cooling and forms a 13 M⊙ Pop III star at 𝑧 = 19.02. +MNRAS 000, 1–12 (2022) + +10 +C. T. D. Jessop +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +102 +Density (cm +3) +101 +102 +103 +104 +105 +Temperature (K) +10 +20 +10 +16 +10 +12 +10 +8 +10 +4 +100 +Metallicity (Z ) +10 +19 +10 +18 +10 +17 +10 +16 +y(H2O) +10 +3 +10 +2 +10 +1 +100 +101 +Density (cm +3) +101 +102 +103 +Temperature (K) +10 +15 +10 +12 +10 +9 +10 +6 +Metallicity (Z ) +10 +18 +10 +17 +y(H2O) +Figure 6. Similar to Fig. 5, but for a normal CCSN. Top: Slices of density, temperature, metallicity, and water abundance centred on the coordinates of SF 120 +Myr post-SNe. Bottom: Density weighted projections centred on the same coordinates at the same epoch. Each figure has a side length of 4 kpc comoving. +We simulate the MSL of the star using the MORAY radiative transfer +algorithm. The Pop III star is modelled to have significant fallback +onto the core at the point of explosion, reducing the energy of the +explosion to 5 × 1050 ergs and the metal ejecta to just 0.097 M⊙. +The CCSN exhibits striking bipolar outflow from the central region +and forces the bulk of the metal ejecta into the void. The simulation +concludes when the dense substructures throughout the remnant be- +gin catastrophic recollapse. A summary of the primary findings from +this simulation are as follows: +(i) The low-mass of the minihalo allows for the HII region gener- +ated by the Pop III star to escape the virial radius near the end of the +stars lifetime, transitioning from a D-type to R-type in the direction +of the voids. +(ii) The star explodes as a CCSN and the metal ejecta is forced +through the regions where LyC leakage occurs, whilst remaining +confined where the minihalo primary structure meets the bordering +filaments. This creates a bipolar outflow of metals and efficiently +transports the metal ejecta out of the minihalo. +(iii) A dense, pristine clump of gas lies close to the SN and is +impacted 4.5 Myr later. This clump is only partially disrupted by the +shock and recovers quickly. The clump falls into the central region ∼ +54 Myr later. +(iv) The metallicity of the core reaches a minimum value of 10−14 +Z⊙ which increases to a maximum value of 10−6 Z⊙ approximately +900 pc from the core. This suggests firstly that the majority of metals +are transported away from the core in the explosion and secondly, that +the mixing of metals into the primordial material that the infalling +clump provides is very inefficient. +(v) 120 Myr post-SNe, the formation of a horizontal filament +permeated by areas of high density with the collapsing infalling +clump lying at the centre becomes distinctly visible. These areas +have a lower metallicity and also have a few orders of magnitude +lower water abundance than that of the surrounding higher metallicity +regions that exist at lower density. This again suggests that mixing is +extremely inefficient. +(vi) Water formation occurs within low density regions at the +outskirts of the remnant, although the abundance peaks no higher than +y_H2O = 10−15 which is almost negligible in the context of second +generation star and protoplanetary disk formation. The pristine clump +lying at the centre of the remnant loiters in a faux equilibrium, not +incorporating metals into itself and actually suppressing complex +chemical reactions. +(vii) An identical simulation up until SN was initialised where the +energy was doubled and the metal ejecta increased by a factor of 10. +The chemo-thermal evolution of the remnant was very similar, with +the majority of metal ejecta being forced into the void and the dense +clump falling into the central region. +MNRAS 000, 1–12 (2022) + +Forming the First Water +11 +100 +101 +102 +103 +Radius (pc) +10-17 +10-16 +Y H2O +100 +101 +102 +103 +Radius (pc) +10-11 +10-10 +Y Oi +100 +101 +102 +103 +Radius (pc) +10-17 +10-16 +10-15 +10-14 +10-13 +10-12 +10-11 +10-10 +Y Ci +100 +101 +102 +103 +Radius (pc) +10-16 +10-14 +10-12 +10-10 +10-8 +10-6 +Zmet +Figure 7. Spherically averages radial profile of the SN remnant 192 Myr post-SNe of the major species and metallicity. +The configuration of this minihalo mass containing within it two +overdense regions at SF time conspire to create a unique situation +where metal mixing in the centre of the remnant is inefficient, and +therefore complex chemical interactions are suppressed. As soon as +the HII region of a Pop III star is able to overcome the minihalo, any +SNe has the ability to blow the majority of its metal ejecta out into +the void. In this instance, we chose to model a faint-CCSN to give +the minihalo the greatest chance of recovery within a Hubble time as +this represents a scenario with the least energy output. Even so, the +halo is blown apart and it takes at least 190 Myr for some semblance +of recollapse to occur. If a minihalo had lay in close proximity to the +host minihalo, EE may be successful in creating the conditions for +water formation (Smith et al. 2015). As this was not the case here, +IE was the only avenue for water formation to occur, however this +never happened as the pristine clump fell onto the central region of +the remnant and diluted the metallicity. Since the majority of metals +were ejected past the virial radius, the presence of the clump may +only have enhanced the water formation rate (or lack thereof) that +we see in the core. Even when increasing the metal ejecta mass by a +factor of 10 from a normal CCSN, the explosion energy is doubled +and the result is the same (Fig. 6). The minihalo is blown out and +the majority of metals are transported into the void. For IE to be +viable and recollapse to occur on short timescales, the HII region +must be confined to the virial radius of the minihalo. This would +trap the SNe, which would reach equilibrium and initiate recollapse +earlier whilst retaining the bulk of the metal ejecta. Chiaki & Wise +(2018) demonstrate that a minihalo with mass 1.77 × 106 M⊙ is able +to confine a 13 M⊙ Pop III stars HII region, suggesting that a lower +limit can be placed on a minihalo that is able to self-enrich itself +via IE as ∼ 106 M⊙. The dynamics of each minihalo are unique, +and therefore this may not be so much a set rule for all minihaloes +at the lower end of the mass spectrum as a rule for this specific +configuration. What is clear however, is that as the minihalo mass +increases so does the likelihood that IE dominates and the chances +of water formation also increases. Future studies are required on the +matter, and will be able to probe what happens when the configuration +changes slightly. Increasing the mass of the minihalo at SF to the +upper half of 105 M⊙, or alternatively increasing the mass of the Pop +III progenitor to inject more metals into the surrounding area may +prove to have profound effects on the end state of the remnant and +the ability of the recollapsing clumps to mix metals into themselves. +Finding the correct balance between SN explosion energy, metal +ejecta mass, and the host minihalo mass may be the key to discovering +the configuration which promotes the formation of a water abundant +protoplanetary disk around a second-generation star, and therefore +the emergence of the first wet rocky planets. +ACKNOWLEDGEMENTS +C. T. D. Jessop was supported by STFC grant 18379. All numerical +simulations were performed on the Sciama HPC cluster, supported +by the University of Portsmouth and the Institute of Cosmology and +MNRAS 000, 1–12 (2022) + +12 +C. T. D. Jessop +Gravitation. 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F., 2014, Faraday Discuss., 168, 9–47 +This paper has been typeset from a TEX/LATEX file prepared by the author. +MNRAS 000, 1–12 (2022) + diff --git a/k9AzT4oBgHgl3EQfNfvd/content/tmp_files/load_file.txt b/k9AzT4oBgHgl3EQfNfvd/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c437fb6e3fa753a1387d4516357191b3501fd73e --- /dev/null +++ b/k9AzT4oBgHgl3EQfNfvd/content/tmp_files/load_file.txt @@ -0,0 +1,865 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf,len=864 +page_content='MNRAS 000, 1–12 (2022) Preprint 4 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='0 Constraining the Minimum Halo Mass that Supports Water Formation in a CCSN Remnant Christopher T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Jessop,1★ 1Institute of Cosmology and Gravitation, 1-8 Burnaby Rd, Portsmouth, PO1 3FX, UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present a simulation probing the formation of water in the remnant of low-mass Population III supernovae in a cosmological minihalo, and provide a tentative lower mass limit on host minihaloes that can recollapse on a short enough timescale and efficiently mix metals at high densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' We start from cosmological initial conditions and end the simulation when the central density undergoes catastrophic recollapse, whereby the water abundance is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' During the Population III stars lifetime, the minihalo (M= 5 × 105 M⊙) becomes blown out, and consequently the faint supernova explosion (ESN = 5 × 1050 ergs) is completely unconfined to the virial radius of the minihalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' At the end of the simulation there is no significant water formation anywhere throughout the remnant, and the central recollapsing region is inefficient at incorporating the first metals into itself, remaining at low metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The majority of metals are ejected from the core via bipolar outflow into the void and reach a peak metallicity of Z ∼ 10−6 Z⊙ at very low densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The mass of the minihalo is low enough such that the recollapse timescale is unreasonable for this configuration to be the primary avenue of water formation in the early universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' We also provide a comparison with a regular CCSN (ESN = 1051 ergs) and find the same effect, but amplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' As such, we can suggest that the minimum minihalo mass required for a confined explosion, and therefore the possibility of water formation is at least 106 M⊙ and the chemo-thermal evolution of a supernova remnant is more sensitive to the mass of the host minihalo than the mass of the Population III star residing within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Key words: hydrodynamics – stars: Population III – astrochemistry – HII regions – stars: low-mass 1 INTRODUCTION The emergence of life in the universe is one of the most fundamental questions that arises to some degree within most areas and disciplines of science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A question of such complexity and depth cannot possibly be answered in a single paper, however, beginning to understand the cosmological conditions that are sufficient or even promote the inception of life is just one way we can begin to partly understand and answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Water is an essential ingredient for life as we know it (Kasting 2012), which is why an understanding of early-universe chemistry with an emphasis on numerous life-giving molecules, in particular water, is a crucial first step to take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' There is currently very little in the way of hydrodynamical simulations or first principle calculations when it comes to modelling early universe water formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Cosmological simulations on the collapse of primordial molecular clouds (Gnedin & Ostriker 1997) suggest that the first generation of stars, known as Population III stars (Pop III henceforth) may have had extremely large masses, anywhere within the range of 10 M⊙ to > 1000 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This first generation of stars are characterised by their extremely low metallicity, of at most 10−6 𝑍⊙ due to the availability of predominantly hydrogen and helium at their formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Formation pathways for Pop III stars can be categorised into high and low ★ E-mail: christopher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='jessop@port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='uk (KTS) mass regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Firstly, metal-free molecular gas clouds cool very inefficiently and may result in Jeans-mass fragments on the order of 1000 M⊙ (Barkana & Loeb 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bromm & Larson 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bromm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This leads to the possibility of either the formation of a free-growing dense core due to unopposed accretion from its host halo (Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Yoshida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2006) or the formation of a secondary core (Turk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Additionally, there exists a prediction for lower-mass Pop III star formation (< 10 M⊙) under photodissociative feedback in systems where a minihalo forms at later redshifts, whereby the accretion rate of emerging stellar systems is seen to be a lot lower than stellar systems within minihaloes that have formed at earlier redshifts (Stacy & Bromm 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Big Bang Nucleosynthesis allows only the production of the light- est elements, namely isotopes of hydrogen and helium, with trace amounts of lithium (Copi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' These elements underwent gravitational collapse and formed the first cosmological minihaloes, inside of which Pop III star formation occurred to mark the end of the cosmological dark ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' These stars are thought to be the first great nucleosyntethic engines in the universe, expelling vast quantities of heavier elements outwards and enriching both their own host mini- halo, and other surrounding minihaloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Depending on the mass of a Pop III star at the end of its lifetime, it is signified by a supernova that takes one of two types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Pop III stars in the mass range of 10 − 25 M⊙ explode as Type-II Core-Collapse Supernovæ (CCSN) with a corre- sponding energy ESN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 × 1051 ergs and leave behind a compact © 2022 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='01151v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='GA] 3 Jan 2023 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Jessop remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' At the higher end of the Pop III mass spectrum, stars within the mass range of 140 − 260 M⊙ end their lives as Pair-Instability Supernovæ (PISN) releasing an order of magnitude greater energy, and is completely disrupted leaving behind no remnant (Heger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The most massive stars at the top of this range, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 250 M⊙ and above, undergo complete photodisintegration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This reaction ab- sorbs excess energy before runaway collapse leads to a hypernova, before collapsing as a black hole (Fryer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Both CCSN and PISN forge the heavier elements crucial for complex molecule formation in their core, such as oxygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Chemical enrichment can be categorised into 3 distinct methods depending on the mechanisms at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Primarily, metals can be incorporated into massive haloes via a hierarchical structure formation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Greif et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2007) explore this scenario where they explode a 200 M⊙ PISN, which expels an interior, metal enriched, bubble that expands adiabatically into the void and inter-galactic medium (IGM) through cavities created by the supernova shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' It is concluded that a dark matter halo with a virialised mass > 108 M⊙ is required to recollect the shocked gas and allow mixing to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The second method of metal enrichment of a halo can be referred to as the IE mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This mechanism refers to the way enriched material falls back into the host halo after a period of recovery, and is applicable to describe the sequence of metal enrichment only when the supernova progenitor is on the order of tens of solar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The reason for this is to ensure that the host halo has not been blown out as described previously, and so that the HII (Singly ionised Hydrogen) region is relatively confined to well within the virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' It is reported that a moderately massed progenitor, and by extension, a moderate energy supernova, initially is only able to photoionise a small region within its host halo, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' a trapped HII region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Following the instability created due to the su- pernova explosion, ejection material begins to fall back to the central region, reaching a steady accretion rate after approximately 5 Myr, and less than half of the ejecta is able to escape the virial radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2012, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A relatively new concept for metal enrich- ment was explored to explain the observations of the most metal-poor stars in a study that looked to describe the chemical abundance pat- terns of Carbon Enhanced Metal Poor (CEMP) stars, in particular, one with an Iron-to-Hydrogen ratio, [Fe/H] = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='54 (Norris et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Minihaloes that lie external to a Pop III stars host minihalo can undergo external enrichment (EE), whereby enriched supernova material escapes the host minihalo and impacts these nearby mini- haloes, potentially paving a way for the next generation of Population II (Pop II) star formation as long as these externally located mini- haloes have not yet formed stars of their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The pioneering study to look specifically at the EE mechanism, concludes that turbulence from the virialisation of the impacted minihalo is able to incorpo- rate enriched material into a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='01 pc radius to a uniform metallicty Z ∼ 2 × 10−5 Z⊙ (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The possibility of early-universe water formation has not been vig- orously tested until relatively recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bialy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2015) demonstrate that in the low-metallicity environments (𝑍 = 10−3 𝑍⊙) typical of the metal enrichment epoch, significant quantities of water vapour could have been present, using idealised one-dimensional calculations in a system of partially shielded gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Water formation in the present day is dominated by reactions that occur on the surface of dust grains, for example through the hydrogenation of oxygen (van Dishoeck 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Due to the lack of metals that constitute dust grains in the early uni- verse, water formation on dust is not a reasonable avenue to explore at these higher redshifts, and we therefore must look at alternative formation pathways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' At temperatures ≥ 300 K, water is able to form via gas-phase neutral-neutral reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Hydroxyl (OH) is initially formed when oxygen reacts with H2, which in turn reacts with H2 forming water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' O + H2 → OH + H (1) OH + H2 → H2O + H (2) The conditions for this formation pathway are thought to be typical of shocks, where most of the oxygen can be driven into water where the gas has heated to higher temperatures (Draine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' To summarise what is required within a region for water formation to occur in the primordial universe, there needs to be a region of warm, dense gas at a relatively high metallicity (there needs to be sufficiently enough oxygen) 2 METHOD We run our simulations using the Enzo v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 simulation code (O’shea et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bryan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Enzo is an open-source, Eulerian, adaptive mesh refinement (AMR) code that solves N-body dark mat- ter dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The equations of hydrodynamics are solved using a second-order piecewise parabolic method (PPM), fully detailed in Colella & Woodward (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Conventionally, Enzo uses a fixed value for the adiabatic index 𝛾 = 5/3 if the selected hydro method is PPM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' However, at very low metallicities 𝑍⊙ < 10−3 and at densi- ties nH ≥ 108 cm−3, the adiabatic index approaches 𝛾 = 7/5 as the gas becomes fully molecular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Therefore, the hydrodynamics solver is extended to consider a variable adiabatic index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='1 Chemistry - Heating and Cooling Enzo natively is limited to the primordial 12-species model: e−, H, H+, He, He+, He++, H2, H+ 2, H−, D, D+, and HD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' These species can be solved either within Enzo’s internal chemistry solver or with the external chemistry and radiative cooling package Grackle, which can be used to advect and evolve non-equilibrium primordial chem- istry (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Metal cooling rates are interpolated from tables using the CLOUDY (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2008) cooling photoioni- sation code, which are four-dimensional tables that interpolate over density, metallicity, electron fraction, and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Any further chemistry, such as molecular chemistry will require a significantly expanded chemical network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A modified Grackle scheme has been implemented (Chiaki & Wise 2018) and extended to include 49 re- actions for the primordial species listed above, in addition to HeH+, D−, and HD+ to create a 15-species primordial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This model includes various processes such as collisional ionisation and recom- bination of H2, in addition to its formation and destruction pathways from three-body reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Primordial heating and cooling processes are included, such as gas heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In the temperature regime T > 8000 K, inverse Compton cooling, brehmsstrahlung, H and He species transitional line cooling, and ionisation/recombination processes are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' For temperatures under 10000 K, the contribution of molecular cooling from H2 and HD are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The hydrogen mass fraction remains constant at 𝑋𝐻 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='76, whereas the helium fraction varies in the region surrounding the SN ejecta where it be- comes enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The effect of metals when first introduced from Pop III stars have a profound impact on the evolution of a halo (Bromm & Loeb 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Therefore to follow these effects in detail, the ex- tra species and reactions are taken from Omukai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2005) and added to the extended version of Grackle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The additional species MNRAS 000, 1–12 (2022) Forming the First Water 3 being considered are: C, C+, O, O+, CH, CH+ 2, CO, OH, H2O, O2, CO+, O+ 2, OH+, H2O+, H3O+, CO2, Si, SiO, and SiO2 for a total of 40 non-primordial reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The cooling due to fine-structure level transitions for C, C+, and O are included, in addition to the rotational transitions of H2O, OH, and CO by interpolating over pre-computed tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='2 Dust Treatment Similarly to the metal species, the chemical network has been extended to eight species of dust: both metallic silicon (Si) and metallic iron (Fe), forstertite (Mg2SiO4), magnesia (MgO), enstatite (MgSiO3), silica (SiO2), amorphous carbon (C), and troilite (FeS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2015) demonstrate the significance that dust has on the thermal properties of a gas cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Dust can very effectively enhance gas cooling which in turn initiates fragmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The thermal effects of dust on a gas cloud can be discretised in four ways: (i) Formation of H2 on dust grains (crucial to consider for water forma- tion) (ii) Gas cooling via thermal emission from dust grains (iii) Accretion of metals in the gas-phase onto dust grain surfaces (iv) Creates a continuum opacity A detailed description of the formulation for each rate in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='2 is provided in Chiaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Each grain species has its rates tabulated in its own table, and these rates are interpolated from the gas temperature T, density 𝜌, density of a grain species 𝑖(𝜌𝑖), and the density of the species corresponding key element 𝑋(𝜌𝑋) at each timestep for a fluid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The continuum opacity is approximated as 𝜏cont = (𝜅p𝜌 + � 𝑖 𝜅𝑖𝜌𝑖)𝑙jeans, where 𝜅p is the Planck mean opacity of primordial gas, 𝜅𝑖 is the mean opacity of a given grain species 𝑖, 𝜌𝑖 is the mass density of again, a grain species 𝑖, and the 𝑙jeans term is the Jeans length, which is used as a shielding length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A diffusion approximation that effects the continuum cooling rate processes such as dust thermal emission and collisionally induced excitation are reduced by a factor 𝛽cont = min{1, 𝜏−2 cont}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='3 Simulation Setup We initialise the primary isolated minihalo runs using the MU- SIC initial conditions generator, an algorithm that generates multi- scale initial conditions with multiple levels of refinement for cos- mological simulations (Hahn & Abel 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' We initialise a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='25 Mpc/h comoving box using the Planck 2016 cosmological parame- ters [Ω𝑚 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='3089, Ω𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='6911, Ω𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='04860, 𝐻0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='74, 𝜎8 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='8159, 𝑛𝑠 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='9667] (Ade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' For our exploratory run we use a resolution of 2563 corresponding to a dark matter mass res- olution of 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='27 M⊙, with a refinement level 𝑙 = 1 from 𝑧 = 200 to 𝑧 = 15 to find a minihalo with a total mass of 106 M⊙ using the ROCKSTAR halo finder (Behroozi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Simulations are then set up using the same random seed generators and initial conditions as the exploratory run, centred around the 106 M⊙ minihalo, and placing a nested grid with a resolution of 1024 zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This target minihalo at SF will be approximately 105 M⊙ and be classed as a low-mass minihalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The simulation is started from 𝑧 = 200 with the added chemistry, radiative feedback and star formation mechanisms described above, with data being output at every time step, initially in code units of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Adaptive mesh refinement of cells occurs by a factor of 2 when the following criteria are met: (i) Gas mass is > 4 × ¯𝑀baryon × 2−𝑙×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='2, where ¯𝑀baryon is the average gas mass in a cell and l is the refinement level (ii) The dark matter in a cell is > 4 × 𝑀initial, where 𝑀initial is the initial mass of a cell (iii) Local Jeans length is < 16 cells wide The reason for employing such criteria is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Firstly, we have a negative exponent in the baryon density function allowing a greater rate of refinement with increasing levels of adaptive mesh re- finement, ensuring super-Lagrangian behaviour (O’Shea & Norman 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The local Jeans length cell width constraint ensures that the Truelove criterion, which states that Jeans length has to be resolved by a minimum of 4 cells on each axis, is satisfied (Truelove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This ensures that there is no occurrence of artificial fragmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A refinement level 𝑙 = 15 is set which is reached before the formation of the first Pop III star in our simulation, and eventually gives the required resolution to adequately show metal mixing and halo enrichment from SNe and consequent water formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Analysis of all data and all images created from the raw data is done through the YT project, a community developed, open source astrophysical analysis and visualization toolkit developed in Python (Turk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='4 Star Formation We model star formation and the resulting feedback of Pop III stars using the Cen & Ostriker (1992) algorithm extension (Wise & Abel 2008) built into Enzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This extension inserts a star particle into a grid cell when the following criteria are met: (i) An overdensity ≥ 106 cm−3 (ii) A converging gas flow/velocity field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (∇ · vgas < 0) (iii) A molecular hydrogen (H2) fraction > 5 × 10−4 (iv) The metallicity is less than some critical metallicity value (𝑍 < 6 × 10−8) (v) The cooling time is less than the dynamical time (Tcool < 𝑡dyn) These conditions are thought to be typical of metal-free clouds undergoing collapse approximately 10 Myr before the birth of a Pop III star (Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The criteria regarding the molecular hydrogen fraction are required to limit star formation to molecular hydrogen clouds with large rates of H2 formation relative to H2 radiative dissociation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Once the conditions are met, a star particle will then form 10 Myr later, whereby the mass of the star is uniformly removed from a sphere containing twice the stellar mass, centred on the star particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The Pop III star mass is sampled from the IMF 𝑓 (log𝑀Pop III) = 𝑀−1exp � − � 𝑀char 𝑀Pop III �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='6� , (3) where Mchar = 20 𝑀⊙, which is the characteristic mass of Pop III stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The radiation field evolution from point sources, as in the case of Pop III stars, can be accurately calculated using adaptive ray tracing (Abel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2007), utilising the MORAY radiation field solver (Wise & Abel 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Each individual star particle is treated as a point source of ionising radiation, where the photons in each ray are equal to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Furthermore, the sum of the photons in the initial rays are equal to the stellar luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' On incidence with a higher resolution AMR grid, the primary ray is split into 4 daughter rays if and when the associated solid angle 𝜃 > 20% of the area of the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' At the end of the stars lifetime, Tlife, the supernova energy, ESN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 × 1051 ergs is uniformly injected into a sphere with a radius of 10 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The metals corresponding to that of a progenitor are likewise assumed to be uniformly mixed within the ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' At a radius of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 pc, a contact discontinuity (CD) is placed which contains all metals from MNRAS 000, 1–12 (2022) 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Jessop 5 10 15 20 25 30 35 Atomic Number [A] 10 5 10 4 10 3 10 2 10 1 Metal Ejecta [M ] C O MgSi Fe Al S CCSN Abundance Patterns (13 M ) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Metal ejecta in solar masses of the major elements for the 13 M⊙ faint-SN model used in this simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Only when the shock undergoes Rayleigh-Taylor (RT) instabilities can the metals break through the CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 Supernovæ Yields The metal and dust abundances, as well as the dust size distribution are derived from Umeda & Nomoto (2002);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Nozawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2007) for our Pop III progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The model considers SN yields for the following core-collapse masses: 13, 20, 25, and 30 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In addition to the pair-instability masses: 170 and 200 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' For our purposes, only the 13 M⊙ model is selected and explodes with energy ESN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 × 1051 ergs and outputs Mmet = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='097 M⊙ of metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This represents a ‘subdued’ SNe, where fallback onto the core is significant and dampens the effect of the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Pop III metal abundance differ somewhat to the present-time ISM, specifically they show an alpha- species enhancement relative to that of the solar abundance ratio of [𝛼/Fe] ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Note that in the case of a 13 M⊙ CCSN, magnesium ([Mg/Fe] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='23) and oxygen ([O/Fe] = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='15) abundances are relatively depleted with respect to the solar ratio, whilst sulphur ([S/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='17) and silicon ([Si/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='11) are enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3 RESULTS The entire sequence of minihalo collapse, Pop III star formation and HII region generation, supernova explosion feedback, and finally re- collapse are followed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2 display slice plots of density, temperature, metallicity, and HII abundance centred on the coor- dinates of SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Each row of the figure corresponds to a significant period in the evolution of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Firstly, the state of the mini- halo immediately prior to SF, after virialisation and gas collapse via HII cooling has created conditions congruent to SF (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3 Top left), shortly followed by SF and evolution itself (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3 Top right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The creation of the HII region resultant of the Pop III star is followed, which includes breakout regions that extend past 1 kpc, in addition to a D-type ionisation front (I-front).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' After the lifetime of the star Tlife, has elapsed the CCSN occurs, whereby the first source of metals and dust are produced and expelled into the immediate environment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3 Middle left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='1 Star Formation and Feedback The top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2 displays the state of the minihalo immediately prior to SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Gas has accreted onto the minihalo and collapsed to densities around nH = 1 cm−3, at which point the cooling process is dominated by H2 through rotational and vibrational transitions (ro- vibrational lines) until the temperature decreases to 200 K, and the density increases to nH ∼ 103 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' At this point, the ro-vibrational levels of H2 are fully populated at their equilibrium levels and an apparent hydrostatic equilibrium is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Additionally, the cool- ing rate becomes independent of density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Early simulations for the collapse of primordial minihaloes (Gnedin & Ostriker 1997) sug- gest that neutral, metal-free pristine gas cools extremely inefficiently, and therefore result in Jeans-mass scale fragments that could reach > 1000M⊙ (Omukai 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Barkana & Loeb 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bromm & Lar- son 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bromm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In this scenario, the fragment results in a Pop III star formed with a mass 𝑀PopIII = 13 M⊙ and corre- sponding lifetime Tlife = 12 Myr that forms at redshift 𝑧 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The Pop III star mass is sampled from the IMF given by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3 in this instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Immediately prior to SF, the mass of the minihalo is 𝑀halo = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='8×105 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' During its main-sequence lifetime (MSL), the star particle isotropically emits ionising and H2 dissociating Lyman- Werner photons (LW) with rates of 𝑄(H) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='09 × 1048 s−1 and 𝑄(LW) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='54 × 1048 s−1 respectively (Schaerer 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The ionisa- tion rates are given by 𝑄(H) = 1043.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='61+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='9𝑥−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='83𝑥2, (4) 𝑄(H0) = 1042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='51+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='69𝑥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='01𝑥2, (5) 𝑄(He) = 1026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='71+18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='14𝑥−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='58𝑥2, (6) 𝑄(LW) = 1044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='03+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='59𝑥−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='77𝑥2, (7) where 𝑥 = log10(Mstar) and 𝑥2 = 𝑥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The effects of the progenitor star are visible in both Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Specifically, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3 shows the formation of the low density, diffuse region surrounding the star, where the temperature increases to T> 105 K, at a minimum den- sity of n𝐻 ∼ 10−3 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Within this region, hydrogen has become fully ionised, and this becomes clearly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2, as it presents itself as the ‘butterfly’ structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The compact D-type I-front is vis- ible within the density slice plot extending to approximately 100 pc in each direction although the structure remains inhomogeneous, whereas it remains somewhat hidden by the breakout regions in the temperature slice plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Along the ‘principal’ axis (main horizontal structure when looking at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2), the density remains relatively high (n𝐻 ∼ 10−1 cm−1) throughout as the edges of the minihalo meets the filamentary structure, compared to the regions above and be- low the minihalo that borders the void (n𝐻 ≤ 10−2 cm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' As the main structure of the minihalo meets the voids, Lyman continuum leakage (LyC leakage) extends the HII region, as LyC photons are able to escape through holes of low column density channels per the ionisation-bound LyC leakage mechanism, however in the direction of the halo-filament intersection, the I-front is bound within the halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Therefore, in this instance a mostly confined HII region has been produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A secondary clump within the host minihalo located in close proximity to the SF region is discernible in the first 3 rows of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This clump is only marginally disrupted by the Pop III star but overall retains its high density throughout the stars MSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' As the Pop III star resides at the lower-end of the Pop III mass range,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' a MNRAS 000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 1–12 (2022) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='Forming the First Water ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='Density (cm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='101 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='104 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='103 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='104 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='105 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='Temperature (K) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='Metallicity (Z ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='y(HII) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' From left to right, slices of density, temperature, metallicity, and HII abundance centred on the coordinates of SF and SNe with a side length of 2 kpc comoving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' From top to bottom, immediately prior to SF (𝑧 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10), SF (𝑧 = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='53), 1 Myr post-SNe (𝑧 = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='17), 18 Myr post-SNe (𝑧 = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='41), and 112 Myr post-SNe (𝑧 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) 中726 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Jessop 10-2 10-1 100 101 102 Density (cm−3) 101 102 103 Temperature (K) Immediately prior to SF (z = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='21) 10-1 100 101 102 103 104 105 Cell Mass (M ⊙ ) 10-3 10-2 10-1 100 101 102 103 Density (cm−3) 101 102 103 104 105 Temperature (K) Immediately prior to SNe (z = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='53) 10-4 10-2 100 102 104 Cell Mass (M ⊙ ) 10-3 10-2 10-1 100 101 102 Density (cm−3) 101 102 103 104 105 106 107 Temperature (K) Post SNe (z = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='17) 10-3 10-1 101 103 105 Cell Mass (M ⊙ ) 10-4 10-3 10-2 10-1 100 101 102 Density (cm−3) 101 102 103 104 105 Temperature (K) 18 Myr Post SNe (z = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='14) 10-2 100 102 104 Cell Mass (M ⊙ ) 10-4 10-3 10-2 10-1 100 101 Density (cm−3) 101 102 103 104 Temperature (K) 112 Myr Post SNe (z = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='34) 10-2 10-1 100 101 102 103 104 Cell Mass (M ⊙ ) 10-3 10-1 101 103 105 107 Density (cm−3) 101 102 103 104 Temperature (K) 192 Myr Post SNe (z = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='35) 10-5 10-3 10-1 101 103 105 107 Cell Mass (M ⊙ ) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Two-dimensional phase plots displaying temperature and density at the same epochs as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) Forming the First Water 7 weaker I-front is driven through the minihalo that is unable to revert to R-type for most of the MSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The bound component of the HII region evolves as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' If it is assumed that the Franco density column is static, 𝑤 = 1 and follows the power-law 𝑟−𝑤 (Franco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 1990), then the radius of the I-front evolves to good approximation as 𝑅(𝑡) = 𝑅𝑠 � 1 + W0 � −exp � − 𝑟𝑐𝑡 𝑅𝑠𝑡rec,core ��� (8) where 𝑅𝑠 = L/K is the Strömgren radius (Strömgren 1939) and W0(𝑥) is the principal branch of the Lambert function W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Addition- ally, 𝐿 ≡ �𝑁𝛾/(4𝜋𝑛𝑐𝑟𝑐) and 𝐾 ≡ 𝑛𝑐𝑟𝑐𝐶𝛼𝛽, where 𝐶 is the clumping factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Whalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2004) demonstrate the solution of W0(𝑥) for a bound I-front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The mass of the minihalo allows for an HII region to form in this manner, consistent with findings of Whalen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' If the minihalo mass was greater, then outward radiative pressure would not be able to compete with the downward baryonic pressure from the larger potential well and the resultant HII region would have a negligible radius (too small to be resolved).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This minihalo represents an intermediate case between a trapped and unconfined HII region, where it remains confined for the majority of the MSL, breaking through the neutral shell as an R-type front only at the end of the MSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The pressure within the HII region evacuates the central region driving the temperature past T= 20, 000 K whilst si- multaneously reducing the density to an average of n𝐻 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='1 cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The density reduction is enhanced at the north and south poles of the minihalo where the structure borders on the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' As the I-front transitions to a R-type front at these locations, the evolution of the I-front may reduce to a simple unbound solution 𝑟𝐼 = 𝑟0(1 + 2𝑡/𝑡rec,core)1/2, (9) where Trec,core is the recombination time in the core if it is assumed that the the photon output per unit time, �𝑁𝛾 = 16𝜋𝑟3 0𝑛2 0𝐶𝛼𝛽/3 and 𝑟(0) = 𝑟0, n0 is the gas number density at the characteristic radius, 𝑟0 (Mellema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Additionally, this holds only for static Franco density fields, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 𝑤 = 2, but provides a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The MSL of the Pop III star ends after Tlife has elapsed, whereby the SN explosion occurs after removal of the star particle, carrying with it enriched material that propagates throughout the diffuse HII region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The energy and metal contents of the SN are deposited into a region 10 pc in radius centred on the coordinates of the star parti- cle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 Myr after the SN explosion, the shock front impacts the secondary clump and violently disrupts it, reducing its density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The clump falls into the recollapsing region of the SN rem- nant after a period of brief recovery ∼ 54 Myr after the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The SN ejecta exhibits distinct bipolar outflow from the central re- gion, as it is forced through regions of lower density characterised by HII breakout that is observed at the north and south poles of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Finally after 190 Myr, the halo begins to recover and initiate the process of recollapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The selected host minihalo represents a unique scenario where the binding energy of the halo and the energy of the SNe are finely balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The minihalo of mass Mhalo = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='8 × 105 M⊙ at 𝑧 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='10 yields a gravitational binding energy 𝐸bind = −3 5 𝐺M2 halo 𝑅vir = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='45 × 1050, ergs (10) where 𝐺 is the gravitational constant and Rvir is the virial radius given as approximately 145 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The SNe energy 𝐸SN = 5 × 1050 ergs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Therefore in this instance 𝐸bind ≲ 𝐸SN, although they can be approximated as the same with regards to the method of enrichment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In theory, this presents an opportunity for both EE and/or internal enrichment (IE) being realised, or at the very least increasing the timescale for recollapse to occur significantly, when compared to the same configuration with a higher mass halo (𝐸bind ≫ 𝐸SN) or a more energetic SNe (𝐸bind ≪ 𝐸SN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The first case would be a strong candidate for complete IE, whereas the second for complete EE rather than a mixture of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='2 Long-term Remnant Evolution The unconfined nature of the HII region gives rise to the visually distinct bipolar outflow of metal, forced through the poles where the minihalo borders the void and extending further than 2 kpc/h after 112 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The consequences of such a blowout are extremely significant for the subsequent chemical enrichment of the remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The metallicity within the central 100 pc region is radially reduced almost uniformly from 𝑍 = 10−15 𝑍⊙ right at the coordinates of the dead star, as the over dense region lying in close proximity falls into this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In fact, the bulk of the metal ejecta is transported out of the minihalo within the bipolar outflow into the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='2, the highest average metallicity at this time is located ∼ 600 pc away in the extrema of the ejecta, and reaches 𝑍 > 10−6 𝑍⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Although the metallicity is relatively high within these regions, the density is so low that the primary reactions governing water formation and other heavy molecule formation are effectively stifled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' For any significant quantities of water to form, a recollapsing region exceeding nH = 103 cm−3 and enriched to a minimum metallicity (𝑍 ∼ 5 × 10−5𝑍⊙ for a fiducial value) is required to initiate runaway recollapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Immediately prior to the recollapse of multiple small regions throughout the remnant, the distribution of metals within this region are visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Perhaps the most obvious feature within the top series of slice plots is the distribution of higher water abundance (although marginal) at the same position as the relatively higher metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Specifically, the darker orange/red of water abundance matches the position of the brighter spots in the adjacent metallicity plots, whereas the centre has effectively 0 water abundance denoted by the dark blue, and is in the same location as the darker region of metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The region of high density that dominates the top left plot is the consequence of the high density region that lay in close proximity to the blast, falling into the central region 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 Myr after the explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 192 Myr post SNe, the clump reaches a faux equi- librium with the surrounding remnant, remaining in essentially the same state from fall-in until the onset of recollapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The series of projection plots in the lower half of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 5 gives a greater representa- tion of the distribution of the dense knots that permeate the remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Forming a filamentary structure with the aforementioned clump at the centre extends horizontally across the figure, the collapsing re- gions remain at low metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The incorporation of metals into the surrounding pristine minihalo structure via mixing has been studied by Cen & Riquelme (2008), and in some instances 90 % of high density gas inside the 106 M⊙ minihalo is enriched to only 3 % of the surrounding metallicity by 𝑧 = 6 suggesting that metal-free Pop III SF may be possible down to lower redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In our minihalo, the mass at SF is an order of magnitude lower and therefore the apparent inability of metal-enriched material to mix with pristine gas may be amplified, as evident in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='3 Water Formation in a Low-Mass Minihalo In our simulation, there is negligible water formation in the remnant of the SN after significant time (192 Myr) has passed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' We define a significant amount of time as the dynamical time when recollapse would be expected in a halo of mass M ∼ 1 × 106 M⊙ which is MNRAS 000, 1–12 (2022) 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Jessop 100 101 102 103 Radius (pc) 10-17 10-15 10-13 10-11 10-9 10-7 10-5 Zmet 100 101 102 103 Radius (pc) 10-16 10-14 10-12 10-10 10-8 10-6 10-4 Zmet Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Spherically averaged radial profile of the SN remnant in both simulations 112 Myr post-SNe displaying the metallicity weighted by cell mass as a function of radius, centred on the coordinates of the SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Top: Main faint-CCSN simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bottom: Normal CCSN simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The increase of metallicity as the radius grows demonstrates the bulk of the metals being transported out of the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Note the CCSN plot that demonstrates a more consistent distribution of metals due to the greater metal ejecta mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) Forming the First Water 9 10 4 10 3 10 2 10 1 100 101 102 Density (cm 3) 101 102 103 104 105 Temperature (K) 10 20 10 16 10 12 10 8 10 4 100 Metallicity (Z ) 10 19 10 18 10 17 10 16 y(H2O) 10 3 10 2 10 1 100 Density (cm 3) 101 102 103 Temperature (K) 10 16 10 14 10 12 10 10 10 8 10 6 Metallicity (Z ) 10 17 y(H2O) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Top: Slices of density, temperature, metallicity, and water abundance centred on the coordinates of SF 192 Myr post-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bottom: Density weighted projections centred on the same coordinates at the same epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Each figure has a side length of 4 kpc comoving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' approximately 80 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' There are regions within the remnant region that have an elevated water abundance with respect to the background value by a few orders of magnitude, indicating that the remnant is not completely devoid of water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' These values however, are at a maximum 𝑦_H2O ∼ 1015 and exceedingly low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The metallicity in these "high" water abundance regions has a minimum value of Z = 10−9 Z⊙, below this metallicity value the water abundance does not increase further than its background initialisation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Furthermore, the high metallicity regions exist at significant distance from the central overdense region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This suggests that a significant fraction of metal would need to halt further outward expansion and fall back onto the core before the potential for mixing could occur, let alone effective mixing of metal with primordial gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A simulation was initialised with identical initial conditions to en- sure the characteristics of the minihalo were unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' An identical mass (13 M⊙) Pop III star was inserted at the same time described above, only in this instance the SN model was altered to represent a ‘nominal’ CCSN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' the energy was ESN = 1051 ergs and the SN model outputs almost 10 times the metals at Mmet = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='784 M⊙ as the faint-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The progenitor star evolves in an identical fashion and so the initial conditions for the explosion to occur in are consistent between the two simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' After the SNe occurs, the dense clump lying in close proximity is impacted by the shock wave and is partially disrupted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The clump falls into the central region in a similar fashion after a period of brief recovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' After 120 Myr, the same filamentary structure that runs horizontally connecting regions of high density becomes visible, with the pristine clump of gas forming the centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 6 displays the state of the remnant and clearly shows this struc- ture forming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Although taken at a different epoch, comparing with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 5 the similarities are obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The bulk of the metals are trans- ported away from the central region, and marginal water formation occurs within the outer extremities of the remnant where the densities are the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In addition, the collapsing dense substructures that make up the horizontal filament have the lowest water abundance values throughout the remnant at y_H2O < 10−18, over two orders of magnitude less than the lower density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 4 DISCUSSION AND CONCLUSIONS Here we present the first results in a series where we attempt to find the ideal conditions for water formation in the remnant of Pop III supernovae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' For the first time, we solve all the relevant metal chem- istry that governs complex molecule formation and include a detailed prescription for dust chemistry, in addition to mass dependant Pop III SN yields in a self-consistent manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In the target minihalo (105 M⊙) initialised from cosmological initial conditions, pristine gas col- lapses via H2 cooling and forms a 13 M⊙ Pop III star at 𝑧 = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Jessop 10 4 10 3 10 2 10 1 100 101 102 Density (cm 3) 101 102 103 104 105 Temperature (K) 10 20 10 16 10 12 10 8 10 4 100 Metallicity (Z ) 10 19 10 18 10 17 10 16 y(H2O) 10 3 10 2 10 1 100 101 Density (cm 3) 101 102 103 Temperature (K) 10 15 10 12 10 9 10 6 Metallicity (Z ) 10 18 10 17 y(H2O) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 5, but for a normal CCSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Top: Slices of density, temperature, metallicity, and water abundance centred on the coordinates of SF 120 Myr post-SNe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Bottom: Density weighted projections centred on the same coordinates at the same epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Each figure has a side length of 4 kpc comoving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' We simulate the MSL of the star using the MORAY radiative transfer algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The Pop III star is modelled to have significant fallback onto the core at the point of explosion, reducing the energy of the explosion to 5 × 1050 ergs and the metal ejecta to just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='097 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The CCSN exhibits striking bipolar outflow from the central region and forces the bulk of the metal ejecta into the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The simulation concludes when the dense substructures throughout the remnant be- gin catastrophic recollapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' A summary of the primary findings from this simulation are as follows: (i) The low-mass of the minihalo allows for the HII region gener- ated by the Pop III star to escape the virial radius near the end of the stars lifetime, transitioning from a D-type to R-type in the direction of the voids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (ii) The star explodes as a CCSN and the metal ejecta is forced through the regions where LyC leakage occurs, whilst remaining confined where the minihalo primary structure meets the bordering filaments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This creates a bipolar outflow of metals and efficiently transports the metal ejecta out of the minihalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (iii) A dense, pristine clump of gas lies close to the SN and is impacted 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='5 Myr later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This clump is only partially disrupted by the shock and recovers quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The clump falls into the central region ∼ 54 Myr later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (iv) The metallicity of the core reaches a minimum value of 10−14 Z⊙ which increases to a maximum value of 10−6 Z⊙ approximately 900 pc from the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This suggests firstly that the majority of metals are transported away from the core in the explosion and secondly, that the mixing of metals into the primordial material that the infalling clump provides is very inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (v) 120 Myr post-SNe, the formation of a horizontal filament permeated by areas of high density with the collapsing infalling clump lying at the centre becomes distinctly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' These areas have a lower metallicity and also have a few orders of magnitude lower water abundance than that of the surrounding higher metallicity regions that exist at lower density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This again suggests that mixing is extremely inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (vi) Water formation occurs within low density regions at the outskirts of the remnant, although the abundance peaks no higher than y_H2O = 10−15 which is almost negligible in the context of second generation star and protoplanetary disk formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The pristine clump lying at the centre of the remnant loiters in a faux equilibrium, not incorporating metals into itself and actually suppressing complex chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' (vii) An identical simulation up until SN was initialised where the energy was doubled and the metal ejecta increased by a factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The chemo-thermal evolution of the remnant was very similar, with the majority of metal ejecta being forced into the void and the dense clump falling into the central region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' MNRAS 000, 1–12 (2022) Forming the First Water 11 100 101 102 103 Radius (pc) 10-17 10-16 Y H2O 100 101 102 103 Radius (pc) 10-11 10-10 Y Oi 100 101 102 103 Radius (pc) 10-17 10-16 10-15 10-14 10-13 10-12 10-11 10-10 Y Ci 100 101 102 103 Radius (pc) 10-16 10-14 10-12 10-10 10-8 10-6 Zmet Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Spherically averages radial profile of the SN remnant 192 Myr post-SNe of the major species and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The configuration of this minihalo mass containing within it two overdense regions at SF time conspire to create a unique situation where metal mixing in the centre of the remnant is inefficient, and therefore complex chemical interactions are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' As soon as the HII region of a Pop III star is able to overcome the minihalo, any SNe has the ability to blow the majority of its metal ejecta out into the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' In this instance, we chose to model a faint-CCSN to give the minihalo the greatest chance of recovery within a Hubble time as this represents a scenario with the least energy output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Even so, the halo is blown apart and it takes at least 190 Myr for some semblance of recollapse to occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' If a minihalo had lay in close proximity to the host minihalo, EE may be successful in creating the conditions for water formation (Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' As this was not the case here, IE was the only avenue for water formation to occur, however this never happened as the pristine clump fell onto the central region of the remnant and diluted the metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Since the majority of metals were ejected past the virial radius, the presence of the clump may only have enhanced the water formation rate (or lack thereof) that we see in the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Even when increasing the metal ejecta mass by a factor of 10 from a normal CCSN, the explosion energy is doubled and the result is the same (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The minihalo is blown out and the majority of metals are transported into the void.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' For IE to be viable and recollapse to occur on short timescales, the HII region must be confined to the virial radius of the minihalo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' This would trap the SNe, which would reach equilibrium and initiate recollapse earlier whilst retaining the bulk of the metal ejecta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Chiaki & Wise (2018) demonstrate that a minihalo with mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content='77 × 106 M⊙ is able to confine a 13 M⊙ Pop III stars HII region, suggesting that a lower limit can be placed on a minihalo that is able to self-enrich itself via IE as ∼ 106 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' The dynamics of each minihalo are unique, and therefore this may not be so much a set rule for all minihaloes at the lower end of the mass spectrum as a rule for this specific configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' What is clear however, is that as the minihalo mass increases so does the likelihood that IE dominates and the chances of water formation also increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Future studies are required on the matter, and will be able to probe what happens when the configuration changes slightly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Increasing the mass of the minihalo at SF to the upper half of 105 M⊙, or alternatively increasing the mass of the Pop III progenitor to inject more metals into the surrounding area may prove to have profound effects on the end state of the remnant and the ability of the recollapsing clumps to mix metals into themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Finding the correct balance between SN explosion energy, metal ejecta mass, and the host minihalo mass may be the key to discovering the configuration which promotes the formation of a water abundant protoplanetary disk around a second-generation star, and therefore the emergence of the first wet rocky planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' ACKNOWLEDGEMENTS C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' Jessop was supported by STFC grant 18379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' All numerical simulations were performed on the Sciama HPC cluster, supported by the University of Portsmouth and the Institute of Cosmology and MNRAS 000, 1–12 (2022) 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/k9AzT4oBgHgl3EQfNfvd/content/2301.01151v1.pdf'} 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0000000000000000000000000000000000000000..cd29fd9a6a22b695bde5b5d79eadbd0ae26eb9f1 --- /dev/null +++ b/mNE4T4oBgHgl3EQftw2O/content/tmp_files/2301.05227v1.pdf.txt @@ -0,0 +1,4053 @@ +Astronomy & Astrophysics manuscript no. aanda +©ESO 2023 +January 13, 2023 +Chrono-chemodynamical analysis of the globular cluster +NGC 6355: Looking for the fundamental bricks of the Bulge ⋆ +S. O. Souza1, 2 +, H. Ernandes3, 2 +, M. Valentini1 +, B. Barbuy2 +, C. Chiappini1 +, A. Pérez-Villegas4 +, S. +Ortolani5, 6, 7 +, A. C. S. Friaça2, A. B. A. Queiroz1, 8 +, and E. Bica9 +1 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, Potsdam, D-14482, Germany +e-mail: ssouza@aip.de +2 Universidade de São Paulo, IAG, Rua do Matão 1226, Cidade Universitária, São Paulo 05508-900, Brazil +e-mail: stefano.souza@usp.br +3 Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, SE-221 00 Lund, Sweden +4 Instituto de Astronomía, Universidad Nacional Autónoma de México, A. P. 106, C.P. 22800, Ensenada, B. C., México +5 Università di Padova, Dipartimento di Astronomia, Vicolo dell’Osservatorio 2, I-35122 Padova, Italy +6 INAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy +7 Centro di Ateneo di Studi e Attività Spaziali "Giuseppe Colombo" - CISAS. Via Venezia 15, 35131 Padova, Italy +8 Institut für Physik und Astronomie, Universität Potsdam, Haus 28 Karl-Liebknecht-Str. 24/25, D-14476 Golm (Potsdam), Germany +9 Universidade Federal do Rio Grande do Sul, Departamento de Astronomia,CP 15051, Porto Alegre 91501-970, Brazil +Received September 15, 1996; accepted March 16, 1997 +ABSTRACT +The information on Galactic assembly time is imprinted on the chemodynamics of globular clusters. This makes them important +probes that help us to understand the formation and evolution of the Milky Way. Discerning between in-situ and ex-situ origin of +these objects is difficult when we study the Galactic bulge, which is the most complex and mixed component of the Milky Way. To +investigate the early evolution of the Galactic bulge, we analysed the globular cluster NGC 6355. We derived chemical abundances +and kinematic and dynamic properties by gathering information from high-resolution spectroscopy with FLAMES-UVES, photometry +with the Hubble Space Telescope, and Galactic dynamic calculations applied to the globular cluster NGC 6355. We derive an age +of 13.2 ± 1.1 Gyr and a metallicity of [Fe/H]= −1.39 ± 0.08 for NGC 6355, with α-enhancement of [α/Fe]= +0.37 ± 0.11. The +abundance pattern of the globular cluster is compatible with bulge field RR Lyrae stars and in-situ well-studied globular clusters. The +orbital parameters suggest that the cluster is currently confined within the bulge volume when we consider a heliocentric distance of +8.54 ± 0.19 kpc and an extinction coefficient of RV = 2.84 ± 0.02. NGC 6355 is highly likely to come from the main bulge progenitor. +Nevertheless, it still has a low probability of being formed from an accreted event because its age is uncertain and because of the +combined [Mg/Mn] [Al/Fe] abundance. Its relatively low metallicity with respect to old and moderately metal-poor inner Galaxy +clusters may suggest a low-metallicity floor for globular clusters that formed in-situ in the early Galactic bulge. +Key words. Galaxy: Bulge – Globular Clusters: individual: NGC 6355 – Stars: Abundances, Atmospheres – Stars: Hertzsprung- +Russell and C–M diagrams – Galaxy: kinematics and dynamics +1. Introduction +The ΛCDM hierarchical theory of galaxy formation predicts that +a galaxy forms from successive mergers of low-mass objects that +are absorbed by more massive objects (Peebles 1974; White & +Rees 1978; Kauffmann et al. 1993; Springel et al. 2006). The less +massive objects are gradually absorbed while orbiting the mas- +sive objects. The Milky Way (MW) contains remnants of this +early history that can be divided into two groups: those still or- +biting the Galaxy, with their structures entirely or almost intact +(e.g. the Magellanic Clouds); and another group of objects that +were already dissolved by the MW after several encounters and +were completely accreted. The latter objects could have retained +the dynamic signatures of their progenitor if the merger event +occurred during the recent evolution of the Galaxy. An example +is Gaia-Sausage-Enceladus (GSE), known as the remnant of the +⋆ Based on observations from ESO Programs 083.D-0063 (A) (PI: +S. Ortolani) and 099.D-0136 (A) (PI: M. Valentini), and HST Project +GO-11628 (PI:Noyola). +last major merger of the MW with a dwarf galaxy (Belokurov et +al. 2018; Helmi et al. 2018). Because the estimated merger time +is ∼ 8 Gyr (Gallart et al. 2019; Montalbán et al. 2021), the rem- +nant stars did not have enough time to change their dynamical +properties completely. +In addition to the dynamic properties, mergers influence the +chemical properties of the Galaxy (e.g. Grand et al. 2020). It is +expected that an old, moderately metal-poor stellar population +will be formed upon the halt in the star formation history. Many +authors have tried to identify the chemodynamical imprints of +the early assembly steps that are left on the Galactic stellar pop- +ulations with the aim to constrain these important events in the +Galaxy history. This is now possible based on the joint informa- +tion from large spectroscopic surveys and the Gaia proper mo- +tion data (e.g. Anders et al. 2014; Hayden et al. 2015; Anders +et al. 2017; Kordopatis et al. 2020; Queiroz et al. 2020, 2021; +Buder et al. 2022, among many others). +The Galactic bulge is one of the most complex regions of +the Galaxy because in addition to the high extinction, it contains +Article number, page 1 of 20 +arXiv:2301.05227v1 [astro-ph.GA] 12 Jan 2023 + +IDA&A proofs: manuscript no. aanda +stellar populations from several parts of the MW. However, a +study of the bulge can provide information about its complex +formation processes (e.g. Barbuy et al. 2018a; Queiroz et al. +2020, 2021; Rojas-Arriagada et al. 2020). In order to distinguish +the different stellar populations, we have to study the object or- +bit (Pérez-Villegas et al. 2018) together with ages and chemical +composition. The orbits are a key ingredient that provides infor- +mation whether the object always lived in the bulge. Queiroz et +al. (2021, hereafter Q21) mapped and analysed the stellar popu- +lations of the bulge from a chemodynamical point of view, which +allowed them to describe the stellar content of the bulge field. +Another way to characterize the Galactic bulge comes from +the old stellar population, such as RR Lyrae. Savino et al. (2020) +analysed the stellar population in the inner spheroid of the +Galaxy and reported that this structure is very old, with an age +of 13.41 ± 0.54 Gyr, and it is also metal poor, with a metallic- +ity of [Fe/H]∼ −1.02 (Pietrukowicz et al. 2015), [Fe/H]∼ −1.0 +(Minniti et al. 2016), [Fe/H]∼ −1.55 (Crestani et al. 2021), and +[Fe/H]∼ −1.35 (Dékány & Grebel 2022). The globular clusters +(GCs) are also important tracers of the formation and evolu- +tion of the Galaxy because they are old and retain the chemo- +dynamical signatures of the first stages of the MW formation. +Some studies have demonstrated that the metallicity distribution +of bulge GCs peaks at [Fe/H]∼ −1.0 (Bica et al. 2016; Pérez- +Villegas et al. 2020, and references therein), and that they are +mostly older than 12.5 Gyr (Miglio et al. 2016; Barbuy et al. +2016, 2018a; Kerber et al. 2019; Ortolani et al. 2019; Oliveira +et al. 2020; Fernández-Trincado et al. 2020, 2021; Souza et al. +2021). +The assignment to which Galactic component a GC belongs +to is made depending on its orbital integration in order to ver- +ify the most probable regions of its trajectory. While part of the +GCs could have formed in the main progenitor of the Galaxy +(e.g. main bulge or main disk; Massari et al. 2019), others could +come from accreted progenitors. To study the origin of a GC, we +therefore need to analyse its chemical, photometric, and dynam- +ical properties (e.g. Souza et al. 2021). +The age-metallicity relation (AMR) of the MW GCs shows +a bifurcation that splits it into two main groups (Marín-Franch +et al. 2009; Forbes & Bridges 2010; Leaman, VandenBerg, & +Mendel 2013). A steeper branch in which more older GCs are +concentrated is associated with an in-situ population. In con- +trast, the other component is broader and includes very young +to old ages. It is also associated with accretion events during the +early evolution of the Galaxy (Kruijssen et al. 2019; Massari et +al. 2019; Forbes 2020; Limberg et al. 2022; Callingham et al. +2022). When an axisymmetric Galactic potential is employed, +the so-called integrals of motion space (IOM) can be used to- +gether with the AMR. By studying the total energy (E) versus +Z-component of the angular momentum (LZ), it is possible to +investigate the dynamic history of the Galaxy. For example, the +region of lower E with almost zero LZ can be associated with the +inner part of the MW, the bounded objects. On the other hand, +the Galactic halo accreted objects (e.g. GSE) are in the region of +high E. Therefore, the combination of the AMR with the IOM +space has improved the knowledge about the origin of the GCs +system, helping us to understand the Galactic evolutionary his- +tory, particularly that of the Galactic bulge. +Observing GCs within the Galactic bulge is difficult be- +cause the extinction tends to hide the objects. One example +is NGC 6355 (also called GCl-63 and ESO 519-SC15), pro- +jected towards the direction of the Galactic bulge (l = 359.58◦, +b = +5.43◦) with a relatively high extinction (E(B − V) = 0.79; +Harris 1996, 2010 edition). NGC 6355 is a well-known cluster +that has been studied since the 1900s. It is classified as a probable +open cluster (Shapley & Shapley 1919). However, it did not take +long before its globular nature was confirmed based on its rela- +tively high mass, which according to Baumgardt & Hilker (2018) +is 1.01×105 M⊙. Djorgovski & King (1986) classified NGC 6355 +as a core-collapse cluster. This result was recently confirmed by +Cohen et al. (2021a) using the Hubble Space Telescope (HST) +filters F606W and F814W from the Advanced Camera for Sur- +vey (ACS). +Ortolani et al. (2003) analysed the horizontal branch (HB) +and the red giant branch (RGB) of NGC 6355 using a [V,V − I] +colour-magnitude diagram (CMD). They obtained a reddening +of E(B − V) = 0.78, a distance of d⊙ = 8.8 kpc, and a metal- +licity of [Fe/H]∼ −1.3. This was deduced by comparing the +cluster mean locus with the mean loci of the well-studied clus- +ters NGC 6171 and M 5. Assuming their distance derivation, +the authors concluded that the cluster is near the Galactic cen- +ter (see also Bica et al. 2006). Valenti et al. (2007) analysed +the RGB slope and the K magnitude of the RGB tip using the +[K,J − K] and [H, J − H] CMDs. They found E(B − V) = 0.82, +d⊙ 8.7kpc, and [Fe/H]= −1.42. Both results agree with the metal- +licity scales of Carretta et al. (2009b) and Zinn & West (1984) +of [Fe/H]= −1.33±0.14 and [Fe/H]= −1.50±0.15, respectively. +Subsequent metallicity derivation by Vásquez et al. (2015) and +Dias et al. (2016) of [Fe/H]∼ −1.49 and [Fe/H]∼ −1.46, respec- +tively, are also within the range of both metallicity scales. +Barbuy et al. (2009) identified NGC 6355 as a blue horizon- +tal branch (BHB) metal-poor GC, located in the ring at −6◦ – +−12◦ around the Galactic centre. This suggested that NGC 6355 +belonged to the BHB moderately metal-poor clusters of the +Galactic bulge, such as NGC 6558 (Barbuy et al. 2007, 2018b), +HP 1 (Barbuy et al. 2006, 2016), AL 3 (Ortolani et al. 2006; +Barbuy et al. 2021), Terzan 9 (Ernandes et al. 2019), and UKS 1 +(Fernández-Trincado et al. 2020). Nevertheless, when examined +from the orbital viewpoint, it was suggested that NGC 6355 is +more compatible with the Galactic thick disk with a probability +of 93% (Rossi et al. 2015; Pérez-Villegas et al. 2018, 2020, here- +after PV20), assuming a distance of 8.70±0.87 kpc. It also has a +probability of 7% to be part of the Galactic bulge. Here we stress +the importance of having a precise distance derivation. +Kharchenko et al. (2016, hereafter KC16) analysed 147 GCs +including NGC 6355 using integrated JHKs magnitudes. They +derived its age as log t = 10.10 (∼ 12.5 Gyr). Assuming this +age derivation and the distances derived by Baumgardt & Hilker +(2018), Massari et al. (2019) found that NGC 6355 may have +been formed from the main-bulge progenitor and might there- +fore be an in-situ cluster. Their result for NGC 6355 was con- +firmed with a more realistic approach adopted in Moreno et +al. (2022), who employed the formalism for dynamical friction. +More recently, Cohen et al. (2021b, hereafter C21) derived a rel- +ative age of 1.1 Gyr by comparing the CMDs of NGC 6355 and +NGC 6205. The authors give an absolute age of ∼ 13.2 Gyr for +NGC 6355 and assume an age of 12.1 Gyr for NGC 6205 (Van- +denberg et al. 2013, hereafter VB13). This relatively older age +compared to the previous one by KC16 was used by Calling- +ham et al. (2022) to reclassify NGC 6355 as compatible with the +main-bulge progenitor and also with the Kraken accreted struc- +ture as an alternative origin. It is worth noting that a possible +accreted structure within the Galactic bulge was hypothesized +also by Massari et al. (2019) (low-energy progenitor), Kruijssen +et al. (2019) (Kraken), Forbes (2020) (Koala), and Horta et al. +(2021) (Heracles). +In the present work, we combine the chemical information +with photometric and dynamical properties of the cluster to +Article number, page 2 of 20 + +Souza et al.: Globular cluster NGC 6355 +constrain its history. The chemical information is based on the +UVES spectrograph (Dekker et al. 2000) in FLAMES-UVES +mode at the ESO-VLT, the photometry on HST data, and the dy- +namical properties are provided by orbital integration employing +the McMillan (2017) Galactic potential. +This paper is organized as follows. The photometry pro- +cessing, reduction of spectra, and membership analysis of the +observed stars are described in Section 2. Section 3 gives the +derivation of fundamental parameters. The analysis of individ- +ual line abundances and the comparison with the literature are +described in Section 4. The orbital analysis and dynamical prop- +erties of NGC 6355 are presented in Section 5. In Section 6 +we discuss the origin of the cluster. Finally, the conclusions are +drawn in Section 7. +2. Data +2.1. HST photometry processing +The photometric data for NGC 6355 were retrieved from the +HST Project (GO-11628, PI:Noyola), which used the Wide Field +Camera for Surveys 3 (WFC3) with the filters F438W and +F555W. The observation consists of three F438W images with +an exposure time of 440 s, and three F555W images with an ex- +posure time of 80 s. Figure 1 shows the colour image composed +of the combined HST images. We performed a further selection +based on the pipeline described in Nardiello et al. (2018) us- +ing the quality-of-fit and photometric error parameters to select +well-measured stars and reject poor measurements (top left panel +of Figure 2). Additionally, we selected stars within a half-light +radius of 0.88 arcmin (Harris 1996, 2010 edition) to avoid a sub- +stantial number of field stars. For the resulting sample, we com- +puted a simple membership probability by combining the stars +offset from the fiducial line on the CMD with the star distance to +the cluster centre. +The extinction towards the cluster is relatively high, and it +increases the CMD spread. To reduce the effect of differential +reddening, we used the same method as was applied to Palomar 6 +in Souza et al. (2021) (adapted from Milone et al. 2012; Bedin +et al. 2017). The differential reddening map (bottom left panel of +Figure 2) shows that δE(B−V) ∼ −0.04, which is approximately +5% of the expected reddening (E(B-V)= 0.79; Harris 1996, 2010 +edition), and which we can convert into a magnitude difference +of δmF438W = +0.17 and δmF555W = +0.13, and into a difference +in colour δ (mF438W − mF555W) = +0.04. +Finally, to scale the photometry to the same zero-point as +in the evolutionary models, we converted the AB magnitudes +into the Vega system. The final sample corrected for differential +reddening shows a smaller spread and a clear morphology from +the RGB and HB to the lower MS (top right panel of Figure 2). +The ACS F438W/F555W photometry is saturated for mag- +nitudes brighter than F555W∼ 17. Therefore, our spectroscopic +targets were not observed for these filters. To estimate the posi- +tion of our stars in the CMD, we derived an approximation of +their F438W and F555W magnitudes. We fixed the reddening, +metallicity, and distance modulus from Harris (1996, 2010 edi- +tion). For each star, we fitted the magnitudes J, KS , G, GBP, and +GRP (green triangles in Figure 2). We also used this method for +a sample of RGB stars of the Gaia EDR3 from the Vasiliev & +Baumgardt (2021) catalogue (open red circles). It is worth not- +ing that the F438W filter is affected by variations in C, N, and O +abundances. Hence, this filter can better be estimated via spectral +convolution and integration with the filter response curve. +Fig. 1. F438W/F555W combined colour image from the HST WFC3 +camera for NGC 6355. +Table 1. Log of the spectroscopic FLAMES-UVES observations of pro- +grams 083.D-0063 (A) and 099.D-0136 (A), carried out in 2009 and +2017, respectively. The reported seeing and airmass are the mean values +in the exposures. The last column contains the corresponding GIRAFFE +setup, in which additional stars were observed. +Date +UT +exp +Airmass +Seeing +SETUP +( s ) +(′′) +GIRAFFE +Program 083.D-0063 (A) +2009-09-02 +02:48:43 +2700 +1.455 +1.88 +H13-1 +2009-09-01 +01:03:00 +2700 +1.184 +0.87 +H13-2 +2009-09-01 +01:50:54 +2700 +1.191 +0.72 +H13-3 +2009-09-13 +23:32:32 +2700 +1.091 +0.91 +H13-4 +2009-09-14 +00:31:12 +2700 +1.182 +0.82 +H14-1 +2009-09-14 +01:17:51 +2700 +1.467 +0.78 +H14-2 +2009-09-14 +02:04:21 +2700 +1.848 +0.75 +H14-3 +Program 099.D-0136 (A) +2017-07-14 +06:21:39 +2400 +1.751 +0.75 +H11-1 +2017-07-14 +04:34:44 +2400 +1.172 +0.67 +H11-2 +2017-09-02 +01:50:12 +2400 +1.279 +0.61 +H11-4 +2017-09-07 +02:53:12 +2400 +1.831 +0.54 +H13-1 +2.2. Spectral data reduction +The UVES spectra were obtained using the FLAMES-UVES +setup centred at 580 nm, covering the wavelength range 480 - +680 nm, from the ESO Programs 083.D-0063 (A) (PI: S. Or- +tolani) and 099.D-0136 (A) (PI: M. Valentini). The latter ESO +program was coordinated with the program GO11126 (PI: M. +Valentini) for campaign 11 of the K2 satellite (K2 is the repur- +posed Kepler mission; Howell et al. 2014): the goal was to obtain +asteroseismology for the giants in the sample GCs. However, ob- +taining reliable light curves for these stars was not possible. The +log of observations is given in Table 1. +We performed the FLAMES-UVES data reduction pro- +cedure using the ESO-Reflex software with the UVES-Fibre +pipeline (Ballester et al. 2000; Modigliani et al. 2004). The cor- +Article number, page 3 of 20 + +N +个 +E +F438W +F555WA&A proofs: manuscript no. aanda +Fig. 2. Photometric data processing. Left panel: All stars within the FOV obtained from the HST Project (GO-11628, PI: Noyola). Middle panel: +Final differential-reddening-corrected CMD with selected stars (black) and discarded stars (grey). The Gaia EDR3 member stars from the Vasiliev +& Baumgardt (2021) catalogue matched with 2MASS to obtain the HST are shown in red. The green triangles represent the observed stars with +HST magnitudes obtained from the isochrone calibration, and the sizes are from the S/N. The HB region is plotted in blue. Right panel: Differential +reddening map for stars within a half-light radius. The resolution of the map is 0.024 arcminutes (1.44 arcseconds). +responding spectra of each star were corrected for the radial ve- +locity computed using the Python library PyAstronomy. The +radial velocities were obtained by cross-correlating the stellar +spectra with the Arcturus spectrum (Hinklen et al. 2000). The +values of the heliocentric radial velocity of each spectrum and +their mean values are presented in Table 2 for the member stars, +selected from the membership analysis (see section 2.3). +The spectra of stars 1546 and 1239 from ESO Program +083.D-0063, have a low signal-to-noise ratio (S/N; < 15), which +is significantly lower than those obtained from ESO Program +099.D-0136. The spectra of these two stars are therefore strongly +affected by noise, which makes it very difficult to distinguish +strong lines and prevents a satisfactory radial velocity deriva- +tion from the cross-correlation method. For consistency, they can +therefore not be confirmed as members of NGC 6355 given the +uncertainties in their radial velocity values even though these +stars are considered members from the proper-motion member- +ship check. Consequently, the final observed star sample is com- +posed of the four stars of ESO Program 099.D-0136. +Based on our final sample, we found a mean heliocentric ra- +dial velocity for NGC 6355 of −193.2±1.1 km s−1, which agrees +well with the value of −194.6±1.2 km s−1 obtained from the indi- +vidual stars of Gaia DR21. Finally, the normalized spectra were +combined and were weighted by the median flux to obtain the +final stellar spectra. +2.3. Membership selection +The power of Gaia astrometry has been demonstrated in differ- +ent ways, such as in the search for new open clusters and in the +selection of the most probable members of a GC. In particular, +regarding the latter, Gaia was not available until recent years, +1 https://people.smp.uq.edu.au/HolgerBaumgardt/ +globular/appendix/ngc6355.txt +Table 2. Heliocentric radial velocity obtained for each extracted spec- +trum and the average value for each star. +Target +Vhel +r +σVr +Target +Vhel +r +σVr +km s−1 +km s−1 +km s−1 +km s−1 +1546_1 +−227.40 +1.40 +1239_1 +−66.95 +0.19 +1546_2 +−29.98 +0.70 +1239_2 +−192.62 +0.56 +1546_3 +−216.96 +0.34 +1239_3 +−68.47 +0.31 +1546_4 +−318.81 +0.42 +1239_4 +−192.17 +0.90 +1546_5 +−166.55 +0.35 +1239_5 +−187.30 +0.22 +1546 +−216.96 +94.72 +1239 +−187.30 +60.28 +133_1 +−192.92 +0.47 +1176_1 +−196.35 +0.56 +133_2 +−191.65 +0.52 +1176_2 +−196.85 +0.65 +133_3 +−192.80 +0.48 +1176_3 +−193.25 +0.69 +133_4 +−192.03 +0.45 +1176_4 +−193.79 +0.68 +133 +−192.41 +0.53 +1176 +−195.07 +1.56 +1539_1 +−192.25 +0.41 +1363_1 +−193.93 +0.41 +1539_2 +−192.36 +0.40 +1363_2 +−194.01 +0.40 +1539_3 +−192.30 +0.41 +1363_3 +−192.44 +0.40 +1539_4 +−191.90 +0.41 +1363_4 +−192.34 +0.39 +1539 +−192.27 +0.18 +1363 +−193.18 +0.79 +and now the membership probabilities should be verified in all +samples preceding the Gaia era, and in particular for our sample +stars. +To remove bias from our sample, we performed a member- +ship analysis to determine which stars observed in both ESO pro- +grams are members of NGC 6355. Considering both programs, +we have a total of nine stars. We selected the Gaia DR3 stars +within 10′ from the cluster center, and we applied the Gaus- +sian mixture models (GMM; Pedregosa et al. 2011) clustering +method to separate the cluster members from the field stars. +The derived mean proper-motion for NGC 6355 is < µ∗ +α >= +Article number, page 4 of 20 + +N star +HB region +0 +HST GO-11628 (PI:E. Noyola) +P>0.9 +O +0 +-80 +0 +Gaia RGBs +16.0 +16.0 +Obs Stars +- 60 +R = 0.024 arcmin ++1.00 +-0.02 +18.0 +18.0 +40 +(arcmin) ++0.50 ++0.00 +DEC +20.0 +20.0 +20 +-0.50 +-0.02 +-1.00 +-0.04 +Q +1.76 arcmin +22.0 +22.0 ++1.00 ++0.00 +-1.00 +ARA (arcmin) +C +O +24.0 +24.0 +0 +2 +0.5 +1.0 +1.5 +2.0 +F438W-F555W +F438W-F555WSouza et al.: Globular cluster NGC 6355 +Fig. 3. Proper-motion density map from Gaia DR3. The stars show all +the observed stars in both programs (members are plotted in white, and +non-members are given in black). The red lines show the position of the +mean proper motion of NGC 6355. +−4.76 ± 0.06 mas yr−1 and < µδ >= −0.58 ± 0.05 mas yr−1. +This agrees very well with the new values computed by Vasiliev +& Baumgardt (2021). +The membership probabilities were computed considering +cluster and field distributions, following the method presented in +Bellini et al. (2009). When we had determined the membership +probability, we cross-matched our sample stars with the Gaia +data (Table 3), which are indicated with stars in Figure 3. We +found that six of nine stars from both programs have member- +ship probabilities above 80%. Combining the information of ra- +dial velocity and the proper-motion membership probability, we +therefore disregard the non-member stars in the following anal- +ysis. +3. Fundamental parameters +3.1. Atmospheric stellar parameters +3.1.1. Stellar magnitudes +The photometric effective temperature (Teff) and surface gravity +(log g) were derived from the VIJHKS magnitudes given in Ta- +ble 3. For comparison purposes, we obtained the Teff from the +Transiting Exoplanet Survey Satellite (TESS) input catalogue +(TIC; Stassun et al. 2018) for our sample. The 2MASS J, H, +and KS magnitudes were taken from Skrutskie et al. (2006). To +obtain the Teff from a wide wavelength range, we calculated +the colour V − I employing the photometric systems relations +G − V = f(GBP − GRP) and G − I = f(GBP − GRP) from Gaia +EDR3 (Riello et al. 2020). +3.1.2. Photometric effective temperatures Teff and gravities +log g +The Teff values were derived from V − I, V − KS , and J − KS +colour-temperature calibrations of Casagrande et al. (2010). To +use the calibrations, we must perform the reddening corrections. +For NGC 6355, we assumed the metallicity [Fe/H] = −1.33, +E(B − V) = 0.77, and (m − M)V = 17.21 from Harris (1996, +2010 edition) . Table 4 lists the derived photometric effective +temperatures. The < Teff > value given in the fifth column is the +mean effective temperature without the TESS values (which are +too hot). +To derive the photometric log g value, we used the classical +ratio log(g∗/g⊙), where log g⊙ = 4.44 is +log g∗ = 4.44 + 4 log Teff∗ +T⊙ ++ 0.4(Mbol − Mbol⊙) + log M∗ +M⊙ +. +(1) +We adopted the values of from Table 4, M∗ = 0.85M⊙ +and Mbol⊙ = 4.75. The derived values of the photometric Teff and +log g are given in the left columns of Table 4. +3.1.3. Spectroscopic stellar parameters +The final spectroscopic stellar parameters Teff, log g, and the mi- +croturbulence velocity vt of NGC 6355 were derived together +with [Fe/H] based on excitation and ionization equilibria. Equiv- +alent widths (EW) for a list of lines of Fe i and Fe ii lines were +measured using DAOSPEC (Stetson & Pancino 2008). Using a +visual inspection of the stellar spectrum, we remeasured some +lines with the IRAF routine to evaluate the impact of blending +lines, mainly for Fe ii, and some lines that were poorly fitted with +DAOSPEC. The employed lines are listed in the appendix (Table +A.1) with the adopted oscillator strengths (log gf) for Fe i lines +obtained from the VALD3 and NIST databases (Piskunov et al. +1995; Martín et al. 2002) and for Fe ii lines from Meléndez & +Barbuy (2009). +We extracted 1D photospheric models for our sample us- +ing the MARCS grid of atmospheric models (Gustafsson et al. +2008). The adopted CN-mild models consider [α/Fe]= +0.20 +for [Fe/H]= −0.50 and [α/Fe]= +0.40 for [Fe/H]≤ −1.00. For +the solar Fe abundance, we adopted ϵ(Fe) = 7.50 (Grevesse & +Sauval 1998). +The mean photometric and log g values calculated in +Section 3.1.2 were assumed as initial guesses to derive the spec- +troscopic parameters. The method consists of obtaining the exci- +tation and ionization equilibrium of Fe i and Fe ii lines. Figure 4 +shows the excitation and ionization equilibrium for star 133. The +derived spectroscopic parameters Teff, log g, [Fe i/H], [Fe ii/H], +[Fe/H], and vt are presented in the right columns of Table 4. +To derive the final metallicity, we generated a Monte Carlo +(MC) sample for each star to construct their [FeI/H] and [FeII/H] +distributions. The distributions composed of the individual MC +sample of each star are shown in Figure 5 as grey and red for +[FeI/H] and [FeII/H], respectively. Finally, the cluster metal- +licity distribution was obtained by combining the two distribu- +tions (grey and red). The best metallicity value, the correspond- +ing standard deviation, and the error of the mean are [Fe/H]= +−1.39±0.15 (0.08). This metallicity agrees well with the Carretta +et al. (2009a) metallicity scale, which gives a value of [Fe/H] = +−1.33 ± 0.02 for NGC 6355. +3.2. Age and distance +We employed the SIRIUS code (Souza et al. 2020) to perform +the isochrone fitting to the CMD [F555W, F438W-F555W] of +NGC 6355. The code can provide a Bayesian view of the fun- +damental parameters age, reddening (E(B − V)), d⊙, and metal- +licity ([Fe/H]). We adopted the isochrones from the Dartmouth +Stellar Evolutionary Database (Dotter et al. 2008) with a further +linear interpolation in age and [Fe/H] with the random values +given by the algorithm. As a Gaussian prior for the metallicity, +we employed the value derived in this work, while for the other +parameters, we adopted uniform priors: 10 Gyr ≤ age ≤ 14 Gyr, +Article number, page 5 of 20 + +NGC6355 +4 +Observed +2 +0 +μs [mas/yr] +-6 +-10 +-10 +-2 +0 +2 +4 +6 +μ*[mas/yr]A&A proofs: manuscript no. aanda +Table 3. Identifications, coordinates, magnitudes from JHKs 2MASS survey, VI, HST/ACS, and matched Gaia DR3 information. The first two +stars are from program 083.D-0063 (A), and the four last stars are from 099.D-0136 (A). +ID +ID +RA +DEC +V +V − I +J +H +KS +F438W +F555W +†µ∗ +α +µδ +G +BP−RP +S/N +2MASS +(deg) +(deg) +2MASS +HST/WFC3 +(mas yr−1) +1546 +17235883 − 2620183 +260.996 +−26.338 +15.06 +2.24 +11.359 +10.45 +10.19 +17.46 +15.42 +−4.747 +−0.523 +14.32 +2.39 +10.33 +1239 +17240227 − 2621267 +261.010 +−26.357 +14.40 +2.50 +10.284 +9.25 +8.92 +16.81 +14.81 +−4.839 +−0.394 +13.51 +2.66 +12.05 +1539 +17235356 − 2620223 +260.973 +−26.339 +14.82 +2.25 +10.942 +10.12 +9.73 +17.30 +15.19 +−4.780 +−0.659 +14.08 +2.40 +79.19 +1363 +17240101 − 2620597 +261.004 +−26.349 +14.74 +2.34 +10.892 +9.944 +9.63 +17.16 +15.09 +−4.942 +−0.591 +13.95 +2.49 +44.93 +1176 +17235712 − 2621441 +260.988 +−26.362 +15.28 +2.11 +11.684 +10.90 +10.59 +17.66 +15.69 +−5.041 +−0.609 +14.60 +2.26 +51.33 +133 +17235528 − 2621088 +260.980 +−26.352 +15.30 +2.24 +11.435 +10.62 +10.21 +17.64 +15.64 +−4.572 +−0.635 +14.56 +2.39 +36.36 +†µ∗ +α = µα cos δ. +Table 4. Photometric parameters derived using calibrations by Casagrande et al. (2010) for V − I, V − K, J − K colours are given in columns 2-8. +In columns 9-14 are given the spectroscopic stellar parameters. +Photometric parameters +Spectroscopic parameters +ID +T(V−I) T(V−KS ) T(J−KS ) +BCV +Mbol +log g +Teff +log g +[Fe i/H] +[Fe ii/H] +[Fe/H] +vt +(K) +(K) +(K) +(K) +(K) +(km s−1) +1539 4359 +4330 +4297 +4330 +−0.615 −3.02 0.74 +4300 ± 65 0.87 ± 0.23 −1.35 ± 0.11 −1.33 ± 0.18 −1.34 ± 0.15 1.0 ± 0.1 +1363 4246 +4315 +4152 +4246 +−0.702 −3.19 0.64 +4296 ± 76 0.84 ± 0.24 −1.36 ± 0.09 −1.35 ± 0.02 −1.36 ± 0.07 1.2 ± 0.1 +1176 4573 +4642 +4660 +4642 +−0.481 −2.43 1.10 +4580 ± 69 1.20 ± 0.26 −1.48 ± 0.08 −1.48 ± 0.23 −1.48 ± 0.17 1.0 ± 0.1 +133 +4373 +4328 +4250 +4328 +−0.606 −2.53 0.94 +4378 ± 76 1.24 ± 0.19 −1.46 ± 0.07 −1.44 ± 0.17 −1.45 ± 0.13 0.9 ± 0.1 +Fig. 4. Ionization and excitation equilibria for NGC 6355 star 133. The +black dots and red squares correspond to the [FeI/H] and [FeII/H] lines, +respectively. The crosses are the FeI lines that were excluded through a +3σ clipping method. +E(B−V) ≥ 0.0, and d⊙ ≤ 20 kpc. We used the CMD structure +constraints similar to the procedure described by VB13 to im- +prove the code. Nevertheless, we kept the Bayesian nature of the +code and used the structure pattern of the CMD as priors. +The direct comparison between observational data and +isochrones cannot give an accurate physical interpretation of the +cluster (D’Antona et al. 2018) because the likelihood in this +case is purely geometrical. Therefore, the prior distributions are +of great importance to improve the method. In that sense, we +adopted a more robust prior to the magnitude of the horizontal +branch (HB). This prior is crucial to give a more precise dis- +tance derivation when it is very close to the magnitude level of +RR Lyrae stars. To constrain the HB magnitude, we employed +the relation by Recio-Blanco et al. (2005), +MZAHB +F555W = 0.981 + 0.410 × [M/H] + 0.061 × [M/H]2, +(2) +where [M/H] = [Fe/H] + log +� +0.638 × 10[α/Fe] + 0.362 +� +. We as- +sumed [α/Fe]= +0.4 because this is the expected value for GCs +with a similar metallicity (Barbuy et al. 2018a). Then, we re- +calculated the magnitude level for each iteration of the Markov +Fig. 5. Metallicity distribution from sample stars of NGC 6355. The +final distribution (black step histogram) considers both [FeI/H] (grey) +and [FeII/H] (red) for all lines of our sample member stars. +chain Monte Carlo (McMC) sampling. For the apparent magni- +tude of the HB, we assumed mZAHB +F555W = 17.9 ± 0.1 by a visual +inspection, which is very close to the value derived by Ortolani +et al. (2003) of VHB = 17.8 ± 0.2. +Another morphological parameter is the magnitude differ- +ence between zero-age HB (ZAHB) and the turn-off point (TO), +also known as vertical parameter (Vandenberg, Bolte, & Stet- +son 1990; Rosenberg et al. 1999). However, this parameter is +strongly dependent on the ZAHB level. Because of this, we de- +cided to use the horizontal parameter (Vandenberg, Bolte, & +Stetson 1990; Rosenberg et al. 1999). The horizontal parame- +ter is the colour difference between the TO and the point at the +RGB that is 2.5 magnitude brighter than the TO. +In order to implement the horizontal method in the observed +CMD, we computed the fiducial or ridge line of NGC 6355 us- +ing the method described in Marín-Franch et al. (2009, hereafter +MF09). The procedure is briefly described as follows. We first +computed a simple fiducial line by binning the cluster magnitude +and calculating the median colour for each bin. We applied a dif- +Article number, page 6 of 20 + +-1.0 +1.2 +I,II/HJ +-1.4 +Fel +-1.6 +-1.8 ++ +α = 0.00011 ; = - 1.46 ± 0.07 +α = 0.00484 ; = - 1.46 ± 0.07 +-2.0 + = - 1.44 ± 0.17 + = - 1.44 ± 0.17 +20 +40 +60 +80 +100 +0 +1 +2 +3 +4 +5 +0 +6 +EW (mA) +Xex [eV]: +[FeI/H] +[FeI,II/H] +<[FeI,II/H]> +4 +[Fe/H]= - 1.39 ±0.15(0.08) +3 +density +2 +0 +-1.8 +-1.6 +-1.4 +-1.2 +-1.0 +[Fe/H]Souza et al.: Globular cluster NGC 6355 +ferential binning method to have more points around the TO. The +second step was to derive the median colour perpendicular to +each bin. This method is most important for the subgiant branch +(SGB) because this sequence is almost horizontal for bluer fil- +ters. Finally, the algorithm computes the horizontal parameter +for the cluster fiducial line and each McMC isochrone. +The posterior distributions of the parameters are given by +the 50th percentile as the best value, and the 16th and 84th per- +centiles to provide the uncertainties (right corner plots of Figure +6). In Figure 6, the NGC 6355 CMD (left panel) is over-plotted +by the best solution of the isochrone fitting composed of the me- +dian value (solid line) and the 1σ region (shaded region). +Because the expected extinction is relatively high, it is neces- +sary to consider the Teff correction to the isochrones. It is worth +noting that the Teff correction effect increases with the tempera- +ture and changes the isochrone morphology. The method is well +described in Oliveira et al. (2020) and Souza et al. (2020). We +found the following equations: +AF438W/AV = 7.688 − 86.606x + 325.254x2 − 407.219x3 (3) +AF555W/AV = 12.043 − 135.394x + 507.496x2 − 634.233x3(4) +AJ/AV = −0.128 + 1.428x − 5.309x2 + 6.573x3 +(5) +AKS/AV = 0.061 − 0.677x + 2.522x2 − 3.134x3 +(6) +AG/AV = 4.346 − 48.867x + 183.277x2 − 229.296x3 +(7) +AGBP/AV = 6.899 − 77.627x + 291.243x2 − 364.345x3 +(8) +AGRP/AV = −0.154 − 1.695x − 6.175x2 + 7.449x3, +(9) +where x is log Teff. The immediate effect on the isochrone is an +offset in the direction of the CMD blue-brighter region. There- +fore, the horizontal (E(438 − 555)) and vertical ((m − M)F555W) +displacements should be different from those without a correc- +tion. In addition, the morphology is defined essentially by the +age and metallicity when the helium mass fraction (Y) is fixed +(see Souza et al. 2020). In our case, the metallicity was con- +strained to the value derived here from high-resolution spec- +troscopy. Therefore, only age changes the isochrone morphol- +ogy. Because of this, the age considering the Teff correction tends +to be older than the simple isochrone fitting. The result is shown +in Figure 6. +In this work, we derived the absolute age of 13.2 ± 1.1 Gyr +for NGC 6355. The considerable uncertainty on the age deriva- +tion is due to the narrow colour baseline adopted in this work +(F438W-F555W), which spread the TO region slightly more. Al- +though we provide the first absolute age for NGC 6355 through +isochrone fitting, KC16 derived an age of ∼ 12.5 Gyr using in- +tegrated magnitudes, and C21 reported the age as 13.2 Gyr for +NGC 6355 by comparing its CMD with that of NGC 6205. The +age derived in this work assuming the Teff correction agrees very +well with the age in C21+VB13. This illustrates the importance +of this correction for highly reddened clusters in the central part +of the Galaxy. +Nataf et al. (2016) discussed the extinction towards GCs lo- +cated in the Galactic bulge, where the RV value can be as low +as 2.5. Pallanca et al. (2021) reported a straightforward method +for determining the best value of RV for highly reddened clus- +ters. The method was also applied by Souza et al. (2020), who +derived a value of 2.6 for Pal6. The method for deriving the RV +consists of simultaneously fitting CMDs with different colour +baselines with the same set of reddening and distance. Here we +fitted (in addition to the HST CMD) the CMDs [J, J − KS ] +from Valenti et al. (2007) and [G, GBP − GRP] from Gaia DR3. +From the HST CMD, we found E(438 − 555) = 0.78 ± 0.03 +and (m − M)F555W = 17.31 ± 0.12. These values were con- +verted into E(B − V) and (m − M)0 for different values of RV, +as shown in Figure 7. The best RV is the mean between the best +values for Valenti et al. (2007) and Gaia DR3 CMDs. We find +RV = 2.84±0.02. Hence, for NGC 6355 with the derived RV, we +find E(B − V) = 0.89 ± 0.03 and d⊙ = 8.54 ± 0.19 kpc. +The distance value is crucial for deriving the orbital parame- +ters of the clusters, as demonstrated by PV20 and illustrated by +the case of Palomar 6, as discussed in Souza et al. (2021). To +verify our distance derivation, we collected the RR Lyrae star +members of NGC 6355 from the fourth data release of the Op- +tical Gravitational Lensing Experiment (OGLE-IV; Soszy´nski et +al. 2019). We adopted the calibrations from Gaia Collaboration +et al. (2017, G17) using the least-squares (LQS) and Bayesian +(BA) methods (Muraveva et al. (2018, M18), and Oliveira et al. +(2022, O22)). All distances are displayed in Figure 8, including +the derivation by Baumgardt & Vasiliev (2021, B22) 2. The value +of 8.54 ± 0.19 kpc derived in this work agrees well with the oth- +ers, particularly the B22 value of 8.65 ± 0.22 kpc, which is the +most recent value. +4. Abundance analysis +We carried out a detailed abundance analysis employing line-by- +line spectrum synthesis. We employed the spectrum synthesis +code PFANT (Barbuy et al. 2018c) to derive the abundances of +the elements C, N, O, Na, Mg, Al, Si, Ca, Ti, V, Mn, Co, Cu, +Zn, Y, Zr, Ba, La, Nd, and Eu. The line list with the abundance +ratios for each line are given in the appendix (Table A.2). The +code PFANT is an update of the Meudon code by M. Spite and +adopts local thermodynamic equilibrium (LTE). The atomic line +list is from VALD3 (Ryabchikova et al. 2015). +The abundance values were derived through the χ2 mini- +mization algorithm described in detail in Souza et al. (2021). +Figure 9 gives a visual illustration of the method for star 1363. +The observed spectrum around the lines Na i 5682.633 Å and +Al i 6698.673 Å is shown in black. The best-fit solution is the +solid red line. For completeness, we also compare the spectrum +without the abundance contribution of the current element (solid +green line), the best fit plus 0.15 (solid magenta line), and the +best fit minus 0.15 (solid cyan line). Finally, we adopted the so- +lar abundances from Grevesse et al. (2015). +4.1. C, N, and O abundances +The CNO abundances were derived through an iterative fitting of +the C2(1,0) Swan bandhead at 5635.3 Å, and CN(6,2) at 6478.48 +Å of the A2ΠX2Σ system band heads and the forbidden oxygen +line [OI] 6300.31 Å. The algorithm fits the three lines simulta- +neously and takes the interdependent continuum variation due to +changes in C, O, and N values into account. Table 5 lists the de- +rived abundances. Because the region of the C2(1,0) bandhead is +strongly affected by the S/N and the line is weak, we assumed +the C abundances as upper limits. Finally, before fitting the [OI] +line, we verified the contamination by telluric lines in this region +and concluded that for our sample, none of the stars has telluric +line contamination on the [OI] line. The spectral fitting for C, N, +and O for star 1363 are shown in Figure 10. +As expected for most GCs (Piotto et al. 2015; Milone et +al. 2017), NGC 6355 seems to host multiple stellar populations +(MPs; see the reviews Gratton, Sneden, & Carretta 2004; Grat- +ton, Carretta, & Bragaglia 2012; Bastian & Lardo 2018; Milone +2 https://people.smp.uq.edu.au/HolgerBaumgardt/ +globular/fits/disfit/ngc6355_dist.pdf +Article number, page 7 of 20 + +A&A proofs: manuscript no. aanda +Fig. 6. Isochrone fitting for NGC 6355. The best solution is composed of the median values of the posterior distributions (solid dark red line), and +the 1σ extrapolation is constructed from the 16th and 84th percentiles (shaded dark red region). The corner plot shows the correlations among the +parameters. +Fig. 7. Simultaneous isochrone fitting to derive the cluster RV using +three CMDs: HST (left panel), 2MASS JKS from Valenti et al. (2007) +(top right panel), and Gaia DR3 (bottom right panel). The isochrones are +coloured according to their RV value. In each panel, the best solution is +represented by the solid dark red isochrone. For the two right panels, +the χ2 analysis is plotted in the inset plot, and the dots are coloured by +the same colour as the corresponding isochrone. +& Marino 2022). The relatively high nitrogen abundance of stars +1176 and 133 with relatively low values of carbon abundances +indicates the presence of MPs in NGC 6355. The other two stars +have a relatively low N abundance and relatively normal (solar) +C one. Because stellar evolution theory predicts an N-C anti- +correlation, we must further investigate to confirm the presence +of MPs in NGC 6355. This is further analysed below. +Fig. 8. Our distance derivation compared with the literature. The violins +show the distance distribution using RR Lyrae stars, the recent distance +derivation by Baumgardt & Vasiliev (2021), and the distance found in +this work through isochrone fitting. For the derived RR Lyrae distances, +four calibrations were adopted that are represented by the first four vio- +lins (see the text). +Table 5. Carbon, nitrogen, and oxygen abundances [X/Fe] from C2, CN +bandhead, and [OI], respectively. +[C/Fe] +[N/Fe] +[O/Fe] +Star +C2 +CN(6,2) +[OI] +5635.50 Å +6478.60 Å +6300.31 Å +1539 +≤ +0.10 ++0.21 ++0.43 +1363 +≤ +0.18 ++0.25 ++0.49 +1176 +≤ +0.00 ++0.87 ++0.37 +133 +≤ −0.09 ++0.70 ++0.24 +Article number, page 8 of 20 + +6 +Age(Gyr) = +1.15 +E(438 - 555) = +0.78 +0.03 +-0.01 +(m - M)F555W = +-0.09 +00 +O +[Fe/H] = -1.39] + +0.08 +10 +0.07 +E(438 - 555) +0.85 +0.80 +? +mF555W +20 +2 +[Fe/H] +-1.4 +g0 +1.2 +Age (Gyr) +E(438 - 555) +(m - M)F555w +mF438W - mF555W10 +17 +12 +0 +2.53.03.5 +Rv +18 +14 +19 +O +16 +20 +18 +mF555W +b. +0.5 +1.0 +1.5 +J-Ks +21 +14 +22 +0 +O +2.53.03.5 +G +Rv +16 +23 +G +O +24 +18 +1.0 +1.5 +1.0 +1.5 +2.0 +2.5 +GBP - GRP +mF438W - mF555W12 +T +11 +10 +(ody) +9 +8 +7 +6 +M18 +B22 +G17 (BA) +This workSouza et al.: Globular cluster NGC 6355 +Fig. 9. Example of line-profile fitting for star 1363. The upper panel +shows the result for the Na i 5682.633 Å, and the bottom panel shows +the fit for the Al i 6698.673 Åline. The black lines correspond to the +observed spectra. The solid red line shows the best-fit solution as the +median. For comparison purposes, we also plot the best-fit solution with +a variation of ±0.15 (solid cyan and magenta lines) and the spectrum +without the element abundance (green line). +Fig. 10. Spectral fitting of C, N, and O for star 1363. The observed +spectrum is given in black. The solid red line is the best fit, and the cyan +and magenta lines show the best fit ±0.15, respectively. The yellow line +shows the line region. For C2 (upper panel) we also show the bandhead +lines in dotted silver lines. +4.2. alpha-elements +The α-elements O and Mg are the most reliable indicators of +enrichment in α-elements from hydrostatic phases of massive +stars (Woosley & Weaver 1995). Together with the explosive α- +elements Si and Ca, they are good indicators of a fast early en- +richment of the proto-cluster gas by supernovae type II (SNII). Ti +is classified as an iron-peak element (Woosley & Weaver 1995), +but shows a similar α-element behaviour and is often included +as another α-element. The spectral fitting results for Mg, Si, Ca, +Fig. 11. Same as figure 10 for Mg, Si, Ca, and Ti. The solid red line +is the best fit, and the cyan and magenta lines show the best fit ±0.15, +respectively. +and Ti of star 1363 are shown in Figure 11, and the results are +presented in Table 6. +4.3. Odd-Z elements +The sodium abundances were derived from Na i 5682.633 Å, +5688.194 Å, 6154.23 Å, and 6160.753 Å lines. The Al abun- +dances were derived from lines Al i 6696.185 Å, 6698.673 Å. +The (anti-)correlations indicating the effect of MPs are +shown in Figure 12. We also calculated the Spearman corre- +lation parameter for each combination. For N-Al, we found a +strong correlation, and the anti-correlation for N-O, Na-O, and +Al-O is high. Moreover, the main correlations come from the ni- +trogen abundances (Fernández-Trincado et al. 2022). However, +[Al/Fe]= +0.30 is also a threshold for second-generation (2G) +stars (Mészáros et al. 2020). The figure shows a visible separa- +tion of our sample into two groups: two stars are moderately rich +in N and Al, and two stars have low values of [Al/Fe]. This af- +fects their mean abundances (Table 6). This is further discussed +below. +4.4. Iron-peak elements +We derived the abundances of the iron-peak elements V, Mn, +Co, Cu, and Zn. While V and Mn are members of the lower +iron-peak element group, Co, Cu, and Zn are considered to be- +Article number, page 9 of 20 + +star 1363 +1.0 +0.8 +[Na/Fel= -0.30 +a = 5682.633 A +Norm flux +5682.0 +5682.5 +5683.0 +5683.5 +star 1363 +1.0 +[A1/Fe]= +0.00 +0.9 +^ = 6698.673 A +6698.0 +6698.5 +6699.0 +6699.5 +Wavelength (A)star13＇1363 +1.00 +0.95 +[C2/Fe] = + 0.03 +[C2/Fe] = + 0.18 +[C2/Fe] = + 0.33 +5634.0 +5634.6 +5635.2 +5635.8 +star13 1363 +Norm flux +1.00 +[CN(6,2)/Fe] = + 0.10 +0.95 +[CN(6,2)/Fe1= + 0.25 +[CN(6,2)/Fe] = + 0.40 +6478.2 +6478.8 +6479.4 +star13 1363 +1.0 +[OI/Fe] = + 0.34 +[Ol/Fe] = + 0.49 +[O]/Fe] = + 0.64 +0.5 +6298.8 +6299.4 +6300.0 +6300.6 +6301.2 +6301.8 +Wavelength (A) star 1363 +1.0 +0.9 +[Mg/Fel= +0.47 +^= 6318.720A +6318.0 +6318.5 +6319.0 +6319.5 + star 1363 +1.0 +0.9 +[Si/Fe]= +0.14 +A=6237.328A + flux +6236.5 +6237.0 +6237.5 +6238.0 +Norm +star 1363 +1.00 +0.75 +TCa/Fel= +0.44 +=6161.295A +0.50 +6160.5 +6161.0 +6161.5 +6162.0 +star 1363 +1.0 +0.8 +[Ti/Fel= +0.30 +2 = 6064.623 +6064.0 +6064.5 +6065.0 +6065.5 +Wavelength (A)A&A proofs: manuscript no. aanda +Fig. 12. (Anti-)Correlations indicating effects of multiple stellar popula- +tions. The dotted orange line in both left panels represents the transition +to the N-rich regime at [N/Fe]∼ 0.7 for [Fe/H] around the NGC 6355 +value (Fernández-Trincado et al. 2022). Additionally, the grey line in +the two bottom panels shows the upper limit for first-generation stars +(Mészáros et al. 2020). The colour bar shows the Mg abundances. +long to the upper iron-peak group (Woosley & Weaver 1995). +The first group is mainly produced in type Ia supernovae (SNIa) +with a contribution from core-collapse supernovae (Nomoto, +Kobayashi, & Tominaga 2013, and refereces therein). In con- +trast, Co, Cu, and Zn are predominantly produced by core- +collapse supernovae (Woosley, Heger, & Weaver 2002, and ref- +erences therein). The atomic lines were adopted from Ernandes +et al. (2018) and Ernandes et al. (2020), together with their hy- +perfine structure. The spectral fitting results for V, Mn, and Co +are shown in Figure 13 for star 1363, and Cu and Zn are given in +Figure 14 for star 1539. +4.5. Heavy elements +The abundances of the heavy neutron-capture s-elements Y, Zr, +Ba, La, and Nd, and the r-element Eu also were derived. For Y, +we measured the Y i 6435.004 Å and the Y ii 6613.73 Å lines, +and we assumed for the mean that the ionized species of Y con- +tributes with 99% to the abundance. For the barium abundance, +we used the Ba ii lines 5853.675 Å, 6141.713 Å, and 6496.897 +Å, with hyperfine structure from Barbuy et al. (2014). The Zr i +6127.47 Å, 6134.58 Å, 6140.535.58 Å, and 6143.25 Å, La ii +6262.287 Å, 6320.376 Å, and 6390.477 Å, Nd ii 6740.078 Å, +6790.372 Å, and 6549.525 Å, and Eu ii 6437.6 Å and 6645.1 Å +were used for Zr, La, Nd, and Eu. The spectral fitting results for +Y, Zr, Ba, La, Eu, and Nd are shown in Figures 15 and 16 for star +1363. +4.6. Errors +The uncertainties in spectroscopic parameters are given in the +last four columns of Table 6 for star 133. For each stellar pa- +Fig. 13. Same as figure 10 for V, Mn, and Co. The solid red line is +the best fit, and the cyan and magenta lines show the best fit ±0.15, +respectively. +Fig. 14. Same as figure 10 for Cu and Zn. The solid red line is the best +fit, and the cyan and magenta lines show the best fit ±0.15, respectively. +rameter, we adopted the usual uncertainties for similar samples +(Barbuy et al. 2014, 2016, 2018b). The sensitivities were com- +puted by employing models with these modified parameters and +recomputing lines of different elements considering changes of +∆Teff = +100 K, ∆log g= +0.2, ∆vt = 0.2 km s−1. The given er- +ror is the difference between the new and the adopted abundance. +The uncertainties due to non-LTE effects are negligible for these +stellar parameters, as discussed in Ernandes et al. (2018). The +same error analysis and estimations can be applied to other stars +in our sample. It is worth noting that star 133 has the lowest S/N +of the four sample stars. The uncertainties given in Table 6 can +therefore be considered as upper limits. The faint La lines appear +to be more reliable than the strong Ba lines. Finally, it is impor- +Article number, page 10 of 20 + +0.5 +[O/Fe] +0.4 +0.3 +- p= - 0.70 +0.2 +p= + 0.39 +0.2 +[Na/Fe] +0.0 +-0.2 +- 0.80 +b= +-0.4 +0.2 +[Al/Fe] +0.0 +-0.2 +p= + 0.97 +p = - 0.78 ++0.60 +0.25 +0.50 +0.75 +0.2 +0.4 +-0.25 +0.00 +0.25 +[N/Fe] +[O/Fe] +[Na/Fe] +0.34 +0.36 +0.38 +0.40 +0.42 +0.44 +0.46 +[Mg/Fe]star 1363 +1.0 +0.8 +[V/Fel= +0.08 +^=6119.520A +0.6 +6118.5 +6119.0 +6119.5 +6120.0 +6120.5 +star 1363 +1.0 +Norm flux +0.8 +[Mn/Fel= -0.25 += 6013.513A +0.6 +6013.0 +6013.5 +6014.0 +star 1363 +1.0 +0.8 +[Co/Fe] = +0.05 + = 5342.708 A +5342.0 +5342.5 +5343.0 +5343.5 +Wavelength (A)star 1363 +1.0 +0.5 +[Cu/Fe]= -0.15 +a= 5105.537A +Norm flux +5104.5 +5105.0 +5105.5 +5106.0 +5106.5 +star 1363 +1.00 +[Zn/Fe]= -0.20 +a= 6362.339 A +0.95卜 +6361.5 +6362.0 +6362.5 +6363.0 +Wavelength (A)Souza et al.: Globular cluster NGC 6355 +Fig. 15. Same as figure 10 for Y, Zr, and Ba. The solid red line is the best +fit, and the cyan and magenta lines show the best fit ±0.15, respectively. +Fig. 16. Same as figure 10 for La, Eu, and Nd. The solid red line is +the best fit, and the cyan and magenta lines show the best fit ±0.15, +respectively. +tant to note that the main uncertainties in stellar parameters are +due to uncertainties in the Teff, as shown in Table 6. +Fig. 17. Density probability map for the x − y and R − z projections of +the set of orbits for NGC 6355. Orange corresponds to higher probabil- +ities, and the black lines show the orbits using the main observational +parameters. +5. Dynamical properties +In order to obtain the orbital parameters of NGC 6355, we em- +ployed an axisymmetric potential McMillan (2017) adopting the +Python package galpy (Bovy 2015). We integrated a set of 1000 +initial conditions forward for 10 Gyr. The set was generated by +using a MC algorithm adopting the observational uncertainties +of the cluster data on proper motions µ∗ +α and µδ, heliocentric ra- +dial velocity, and the heliocentric distance. The McMillan (2017) +Galactic potential was adopted to compare our results with those +of Massari et al. (2019) and to relate NGC 6355 with its plausible +progenitor. A more realistic potential, including a contribution of +the Galactic bar (Pérez-Villegas et al. 2018, 2020), could provide +a farther inward orbit for the GC members of the Galactic bulge. +The orbital parameters are listed in Table 7, including the values +of the IOM. +Figure 17 shows the density probability map of the orbits of +NGC 6355 in the x−y and R−z projections. The space region in +which the orbits of NGC 6355 cross more frequently are shown +in orange, and the black curves are the orbits considering the cen- +tral values of the observational parameters. NGC 6355 is mostly +confined within ∼ 2.6 kpc and therefore has a high probability of +belonging to the bulge component (> 95%) when we adopt the +distance of 8.54 ± 0.22 kpc that we estimated in this work. Our +new distance derivation indicates that the cluster NGC 6355 lies +far inward based on its maximum height of |z| < 2.1 kpc and the +high eccentric orbit > 0.8. It may well be that this perigalactic +distance is one the closest distances to the Galactic center. +6. Discussion +6.1. Kinematic classification +The orbital analysis shows that the orbit of NGC 6355 is com- +patible with a location at the Galactic bulge volume according +to the classification of PV20, who presented the probability dis- +tribution of belonging to each Galactic component through the +values of rapo and |z|max (Figure 17 and Table 7). It is essential +to mention that their classification is based on a Galactic po- +tential that includes the contribution of the Galactic bar. Another +Article number, page 11 of 20 + +star 1363 +1.00 +0.95 +[Y/Fel= +0.00 +=6795.414A +6794.5 +6795.0 +6795.5 +6796.0 +star 1363 +1.0 +Norm flux +[Zr/Fel= ±0.05 +0.8 +2=6127.475A +6127.0 +6127.5 +6128.0 +star 1363 +1.0 +0.5 +[Ba/Fe]= + 1.00 + = 6496.897 A +6496.0 +6496.5 +6497.0 +6497.5 +Wavelength (A) star 1363 +1.0 +0.9 +[La/Fel= +0.17 +Λ = 6390.477 A +0.8 +6390.0 +6390.5 +6391.0 +star 1363 +1.0 +Norm flux +0.9 +[Eu/Fe]= +0.55 +^= 6437.640A +6437.0 +6437.5 +6438.0 +6438.5 +star 1363 +1.0 +0.9 +[Nd/Fe] =+0.15 +a= 6740.078 A +6739.5 +6740.0 +6740.5 +6741.0 +Wavelength (A)2 +2 +1 +[kpc] +0 +0 +y +-1 +-1 +-2 +-2 +0 +2 +0 +2 +[kpc] +0 +0 +Z +-1 +-1 +-2 +2 +-2 +-1 +0 +1 +2 +0 +1 +2 +x [kpc] +R [kpc]A&A proofs: manuscript no. aanda +Table 6. Abundances in the four UVES member stars. The mean values were computed considering all four stars (< all >), considering only 1G +stars (< 1G >), and only 2G stars (< 2G >). The last four columns show the abundance sensitivity due to variation in atmospheric parameters for +star 15 (133) considering uncertainties of ∆Teff = 100 K, ∆log g = 0.2, and ∆vt = 0.2 km s−1, and the last column is the total error. These errors +were taken into account when we composed the final reported abundances. +[X/Fe] +star 1539 +star 1363 +star 1176 +star 133 +< all > +< 1G > +< 2G > +∆T +∆ log g +∆vt +( 1 +3 +�x2)1/2 +1G +2G +K +kms−1 +C ++0.10 ± 0.05 ++0.18 ± 0.05 ++0.00 ± 0.05 +−0.09 ± 0.05 ++0.05 ± 0.11 ++0.14 ± 0.06 +−0.04 ± 0.07 ++0.02 ++0.03 ++0.00 ++0.03 +N ++0.21 ± 0.05 ++0.25 ± 0.05 ++0.87 ± 0.05 ++0.70 ± 0.05 ++0.51 ± 0.29 ++0.23 ± 0.05 ++0.78 ± 0.10 ++0.12 ++0.08 ++0.00 ++0.08 +O ++0.43 ± 0.05 ++0.49 ± 0.05 ++0.37 ± 0.05 ++0.24 ± 0.05 ++0.38 ± 0.11 ++0.46 ± 0.06 ++0.30 ± 0.08 ++0.00 ++0.03 ++0.00 ++0.03 +Mg ++0.33 ± 0.05 ++0.44 ± 0.05 ++0.38 ± 0.05 ++0.47 ± 0.05 ++0.41 ± 0.07 ++0.39 ± 0.07 ++0.42 ± 0.07 ++0.02 +−0.02 +−0.03 ++0.03 +Si ++0.27 ± 0.10 ++0.25 ± 0.12 ++0.33 ± 0.15 ++0.28 ± 0.27 ++0.28 ± 0.16 ++0.26 ± 0.11 ++0.30 ± 0.21 +−0.02 +−0.02 +−0.07 ++0.04 +Ca ++0.48 ± 0.16 ++0.47 ± 0.10 ++0.34 ± 0.26 ++0.57 ± 0.12 ++0.46 ± 0.18 ++0.47 ± 0.13 ++0.45 ± 0.22 ++0.26 ++0.04 +−0.08 ++0.16 +Ti ++0.30 ± 0.12 ++0.34 ± 0.12 ++0.28 ± 0.12 ++0.38 ± 0.10 ++0.33 ± 0.12 ++0.32 ± 0.12 ++0.33 ± 0.12 +−0.03 ++0.09 +−0.06 ++0.06 +Na +−0.29 ± 0.08 +−0.15 ± 0.15 +−0.22 ± 0.15 ++0.20 ± 0.12 +−0.11 ± 0.23 +−0.22 ± 0.14 +−0.01 ± 0.25 ++0.10 +−0.00 +−0.05 ++0.06 +Al +−0.29 ± 0.05 +−0.15 ± 0.15 +< +0.30 ± 0.05 +< +0.30 ± 0.05 +< +0.04 ± 0.28 +−0.22 ± 0.12 +< +0.30 ± 0.06 ++0.08 +−0.00 +−0.02 ++0.05 +Y ++0.20 ± 0.07 +−0.00 ± 0.07 +−0.00 ± 0.07 +— ++0.06 ± 0.12 ++0.10 ± 0.12 +−0.00 ± 0.07 ++0.24 ++0.09 +−0.14 ++0.17 +Zr +−0.06 ± 0.08 ++0.09 ± 0.08 +— +−0.11 ± 0.26 +−0.02 ± 0.16 ++0.02 ± 0.11 +−0.11 ± 0.26 ++0.20 ++0.02 +−0.01 ++0.12 +Ba ++0.84 ± 0.17 ++0.93 ± 0.09 ++0.92 ± 0.19 ++1.02 ± 0.16 ++0.93 ± 0.17 ++0.89 ± 0.14 ++0.97 ± 0.19 ++0.02 ++0.03 +−0.13 ++0.08 +La ++0.08 ± 0.12 ++0.06 ± 0.08 ++0.10 ± 0.07 ++0.27 ± 0.05 ++0.13 ± 0.12 ++0.07 ± 0.10 ++0.19 ± 0.11 ++0.03 ++0.09 +−0.02 ++0.06 +Eu ++0.53 ± 0.05 ++0.55 ± 0.05 ++0.57 ± 0.08 ++0.60 ± 0.10 ++0.56 ± 0.07 ++0.54 ± 0.05 ++0.59 ± 0.09 +−0.03 ++0.07 +−0.02 ++0.05 +Nd ++0.47 ± 0.06 ++0.28 ± 0.10 ++0.06 ± 0.08 +−0.30 ± 0.05 ++0.13 ± 0.30 ++0.38 ± 0.12 +−0.12 ± 0.19 ++0.03 ++0.09 +−0.03 ++0.06 +V ++0.03 ± 0.06 ++0.19 ± 0.10 +−0.33 ± 0.06 ++0.00 ± 0.08 +−0.03 ± 0.20 ++0.11 ± 0.11 +−0.17 ± 0.18 ++0.20 ++0.02 +−0.07 ++0.12 +Mn +−0.34 ± 0.05 +−0.42 ± 0.10 +−0.39 ± 0.13 +−0.43 ± 0.08 +−0.39 ± 0.10 +−0.38 ± 0.09 +−0.41 ± 0.11 ++0.11 +−0.00 +−0.02 ++0.06 +Co ++0.03 ± 0.05 ++0.07 ± 0.05 ++0.07 ± 0.09 ++0.16 ± 0.11 ++0.08 ± 0.09 ++0.05 ± 0.06 ++0.11 ± 0.11 ++0.15 ++0.04 +−0.00 ++0.09 +Cu +−0.35 ± 0.05 +−0.07 ± 0.07 +−0.12 ± 0.17 +−0.17 ± 0.17 +−0.18 ± 0.16 +−0.21 ± 0.15 +−0.15 ± 0.18 ++0.13 ++0.04 +−0.09 ++0.09 +Zn +−0.30 ± 0.05 +−0.20 ± 0.05 +−0.30 ± 0.05 +−0.10 ± 0.05 +−0.23 ± 0.10 +−0.25 ± 0.07 +−0.20 ± 0.11 ++0.01 ++0.03 +−0.06 ++0.04 +[Fe/H] +−1.34 ± 0.15 +−1.36 ± 0.07 +−1.48 ± 0.17 +−1.45 ± 0.13 +−1.39 ± 0.08 +−1.35 ± 0.09 +−1.46 ± 0.13 ++0.10 ++0.10 ++0.04 ++0.08 +Table 7. Orbital parameters, velocities, and membership probabilities. +Parameter +Mean +Unit +E +−2.31 ± 0.03 +×105km2 s−2 +LZ +−31.28 ± 24.42 +km s−1kpc +rperi +0.25 ± 0.08 +kpc +rapo +2.46 ± 0.14 +kpc +|z|max +1.91 ± 0.08 +kpc +ecc +0.82 ± 0.05 +— +vR +−218.27 ± 66.25 +km s−1 +vφ +−192.39 ± 34.54 +km s−1 +Pbulge +95.00 +% +Pdisk +5.00 +% +Pinner halo +0.00 +% +Pouter halo +0.00 +% +robust Galactic potential, taking into account the friction dynam- +ics in addition to the contribution of the Galactic bar, was applied +by Moreno et al. (2022). Their orbital parameters are essentially +compatible with our results. The values of E are precisely the +same. The LZ and rperi are compatible within 1σ, while our value +of rapo is higher than that of Moreno et al. (2022). This indicates +that a more realistic Galactic potential confines NGC 6355 even +more within the Galactic bulge volume. With the results using +the McMillan (2017) Galactic potential, NGC 6355 is a Galactic +bulge GC with a probability of about 95%, and a 5% probability +of belonging to the Galactic disk. +After establishing that NGC 6355 currently is a member of +the Galactic bulge, we investigated whether this cluster origi- +nated from the primordial material of the Galaxy or if it is a +remnant of the first mergers of the MW. To do this, we stud- +ied the chemodynamical and photometric information derived in +previous sections. +6.2. Comparison with bulge field stars +To study NGC 6355 in the context of the Galactic bulge, we +compared the orbital parameters derived in this work with the +field star population composed by the reduced proper motion +(RPM) sample from Q21 and the bulge RR Lyrae from the +OGLE Galaxy Variability Survey (Soszy´nski et al. 2019). +We matched the OGLE sample with APOGEE DR17, which +already provides the abundances, radial velocities, and proper +motions (previously obtained from Gaia EDR3). After this, the +second sample was matched with Starhorse (Queiroz et al. 2020) +in order to obtain the distance values. Finally, the resulting sam- +ple consisted of 4132 stars. +NGC 6355 has a relatively high |Z|max and e, placing it in +cell F of Figure 20 in Q21. This is reproduced here in the upper +left panel of Figure 18 for the RPM sample and in the upper +right panel for the RR Lyrae sample. The normalized population +density as a function of [Fe/H] (MDF), Rmean (mean between rapo +and rperi), and vφ is shown in the three bottom panels of Figure +18, respectively. Based on the MDF (lower left panel ), the RPM +sample comprises the moderately metal-rich bulge MDF, while +the RR Lyrae sample is the metal-poor tail one. As expected, +NGC 6355, an old GC, is located together with the peak of the +RR Lyrae MDF and Rmean distribution. Nevertheless, the vφ of +the cluster is in the tail of both samples. +The comparison with the bulge field populations shows that +NGC 6355 is likely an in-situ GC compatible with the old and +metal-poor RR Lyrae component of the Galactic bulge. +6.3. Comparison with chemodynamical models +To investigate the chemical abundances in the context of nucle- +osynthesis, we compared our results with chemical evolution +models. The models for O, Mg, Si, Ca, V, Mn, Co, Cu, and +Zn were computed with the code described in Friaça & Bar- +buy (2017) (see also Barbuy et al. 2015; Ernandes et al. 2020, +2022, for V, Mn, Co, Cu, and Zn). The star formation rate (SFR) +was found to be best suited with ν = 1 Gyr−1 in order to fit the +abundances of a selected sample of bulge stars in Razera et al. +Article number, page 12 of 20 + +Souza et al.: Globular cluster NGC 6355 +Fig. 18. NGC6355 compared with the RPM bulge sample of Q21 (left panel) and Galactic bulge RR Lyrae population (right panel). Upper panels: +|Z|max as a function of the eccentricity plane divided into nine frames defined by the letter close to the horizontal lines. The golden star represents +the locus of NGC 6355. The bottom panels show the population density of [Fe/H], Rmean, and vφ for cell F. The gold lines represent the position of +NGC 6355 in each panel, and the shaded gold region shows the 1σ distribution. +(2022). Therefore, we adopted this SFR for all elements. The +SFR is the rate at which the available gas mass is turned into +stars. Consequently, it measures the inverse of the system for- +mation timescale: Our adopted ν = 1.0 Gyr−1 represents a rather +fast star formation of 1.0 Gyr. The models assume a baryonic +mass of 2 × 109 M⊙, a dark halo mass 1.3 × 1010 M⊙, and the +cosmological parameters from Planck Collaboration (2016). The +bulge is considered a classical spheroidal component. Finally, +the models project the chemical abundance distribution at dif- +ferent radius ranges: r< 0.5 kpc (dash-dotted line in Figure 19), +0.5 < r < 1 kpc (dashed line), 1 < r < 2 kpc (dotted line), and +2 < r < 3 kpc (solid line). For Na and Al, we used the Kobayashi +et al. (2020) models for the Galactic bulge. These models also +assume an SFR ν ∼ 1.0 Gyr−1. +In addition to the chemodynamical models, in the follow- +ing analysis, we also compare the abundance ratios obtained in +this work with the bulge GCs Palomar 6 (Souza et al. 2021), +HP 1 (Barbuy et al. 2018a), NGC 6558 (Barbuy et al. 2018b), +and NGC 6522 (Barbuy et al. 2021). Furthermore, we compare +NGC 6355 with the RPM and RR Lyrae samples (for the case of +α and odd-Z elements) inside cell F of Figure 18. +Figure 19 shows the abundances of O, Mg, Si, and Ca as a +function of [Fe/H]. To better illustrate the comparison, the mean +locus of the RPM sample is shown as a solid black line. We de- +rived [α/Fe]= +0.36 ± 0.09 considering all α elements O, Mg, +Si, Ca, and Ti. Our mean [α/Fe] is compatible within 1σ with +the assumed value for the isochrone fitting. This reinforces the +consistency of the analysis. In all cases, NGC 6355 is compatible +with the RR Lyr locus. Additionally, NGC 6355 is also compat- +ible with the other GCs, execpt for Ca, in which the cluster is +relatively richer than the others. +As a result of the presence of MPs, the spread in [Na/Fe] is +higher than for the other elements. This effect can be observed +in the top panel of Figure 20 with the discrepancy between the +two models and the mean locus for the case of low metallici- +ties. Souza et al. (2021) found that Pal 6 is not compatible with +a bulge [Na/Fe]. They argued that the reason is due to the pres- +ence of a 2G star in their sample. The same effect is expected +for [Al/Fe] because the Al abundance is a good indicator of the +presence of 2G stars. The lower panel of the same figure shows +the high error bars of [Al/Fe] for NGC 6355 that are due to the +presence of two moderately Al-rich stars. +In Figure 21 we investigate the iron-peak elements V, Mn, +Co, and Cu. To increase the bulge sample, we also compared +our results with bulge GC stars from Ernandes et al. (2018) and +the bulge field stars from Ernandes et al. (2020). The chemical +evolution model fits NGC 6355 perfectly. For Cu abundances, +the selected bulge clusters have relatively lower values than +NGC 6355, indicating a different possible scenario for its early +evolution. In the case of V (top left panel), the evolution model +is shifted to lower abundances for all metallicities than the mean +locus. The models suitably fit the abundances of NGC 6355, the +selected clusters, and the bulge GC stars for Mn, Co, and Cu. +The Zn abundances derived in this work are based only on +the line Zn i 6362.339 Å. In Figure 22, NGC 6355 is perfectly +fitted by the models and is compatible with all reference clusters. +Here it is worth noting that the models predict supersolar zinc +abundances for metallicities above −1.0 and subsolar for values +Article number, page 13 of 20 + +RPM Bulge sample +Bulge RR Lyrae stars +3.0 +2.5 +2.5 +2.0 +2.0 +(kpc) +1.0 +1.0 +0.5 +0.5 +00 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ecc +ecc +NGC6355 +RPM +RR Lyr +2 +4 +0 +0 +-200 +200 +[Fe/H] + (kpc) +Vμ (km s-1)A&A proofs: manuscript no. aanda +Fig. 19. O, Mg, Si, and Ca abundance as a function of [Fe/H]. The +KDE plot represents the RPM bulge selection from cell F, and the blue +contours represent the RR Lyrae sample. The stars are abundances of +bulge GCs: NGC 6558 (cyan), NGC 6522 (pink), HP1 (green), and Pal +6 (magenta). The golden star represents the mean abundance of NGC +6355. The chemodynamical evolution models are shown in different +radii ranges: r< 0.5 kpc (dash-dotted line), 0.5 < r < 1 kpc (dashed +line), 1 < r < 2 kpc (dotted line), and 2 < r < 3 kpc (solid line). +below −1.0. The low Zn as an indicator of an ex-situ origin was +suggested only for the case of near-solar metal-rich stars Minelli +et al. (2021). +Fig. 20. Same as Figure 19 for odd-Z elements Na (upper) and (bottom). +The solid red line is the chemical evolution model from Kobayashi et +al. (2020). +Fig. 21. Same as Figure 19 for V, Mn, Co, and Cu. The black squares +are bulge GC stars from Ernandes et al. (2018), and black crosses show +bulge field stars from Ernandes et al. (2020). The chemodynamical evo- +lution models are shown in different radius ranges: r< 0.5 kpc (dash- +dotted line), 0.5 < r < 1 kpc (dashed line), 1 < r < 2 kpc (dotted line), +and 2 < r < 3 kpc (solid line). +Fig. 22. Same as Figure 19 for Zn. The Friaça & Barbuy (2017) evo- +lution models are shown in different radius ranges: r< 0.5 kpc (dash- +dotted line), 0.5 < r < 1 kpc (dashed line), 1 < r < 2 kpc (dotted line), +and 2 < r < 3 kpc (solid line). +The comparison of heavy-element abundances of NGC 6355 +with literature GCs is shown in Figure 23. NGC 6355 abun- +dances are compatible with HP 1 in almost all heavy elements +except for Ba, for which NGC 6355 has higher values. There is +a rather large scatter in the abundances of n-capture elements, +especially Y, Zr, and Ba. This pattern is better explained in Chi- +appini et al. (2011), Cescutti & Chiappini (2014), and Barbuy et +al. (2018a). +6.4. Analysis of abundance discriminators +The [Mg/Mn]-[Al/Fe] plane is often used in the context of the +Galactic halo to split the original MW population from merger +remnants (Hawkins et al. 2015; Limberg et al. 2022) because the +accreted population shows lower [Al/Fe] abundances and high +α abundances due to the abrupt evolution interruptions of the +merger progenitor. Horta et al. (2021) applied the same idea for +a star sample located in the Galactic centre to find debris stars +within the Galactic bulge. They called this inner Galaxy struc- +Article number, page 14 of 20 + +Mg +0.50 +0.25 +0.00 +[X/Fe] +Si +Ca +0.50 +10 +0.25 +0.00 +-2 +-1 +0 +-2 +-1 +0 +[Fe/H] +Bulge RR Lyrae stars +★★★★木 +NGC 6558 - Barbuy et al. (2018) +v= 1.0 Gyr-1: r<0.5 kpc +NGC 6522 - Barbuy et al. (2020) +HP 1 - Barbuy et al. (2018) +v= 1.0 Gyr-1: 0.5 +0.25 +[Mg/Mn] > 5 × [Al/Fe] + 0.5. +(10) +In the context of the [Mg/Mn]-[Al/Fe] plane (Figure 24), the +reference bulge GCs have no preferential positioning. However, +Pal 6 and NGC 6355 show interesting behaviours. Souza et al. +(2021) found that Pal 6 is an in-situ member. We confirm this +through the [Mg/Mn]-[Al/Fe] plane, with APOGEE abundances +for Pal 6 members, because this cluster is located perfectly in +the high-α region. In contrast, the NGC 6355 is located at the +border between Heracles and the in-situ high-α region. Two of +our four stars present a maximum Al abundance of +0.30, which +could be 2G stars (2G Mészáros et al. 2020; Fernández-Trincado +et al. 2022). Figure 12 shows the (anti-)correlations that indicate +the presence of MPs. We do not find a Mg-Al anticorrelation, +although this is expected mainly for massive clusters because +of the metallicity multimodality (Mészáros et al. 2020). We can +also observe a slight difference between the 1G and 2G [La/Fe] +mean abundances. Marino et al. (2021) showed that it is not pos- +sible to separate the MPs using La abundances. However, the +mean abundance value is higher for the anomalous than for the +normal stars. +The mean on the [Mg/Mn]-[Al/Fe] plane changes in the di- +agonal direction going to the left or right circles of Figure 24 +when only the 1G or 2G stars are considered to compute the +cluster mean abundance. Then, assuming the mean abundance +of the 1G stars, NGC 6355 is placed inside the Heracles re- +gion on the [Mg/Mn]-[Al/Fe] plane. It is worth pointing out that +Q21 showed that a sample of counter-rotating stars in the RPM +sample presents no preferential location in the [Mg/Mn]-[Al/Fe] +plane, suggesting that this region may not be entirely composed +of accreted objects. Therefore, even though NGC 6355 is placed +in the accreted region, this by itself does not signify an ex-situ +origin. +Fig. 24. [Mg/Mn]-[Al/Fe] plane with the identification of Heracles in +red contours. The colours and density map are the same as Figure 20. +The circles represent the mean considering only 1G (left) and 2G (right) +stars in NGC 6355. +6.5. Age-metallicity relation and integral-of-motion space +The AMR is an interesting tool for investigating the origin of the +MW GCs (Massari et al. 2019). Figure 25 shows the MW GCs +AMR collected from Kruijssen et al. (2019). We overplot the +mean locus of the two branches (in-situ and ex-situ) as defined +by Forbes (2020), which are defined as follows: +Z = −p ln +� t +t f +� +, +(11) +where Z is the mass metallicity, p is the effective yield, and t f +is the time to the initial formation of the system. The red line +is the ex-situ population obtained using the parameters found by +Limberg et al. (2022). To represent the in-situ population, we +only changed the parameters by hand to place the line above the +older branch (blue line). +In Figure 25 we show NGC 6355 as the gold star together +with the bulge GCs. It is clear that age is hugely crucial for the +progenitor of a GC classification. For the case of NGC 6355, the +isochrone fitting considering the Teff correction clearly indicates +an in-situ candidate. Nevertheless, the uncertainty on age still +gives NGC 6355 a low probability of having an ex-situ origin. +The dynamics based on the orbital parameters of Table 7 +show that E and LZ of NGC 6355 are −2.31 ± 0.03 ×105km2 s−2 +and −31.28 ± 24.42 km s−1kpc, respectively. These values place +the cluster in the low-energy and low absolute LZ region. Horta +et al. (2021) also analysed the IOM space in the context of the +inner Galaxy separating Heracles (as defined above) from the +bulge selection. We reproduced their Figure 5 in our Figure 26 +and removed their high-E stars with e < 0.4. The main-bulge +progenitor is shown in the left panel, and the Heracles (suppos- +edly ex-situ) progenitor is shown in the right panel. NGC 6355 +and the selected GCs are placed almost in the same region in the +IOM space (Figure 26). However, it is not possible to distinguish +a specific region for each progenitor based on the contour curves +alone. +Although NGC 6355 has properties of ex-situ GCs such as +the [Mg/Mn] and [Al/Fe] abundances, which are compatible +Article number, page 15 of 20 + +1.0 +0.8 +NGC6558 +0.6 +[X/Fe] +Pa16 +0.4 +NGC6522 +0.2 +HPI +NGC6355 +0.0 +-0.2 +Y +Zr +Ba +La +Nd +Eu +39 +40 +56 +57 +60 +63 +ZEx situ +In 'situ high α +1.0 +<2G> +0.8 +KIG> +0.6 +[Mg/Mn] +0.4 +0.2 +0.0 +-0.2 +In situ Iow α +-0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +[Al/Fe]A&A proofs: manuscript no. aanda +Fig. 25. AMR for the Galactic GCs system. The grey dots are the values +reported in Kruijssen et al. (2019). For comparison criteria, the mean +locus of in-situ and ex-situ populations are shown (see text for details). +The symbols are the same as in Figure 20. +Fig. 26. IOM space for the bulge stars selected by Horta et al. (2021). +The left panel shows the contours for the main-bulge progenitor stars. +The right panel shows the contours of the Heracles progenitor. The stars +are coulored as in Figure 19. +with Heracles, we can confirm the in-situ origin of NGC 6355 +because it is confined to the volume of the Galactic bulge. From +the point of view of chemical abundances, most of its element +abundances follow the in-situ clusters and bulge RR Lyrae pop- +ulation, including its low Zn abundance, which appears to be +compatible with the chemodynamical evolution models. The old +age of NGC 6355 completes the in-situ scenario for the cluster +because its age fits the predictions for the early evolution of the +Galactic bulge perfectly. +7. Conclusions +We analysed the globular cluster NGC 6355 in the context +of the evolution of the Galactic bulge. This task required a +deep and careful analysis, including photometry, chemical abun- +dances, and dynamics. To do this, we gathered high-resolution +spectroscopy from FLAMES-UVES, photometry from the HST +F438W/F555W filters, and Galactic dynamics calculations. +The spectroscopic analysis resulted in a metallicity of +[Fe/H]= −1.39 ± 0.08 for NGC 6355. This means that the GC +is one of the metal-poor clusters of the Galactic bulge. A mean +[α/Fe]= +0.36 ± 0.09 was derived based on O, Mg, Si, and Ca +abundances, indicating that NGC 6355 is characterized by en- +richment from supernovae type II. We found good agreement +among NGC 6355 abundances, bulge GCs, and the bulge RR +Lyrae sample, except for the cases of Ca, Zn, and Ba, for which +Ca and Ba appear to be enhanced, and Zn, which is deficient +relative to the comparison clusters. At the same time, Ca is com- +patible with the RR Lyrae sample. We tested the hypothesis of a +different origin for NGC 6355 with the [Mg/Mn]-[Al/Fe] plane +and found that NGC 6355 is in the accreted region of the plane +when we assume only the Al-depleted (1G) stars. We found that +two of the four stars are Al-depleted and N-normal, while the +others are N- and Al-rich. The detection of MPs was confirmed +through the (anti-)correlations Al-N/Na (Mészáros et al. 2020), +and Na-O (Carretta et al. 2009a). +From isochrone fitting, we derive an age of 13.2 ± 1.1 Gyr, +reinforced by the fact that the cluster has a BHB, similar to other +old and moderately metal-poor bulge GCs such as NGC 6522, +NGC 6558, HP 1, AL 3, Terzan 9, and UKS 1. This old age +places NGC 6355 perfectly at the in-situ branch of the AMR. Be- +cause of the rather low metallicity of NGC 6355 relative to these +other clusters, these results raise the question which would be the +lowest metallicity for the family of metal-poor globular clusters +of the Galactic bulge, as discussed in Geisler et al. (2022). +Finally, we investigated the IOM space. This is not conclu- +sive because as observed by Horta et al. (2021), the internal re- +gion of the Galaxy is dynamically mixed, and it is not possible +to separate the ex-situ stars from the in-situ population. As ev- +idence of this, Kruijssen et al. (2019) concluded that the low- +energy progenitor called Kraken must have been formed at red- +shift z = 4, or ∼ 12.3 Gyr ago, indicating that the early merger +of the Kraken galaxy with the Galaxy had enough time to mix +dynamically. +In conclusion, according to our photochemodynamical anal- +ysis, NGC 6355 is currently a member of the Galactic bulge and +seems to have originated from the main-bulge progenitor, with a +low probability of an ex-situ origin. More spectroscopic data of +cluster stars could provide more consolidated abundance values, +giving us more constraints on the NGC 6355 origin and its MPs +formation scenario. +Acknowledgements. We thank an anonymous referee for suggestions that have +improved the quality of this paper. SOS acknowledges a FAPESP PhD fellow- +ship no. 2018/22044-3. HE acknowledges a CAPES PhD fellowship. SOS and +MV acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG, +project number: 428473034). 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A., 2002, RvMP, 74, 1015. +Zinn, R. & West, M. J. 1984, ApJS, 55, 45. +Article number, page 17 of 20 + +A&A proofs: manuscript no. aanda +Appendix A: Line lists +Table A.1. Equivalent widths for Fe i and Fe ii lines. +Ion +λ +χex +loggf +star 1539 +star 1363 +star 1176 +star 133 +[Å] +[eV] +[m Å] +Fe ii +5991.38 +3.10 +−3.65 +— +21.1 +33.4 +17.0 +Fe ii +6084.11 +3.20 +−3.97 +— +16.8 +13.2 +11.8 +Fe ii +6149.25 +3.80 +−2.69 +— +62.4 +28.1 +17.5 +Fe ii +6247.56 +3.80 +−2.30 +41.6 +— +47.1 +29.9 +Fe ii +6416.93 +3.80 +−2.64 +29.7 +34.6 +25.5 +18.1 +Fe ii +6432.68 +2.80 +−3.57 +42.2 +42.4 +40.2 +33.1 +Fe ii +6456.39 +3.90 +−2.31 +— +48.2 +61.7 +40.8 +Fe ii +6516.08 +2.80 +−3.31 +53.8 +17.1 +47.0 +42.7 +Fe i +5905.67 +4.60 +−0.73 +— +30.3 +— +27.2 +Fe i +5916.25 +2.40 +−2.97 +87.1 +— +55.6 +— +Fe i +5927.79 +4.60 +−1.09 +— +23.7 +17.1 +19.3 +Fe i +5929.67 +4.50 +−1.41 +23.9 +17.7 +14.4 +16.0 +Fe i +5930.18 +4.60 +−0.23 +66.3 +62.6 +46.7 +49.9 +Fe i +5934.65 +3.90 +−1.17 +70.3 +— +49.0 +54.2 +Fe i +5952.73 +3.90 +−1.44 +50.9 +49.2 +38.9 +43.3 +Fe i +5956.69 +0.80 +−4.60 +119.8 +— +78.4 +— +Fe i +5975.35 +4.80 +−0.69 +33.6 +35.7 +23.6 +— +Fe i +5983.69 +4.50 +−1.47 +— +44.4 +34.0 +— +Fe i +5987.06 +4.80 +−0.43 +39.7 +— +25.3 +— +Fe i +6003.01 +3.80 +−1.12 +74.9 +74.5 +54.2 +65.3 +Fe i +6005.54 +2.50 +−3.61 +37.8 +41.6 +19.2 +24.9 +Fe i +6008.56 +3.80 +−0.99 +84.3 +— +60.1 +67.0 +Fe i +6020.17 +4.60 +−0.27 +73.3 +— +50.0 +54.5 +Fe i +6024.05 +4.50 +−0.12 +82.2 +85.0 +67.4 +72.0 +Fe i +6027.06 +4.00 +−1.09 +55.7 +57.8 +34.0 +38.6 +Fe i +6054.08 +4.30 +−2.31 +— +— +— +— +Fe i +6065.48 +2.60 +−1.53 +148.7 +— +124.5 +— +Fe i +6078.49 +4.80 +−0.32 +43.5 +45.5 +26.9 +37.8 +Fe i +6079.01 +4.60 +−1.12 +21.2 +23.2 +13.6 +17.0 +Fe i +6082.71 +2.20 +−3.57 +67.4 +69.1 +33.1 +50.0 +Fe i +6093.64 +4.60 +−1.50 +20.3 +14.6 +20.1 +12.8 +Fe i +6137.70 +2.50 +−1.40 +— +— +137.6 +— +Fe i +6151.62 +2.10 +−3.30 +82.6 +— +53.7 +72.7 +Fe i +6165.36 +4.10 +−1.47 +32.4 +30.1 +16.6 +17.6 +Fe i +6180.21 +2.70 +−2.59 +82.4 +84.8 +49.6 +64.2 +Fe i +6187.99 +3.90 +−1.72 +40.0 +37.3 +22.4 +31.8 +Fe i +6213.44 +2.20 +−2.48 +122.4 +— +96.4 +— +Fe i +6219.29 +2.20 +−2.43 +131.7 +— +109.2 +— +Fe i +6220.78 +3.80 +−2.46 +10.3 +— +— +— +Fe i +6226.73 +3.80 +−2.22 +17.5 +23.1 +13.8 +16.1 +Fe i +6252.57 +2.40 +−1.69 +— +— +134.3 +— +Fe i +6254.25 +2.20 +−2.44 +133.7 +— +114.8 +— +Fe i +6265.14 +2.10 +−2.55 +128.2 +— +110.6 +— +Fe i +6270.23 +2.80 +−2.46 +71.7 +76.1 +42.6 +58.0 +Fe i +6271.28 +3.30 +−2.70 +27.4 +27.3 +15.2 +16.9 +Fe i +6301.51 +3.60 +−0.72 +104.0 +— +86.7 +— +Fe i +6311.50 +2.80 +−3.14 +43.6 +42.7 +23.3 +28.9 +Fe i +6315.31 +4.10 +−1.23 +45.3 +41.9 +28.3 +— +Fe i +6315.81 +4.00 +−1.71 +29.2 +28.7 +10.3 +18.2 +Fe i +6335.34 +2.20 +−2.18 +142.7 +— +122.2 +— +Fe i +6344.16 +2.40 +−2.92 +91.2 +— +62.8 +72.9 +Fe i +6380.75 +4.10 +−1.38 +42.2 +35.9 +26.3 +33.5 +Fe i +6392.54 +2.20 +−4.03 +36.9 +37.5 +— +— +Fe i +6393.61 +2.40 +−1.43 +— +— +140.8 +— +Fe i +6411.66 +3.60 +−0.60 +115.8 +— +98.5 +101.7 +Fe i +6419.94 +4.70 +−0.24 +57.5 +55.9 +40.6 +45.1 +Fe i +6421.35 +2.20 +−2.03 +— +— +132.5 +— +Article number, page 18 of 20 + +Souza et al.: Globular cluster NGC 6355 +Table A.1 – continued +Ion +λ +χex +loggf +star 1539 +star 1363 +star 1176 +star 133 +Fe i +6430.86 +2.10 +−2.01 +— +— +132.8 +— +Fe i +6469.21 +4.80 +−0.77 +— +— +— +— +Fe i +6475.63 +2.50 +−2.94 +82.5 +— +52.8 +68.5 +Fe i +6546.25 +2.70 +−1.54 +136.6 +— +116.3 +— +Fe i +6569.22 +4.70 +−0.42 +54.3 +55.8 +42.5 +46.4 +Fe i +6581.21 +1.40 +−4.68 +52.6 +63.0 +16.7 +39.1 +Fe i +6593.87 +2.40 +−2.42 +125.9 +— +98.8 +— +Fe i +6597.56 +4.80 +−1.07 +21.0 +21.1 +— +14.9 +Fe i +6608.04 +2.20 +−4.03 +37.5 +37.9 +— +— +Fe i +6609.12 +2.50 +−2.69 +97.6 +— +64.0 +81.1 +Fe i +6627.54 +4.50 +−1.68 +— +14.5 +— +11.2 +Fe i +6678.00 +2.60 +−1.42 +— +— +134.5 +— +Table A.2. Line-by-line abundances ratios in the six UVES sample stars for the +odd-Z (Na and Al), alpha- (Mg, Si, Ca) + Ti, iron-peak (V, Mn, Co, Cu, and Zn), +and heavy elements (Y, Zr, Ba, La, Nd, and Eu). +Species +λ +χex +loggf +star 1539 +star 1363 +star 1176 +star 133 +[Å] +[eV] +[X/Fe] +Mg i +6318.720 +5.11 +−2.36 ++0.35 ++0.47 ++0.38 ++0.47 +Mg i +6319.242 +5.11 +−2.80 ++0.32 ++0.41 +— +— +Si i +5665.555 +4.92 +−2.04 ++0.20 ++0.20 ++0.25 ++0.40 +Si i +5666.690 +5.62 +−1.74 +— +— ++0.56 ++0.10 +Si i +5690.425 +4.93 +−1.87 ++0.35 ++0.33 ++0.40 ++0.30 +Si i +5948.545 +5.08 +−1.30 ++0.30 ++0.30 ++0.30 ++0.35 +Si i +6142.494 +5.62 +−1.50 ++0.45 ++0.30 +— +−0.40 +Si i +6145.020 +5.61 +−1.45 ++0.30 ++0.50 ++0.15 ++0.50 +Si i +6155.142 +5.62 +−0.85 ++0.25 ++0.19 ++0.35 ++0.30 +Si i +6237.328 +5.61 +−1.01 ++0.06 ++0.14 ++0.25 ++0.35 +Si i +6243.823 +5.61 +−1.30 ++0.26 ++0.21 ++0.35 ++0.35 +Si i +6414.987 +5.87 +−1.13 ++0.23 ++0.06 ++0.56 ++0.10 +Si i +6721.844 +5.86 +−1.17 +— +— ++0.10 ++0.69 +Ca i +5601.277 +2.53 +−0.52 ++0.43 ++0.47 ++0.09 ++0.68 +Ca i +5867.562 +2.93 +−1.55 ++0.26 ++0.31 ++0.35 ++0.45 +Ca i +6156.030 +2.52 +−2.39 +— +— +— +— +Ca i +6102.723 +1.88 +−0.79 ++0.60 ++0.50 ++0.70 ++0.70 +Ca i +6122.217 +1.89 +−0.20 ++0.50 ++0.50 ++0.70 ++0.70 +Ca i +6161.295 +2.51 +−1.02 ++0.36 ++0.44 ++0.20 ++0.44 +Ca i +6162.167 +1.90 +−1.09 ++0.30 ++0.30 ++0.70 ++0.50 +Ca i +6166.440 +2.52 +−0.90 ++0.15 ++0.30 ++0.09 ++0.30 +Ca i +6169.044 +2.52 +−0.54 ++0.61 ++0.50 ++0.05 ++0.55 +Ca i +6169.564 +2.52 +−0.27 ++0.65 ++0.58 ++0.15 ++0.71 +Ca i +6439.080 +2.52 ++0.30 ++0.75 ++0.55 ++0.70 ++0.75 +Ca i +6455.605 +2.52 +−1.35 ++0.30 ++0.44 ++0.20 ++0.53 +Ca i +6464.679 +2.52 +−2.10 ++0.60 ++0.70 ++0.55 ++0.70 +Ca i +6493.788 +2.52 +−2.44 ++0.50 ++0.50 ++0.55 ++0.50 +Ca i +6499.654 +2.52 +−0.85 ++0.50 ++0.50 ++0.10 ++0.50 +Ca i +6572.779 +0.00 +−4.32 ++0.60 ++0.49 +−0.01 ++0.55 +Ca i +6717.687 +2.71 +−0.61 ++0.50 ++0.50 ++0.25 ++0.50 +Ti i +5922.108 +1.05 +−1.46 ++0.51 ++0.55 ++0.18 ++0.48 +Ti i +5941.750 +1.05 +−1.50 ++0.36 ++0.40 ++0.28 ++0.33 +Ti i +5965.825 +1.88 +−0.42 ++0.30 ++0.45 ++0.08 ++0.34 +Ti i +5978.539 +1.87 +−0.53 ++0.40 ++0.30 ++0.13 ++0.37 +Ti i +6064.623 +1.05 +−1.94 ++0.31 ++0.30 ++0.05 ++0.28 +Ti i +6091.169 +2.27 +−0.42 ++0.14 ++0.26 +— ++0.23 +Ti i +6126.214 +1.07 +−1.43 ++0.30 ++0.40 ++0.10 ++0.30 +Ti i +6258.110 +1.44 +−0.36 ++0.10 ++0.15 ++0.19 ++0.10 +Ti i +6261.106 +1.43 +−0.48 ++0.30 ++0.50 ++0.10 ++0.30 +Ti i +6312.240 +1.46 +−1.60 ++0.24 ++0.40 ++0.09 ++0.35 +Article number, page 19 of 20 + +A&A proofs: manuscript no. aanda +Table A.2 – continued +Species +λ +χex +loggf +star 1539 +star 1363 +star 1176 +star 133 +Ti i +6336.113 +1.44 +−1.74 ++0.20 ++0.34 ++0.26 ++0.18 +Ti i +6554.238 +1.44 +−1.22 ++0.10 ++0.15 ++0.06 ++0.26 +Ti i +6556.077 +1.46 +−1.07 ++0.30 ++0.40 ++0.22 ++0.32 +Ti i +6599.113 +0.90 +−2.09 ++0.28 ++0.30 ++0.10 ++0.30 +Ti i +6743.127 +0.90 +−1.73 ++0.20 ++0.26 +−0.19 ++0.20 +Ti ii +5418.751 +1.58 +−2.13 ++0.30 ++0.38 ++0.30 ++0.43 +Ti ii +6491.580 +2.06 +−2.10 ++0.30 ++0.38 ++0.30 ++0.43 +Ti ii +6559.576 +2.05 +−2.35 ++0.26 ++0.30 ++0.23 ++0.36 +Ti ii +6606.970 +2.06 +−2.85 ++0.35 ++0.30 ++0.31 ++0.30 +Na i +5682.633 +2.10 +−0.71 +−0.32 +−0.30 +−0.00 ++0.31 +Na i +5688.194 +2.10 +−1.40 +−0.35 +−0.30 +−0.30 ++0.20 +Na i +6154.230 +2.10 +−1.56 +−0.15 +−0.00 +— ++0.30 +Na i +6160.753 +2.10 +−1.26 +−0.35 +−0.00 +−0.35 ++0.00 +Al i +6696.185 +4.02 +−1.58 +−0.30 +−0.30 +< +0.30 +< +0.30 +Al i +6698.673 +3.14 +−1.65 +−0.29 ++0.00 +< +0.30 +< +0.30 +V i +5703.560 +1.05 +−0.21 ++0.05 ++0.17 +— ++0.14 +V i +6081.440 +1.05 +−0.58 +−0.02 ++0.17 +— ++0.02 +V i +6090.220 +1.08 +−0.16 ++0.05 ++0.14 +−0.32 ++0.08 +V i +6119.520 +1.06 +−0.47 +−0.05 ++0.08 +−0.26 +−0.05 +V i +6199.190 +0.29 +−1.48 ++0.05 ++0.17 +— +−0.11 +V i +6243.100 +0.30 +−0.88 ++0.11 ++0.38 +— ++0.05 +V i +6251.820 +0.29 +−1.44 ++0.11 ++0.32 +−0.41 +−0.05 +V i +6274.650 +0.27 +−1.72 +−0.05 ++0.11 +— +−0.08 +Mn i +5394.669 +0.00 +−3.55 +−0.40 +−0.50 +−0.40 +−0.50 +Mn i +6013.513 +3.07 +−0.40 +−0.30 +−0.25 +−0.30 +−0.30 +Mn i +6016.640 +3.07 +−0.22 +−0.35 +−0.50 +−0.60 +−0.40 +Mn i +6021.800 +3.08 +−0.10 +−0.30 +−0.45 +−0.25 +−0.50 +Co i +5212.691 +3.51 +−0.11 +— ++0.15 +— ++0.15 +Co i +5301.047 +1.71 +−2.00 ++0.00 ++0.10 +— ++0.10 +Co i +5342.708 +4.02 ++0.69 ++0.05 ++0.05 +— ++0.00 +Co i +5454.572 +4.07 ++0.24 +— ++0.10 ++0.20 ++0.10 +Co i +5647.234 +2.28 +−1.56 ++0.05 ++0.05 ++0.00 ++0.30 +Co i +6188.996 +1.71 +−2.45 ++0.00 ++0.00 ++0.00 ++0.30 +Cu i +5105.537 +1.39 +−1.52 +−0.40 +−0.15 +−0.30 +−0.35 +Cu i +5218.197 +3.82 ++0.00 +−0.30 ++0.00 ++0.05 ++0.00 +Zn i +6362.339 +5.79 +−0.30 +−0.30 +−0.20 ++0.30 +−0.10 +Y i +6435.004 +0.07 +−0.82 +−0.30 +−0.32 +−0.30 ++0.00 +Y ii +6795.414 +1.74 +−1.19 ++0.20 ++0.00 ++0.00 +— +Zr i +6127.475 +0.15 +−1.06 +−0.08 ++0.05 +— +−0.08 +Zr i +6134.585 +0.00 +−1.42 ++0.05 ++0.20 +— ++0.23 +Zr i +6140.535 +0.52 +−1.60 +— +— +— +−0.50 +Zr i +6143.252 +0.07 +−1.10 +−0.14 ++0.02 +— +−0.08 +Ba ii +5853.675 +0.60 +−1.10 ++0.92 ++1.00 ++1.00 ++1.05 +Ba ii +6141.713 +0.70 +−0.08 ++0.60 ++0.80 ++0.65 ++0.80 +Ba ii +6496.897 +0.60 +−0.32 ++1.00 ++1.00 ++1.10 ++1.20 +La ii +6262.287 +0.40 +−1.60 ++0.00 ++0.00 ++0.17 ++0.26 +La ii +6320.376 +0.17 +−1.56 ++0.00 ++0.00 ++0.14 ++0.30 +La ii +6390.477 +0.32 +−1.41 ++0.25 ++0.17 ++0.00 ++0.26 +Nd ii +6740.078 +0.06 +−1.53 ++0.45 ++0.15 ++0.17 +−0.30 +Nd ii +6790.372 +0.18 +−1.77 ++0.55 ++0.40 ++0.00 +−0.30 +Nd ii +6549.525 +0.06 +−2.01 ++0.40 ++0.30 ++0.00 +— +Eu ii +6437.640 +1.32 +−0.32 ++0.55 ++0.55 ++0.65 ++0.50 +Eu ii +6645.064 +1.38 ++0.12 ++0.50 ++0.55 ++0.50 ++0.70 +Article number, page 20 of 20 + diff --git a/mNE4T4oBgHgl3EQftw2O/content/tmp_files/load_file.txt b/mNE4T4oBgHgl3EQftw2O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d66290ebf37d2c08ea4028588522cac39cb13d34 --- /dev/null +++ b/mNE4T4oBgHgl3EQftw2O/content/tmp_files/load_file.txt @@ -0,0 +1,3981 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf,len=3980 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda ©ESO 2023 January 13, 2023 Chrono-chemodynamical analysis of the globular cluster NGC 6355: Looking for the fundamental bricks of the Bulge ⋆ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Souza1, 2 , H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ernandes3, 2 , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Valentini1 , B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Barbuy2 , C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Chiappini1 , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Pérez-Villegas4 , S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ortolani5, 6, 7 , A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Friaça2, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Queiroz1, 8 , and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Bica9 1 Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, Potsdam, D-14482, Germany e-mail: ssouza@aip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='de 2 Universidade de São Paulo, IAG, Rua do Matão 1226, Cidade Universitária, São Paulo 05508-900, Brazil e-mail: stefano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='souza@usp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='br 3 Lund Observatory, Department of Astronomy and Theoretical Physics, Lund University, Box 43, SE-221 00 Lund, Sweden 4 Instituto de Astronomía, Universidad Nacional Autónoma de México, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 106, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 22800, Ensenada, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=', México 5 Università di Padova, Dipartimento di Astronomia, Vicolo dell’Osservatorio 2, I-35122 Padova, Italy 6 INAF-Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy 7 Centro di Ateneo di Studi e Attività Spaziali "Giuseppe Colombo" - CISAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Via Venezia 15, 35131 Padova, Italy 8 Institut für Physik und Astronomie, Universität Potsdam, Haus 28 Karl-Liebknecht-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 24/25, D-14476 Golm (Potsdam), Germany 9 Universidade Federal do Rio Grande do Sul, Departamento de Astronomia,CP 15051, Porto Alegre 91501-970, Brazil Received September 15, 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' accepted March 16, 1997 ABSTRACT The information on Galactic assembly time is imprinted on the chemodynamics of globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This makes them important probes that help us to understand the formation and evolution of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Discerning between in-situ and ex-situ origin of these objects is difficult when we study the Galactic bulge, which is the most complex and mixed component of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To investigate the early evolution of the Galactic bulge, we analysed the globular cluster NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We derived chemical abundances and kinematic and dynamic properties by gathering information from high-resolution spectroscopy with FLAMES-UVES, photometry with the Hubble Space Telescope, and Galactic dynamic calculations applied to the globular cluster NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We derive an age of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 Gyr and a metallicity of [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 for NGC 6355, with α-enhancement of [α/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The abundance pattern of the globular cluster is compatible with bulge field RR Lyrae stars and in-situ well-studied globular clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The orbital parameters suggest that the cluster is currently confined within the bulge volume when we consider a heliocentric distance of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 kpc and an extinction coefficient of RV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' NGC 6355 is highly likely to come from the main bulge progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Nevertheless, it still has a low probability of being formed from an accreted event because its age is uncertain and because of the combined [Mg/Mn] [Al/Fe] abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Its relatively low metallicity with respect to old and moderately metal-poor inner Galaxy clusters may suggest a low-metallicity floor for globular clusters that formed in-situ in the early Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Galaxy: Bulge – Globular Clusters: individual: NGC 6355 – Stars: Abundances, Atmospheres – Stars: Hertzsprung- Russell and C–M diagrams – Galaxy: kinematics and dynamics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Introduction The ΛCDM hierarchical theory of galaxy formation predicts that a galaxy forms from successive mergers of low-mass objects that are absorbed by more massive objects (Peebles 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' White & Rees 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Kauffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Springel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The less massive objects are gradually absorbed while orbiting the mas- sive objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The Milky Way (MW) contains remnants of this early history that can be divided into two groups: those still or- biting the Galaxy, with their structures entirely or almost intact (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' the Magellanic Clouds);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' and another group of objects that were already dissolved by the MW after several encounters and were completely accreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The latter objects could have retained the dynamic signatures of their progenitor if the merger event occurred during the recent evolution of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' An example is Gaia-Sausage-Enceladus (GSE), known as the remnant of the ⋆ Based on observations from ESO Programs 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0063 (A) (PI: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ortolani) and 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0136 (A) (PI: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Valentini), and HST Project GO-11628 (PI:Noyola).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' last major merger of the MW with a dwarf galaxy (Belokurov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Helmi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Because the estimated merger time is ∼ 8 Gyr (Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Montalbán et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2021), the rem- nant stars did not have enough time to change their dynamical properties completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In addition to the dynamic properties, mergers influence the chemical properties of the Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Grand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is expected that an old, moderately metal-poor stellar population will be formed upon the halt in the star formation history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Many authors have tried to identify the chemodynamical imprints of the early assembly steps that are left on the Galactic stellar pop- ulations with the aim to constrain these important events in the Galaxy history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This is now possible based on the joint informa- tion from large spectroscopic surveys and the Gaia proper mo- tion data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Hayden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Anders et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Kordopatis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Queiroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Buder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2022, among many others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The Galactic bulge is one of the most complex regions of the Galaxy because in addition to the high extinction, it contains Article number, page 1 of 20 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05227v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='GA] 12 Jan 2023 IDA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda stellar populations from several parts of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' However, a study of the bulge can provide information about its complex formation processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Queiroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Rojas-Arriagada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In order to distinguish the different stellar populations, we have to study the object or- bit (Pérez-Villegas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018) together with ages and chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The orbits are a key ingredient that provides infor- mation whether the object always lived in the bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Queiroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021, hereafter Q21) mapped and analysed the stellar popu- lations of the bulge from a chemodynamical point of view, which allowed them to describe the stellar content of the bulge field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Another way to characterize the Galactic bulge comes from the old stellar population, such as RR Lyrae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Savino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020) analysed the stellar population in the inner spheroid of the Galaxy and reported that this structure is very old, with an age of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='54 Gyr, and it is also metal poor, with a metallic- ity of [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 (Pietrukowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2015), [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 (Minniti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2016), [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='55 (Crestani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2021), and [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='35 (Dékány & Grebel 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The globular clusters (GCs) are also important tracers of the formation and evolu- tion of the Galaxy because they are old and retain the chemo- dynamical signatures of the first stages of the MW formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Some studies have demonstrated that the metallicity distribution of bulge GCs peaks at [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 (Bica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Pérez- Villegas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020, and references therein), and that they are mostly older than 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Gyr (Miglio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2016, 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Kerber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ortolani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fernández-Trincado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The assignment to which Galactic component a GC belongs to is made depending on its orbital integration in order to ver- ify the most probable regions of its trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' While part of the GCs could have formed in the main progenitor of the Galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' main bulge or main disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019), others could come from accreted progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To study the origin of a GC, we therefore need to analyse its chemical, photometric, and dynam- ical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The age-metallicity relation (AMR) of the MW GCs shows a bifurcation that splits it into two main groups (Marín-Franch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Forbes & Bridges 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Leaman, VandenBerg, & Mendel 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' A steeper branch in which more older GCs are concentrated is associated with an in-situ population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In con- trast, the other component is broader and includes very young to old ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is also associated with accretion events during the early evolution of the Galaxy (Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Forbes 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Limberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Callingham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' When an axisymmetric Galactic potential is employed, the so-called integrals of motion space (IOM) can be used to- gether with the AMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' By studying the total energy (E) versus Z-component of the angular momentum (LZ), it is possible to investigate the dynamic history of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For example, the region of lower E with almost zero LZ can be associated with the inner part of the MW, the bounded objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' On the other hand, the Galactic halo accreted objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' GSE) are in the region of high E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Therefore, the combination of the AMR with the IOM space has improved the knowledge about the origin of the GCs system, helping us to understand the Galactic evolutionary his- tory, particularly that of the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Observing GCs within the Galactic bulge is difficult be- cause the extinction tends to hide the objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' One example is NGC 6355 (also called GCl-63 and ESO 519-SC15), pro- jected towards the direction of the Galactic bulge (l = 359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='58◦, b = +5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='43◦) with a relatively high extinction (E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='79;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Harris 1996, 2010 edition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' NGC 6355 is a well-known cluster that has been studied since the 1900s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is classified as a probable open cluster (Shapley & Shapley 1919).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' However, it did not take long before its globular nature was confirmed based on its rela- tively high mass, which according to Baumgardt & Hilker (2018) is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='01×105 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Djorgovski & King (1986) classified NGC 6355 as a core-collapse cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This result was recently confirmed by Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021a) using the Hubble Space Telescope (HST) filters F606W and F814W from the Advanced Camera for Sur- vey (ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ortolani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2003) analysed the horizontal branch (HB) and the red giant branch (RGB) of NGC 6355 using a [V,V − I] colour-magnitude diagram (CMD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' They obtained a reddening of E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='78, a distance of d⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 kpc, and a metal- licity of [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This was deduced by comparing the cluster mean locus with the mean loci of the well-studied clus- ters NGC 6171 and M 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Assuming their distance derivation, the authors concluded that the cluster is near the Galactic cen- ter (see also Bica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Valenti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2007) analysed the RGB slope and the K magnitude of the RGB tip using the [K,J − K] and [H, J − H] CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' They found E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='82, d⊙ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='7kpc, and [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Both results agree with the metal- licity scales of Carretta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2009b) and Zinn & West (1984) of [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='33±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='14 and [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Subsequent metallicity derivation by Vásquez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2015) and Dias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2016) of [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='49 and [Fe/H]∼ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46, respec- tively, are also within the range of both metallicity scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2009) identified NGC 6355 as a blue horizon- tal branch (BHB) metal-poor GC, located in the ring at −6◦ – −12◦ around the Galactic centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This suggested that NGC 6355 belonged to the BHB moderately metal-poor clusters of the Galactic bulge, such as NGC 6558 (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2007, 2018b), HP 1 (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2006, 2016), AL 3 (Ortolani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2021), Terzan 9 (Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019), and UKS 1 (Fernández-Trincado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Nevertheless, when examined from the orbital viewpoint, it was suggested that NGC 6355 is more compatible with the Galactic thick disk with a probability of 93% (Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Pérez-Villegas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018, 2020, here- after PV20), assuming a distance of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='87 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It also has a probability of 7% to be part of the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Here we stress the importance of having a precise distance derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Kharchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2016, hereafter KC16) analysed 147 GCs including NGC 6355 using integrated JHKs magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' They derived its age as log t = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 (∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Gyr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Assuming this age derivation and the distances derived by Baumgardt & Hilker (2018), Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2019) found that NGC 6355 may have been formed from the main-bulge progenitor and might there- fore be an in-situ cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Their result for NGC 6355 was con- firmed with a more realistic approach adopted in Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2022), who employed the formalism for dynamical friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' More recently, Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021b, hereafter C21) derived a rel- ative age of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 Gyr by comparing the CMDs of NGC 6355 and NGC 6205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The authors give an absolute age of ∼ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 Gyr for NGC 6355 and assume an age of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 Gyr for NGC 6205 (Van- denberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2013, hereafter VB13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This relatively older age compared to the previous one by KC16 was used by Calling- ham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2022) to reclassify NGC 6355 as compatible with the main-bulge progenitor and also with the Kraken accreted struc- ture as an alternative origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is worth noting that a possible accreted structure within the Galactic bulge was hypothesized also by Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2019) (low-energy progenitor), Kruijssen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2019) (Kraken), Forbes (2020) (Koala), and Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021) (Heracles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In the present work, we combine the chemical information with photometric and dynamical properties of the cluster to Article number, page 2 of 20 Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' : Globular cluster NGC 6355 constrain its history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The chemical information is based on the UVES spectrograph (Dekker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2000) in FLAMES-UVES mode at the ESO-VLT, the photometry on HST data, and the dy- namical properties are provided by orbital integration employing the McMillan (2017) Galactic potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The photometry pro- cessing, reduction of spectra, and membership analysis of the observed stars are described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Section 3 gives the derivation of fundamental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The analysis of individ- ual line abundances and the comparison with the literature are described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The orbital analysis and dynamical prop- erties of NGC 6355 are presented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In Section 6 we discuss the origin of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, the conclusions are drawn in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' HST photometry processing The photometric data for NGC 6355 were retrieved from the HST Project (GO-11628, PI:Noyola), which used the Wide Field Camera for Surveys 3 (WFC3) with the filters F438W and F555W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The observation consists of three F438W images with an exposure time of 440 s, and three F555W images with an ex- posure time of 80 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Figure 1 shows the colour image composed of the combined HST images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We performed a further selection based on the pipeline described in Nardiello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018) us- ing the quality-of-fit and photometric error parameters to select well-measured stars and reject poor measurements (top left panel of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Additionally, we selected stars within a half-light radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='88 arcmin (Harris 1996, 2010 edition) to avoid a sub- stantial number of field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For the resulting sample, we com- puted a simple membership probability by combining the stars offset from the fiducial line on the CMD with the star distance to the cluster centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The extinction towards the cluster is relatively high, and it increases the CMD spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To reduce the effect of differential reddening, we used the same method as was applied to Palomar 6 in Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021) (adapted from Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Bedin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The differential reddening map (bottom left panel of Figure 2) shows that δE(B−V) ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04, which is approximately 5% of the expected reddening (E(B-V)= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='79;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Harris 1996, 2010 edition), and which we can convert into a magnitude difference of δmF438W = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 and δmF555W = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='13, and into a difference in colour δ (mF438W − mF555W) = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, to scale the photometry to the same zero-point as in the evolutionary models, we converted the AB magnitudes into the Vega system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The final sample corrected for differential reddening shows a smaller spread and a clear morphology from the RGB and HB to the lower MS (top right panel of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The ACS F438W/F555W photometry is saturated for mag- nitudes brighter than F555W∼ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Therefore, our spectroscopic targets were not observed for these filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To estimate the posi- tion of our stars in the CMD, we derived an approximation of their F438W and F555W magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We fixed the reddening, metallicity, and distance modulus from Harris (1996, 2010 edi- tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For each star, we fitted the magnitudes J, KS , G, GBP, and GRP (green triangles in Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We also used this method for a sample of RGB stars of the Gaia EDR3 from the Vasiliev & Baumgardt (2021) catalogue (open red circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is worth not- ing that the F438W filter is affected by variations in C, N, and O abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Hence, this filter can better be estimated via spectral convolution and integration with the filter response curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' F438W/F555W combined colour image from the HST WFC3 camera for NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Log of the spectroscopic FLAMES-UVES observations of pro- grams 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0063 (A) and 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0136 (A), carried out in 2009 and 2017, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The reported seeing and airmass are the mean values in the exposures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The last column contains the corresponding GIRAFFE setup, in which additional stars were observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Date UT exp Airmass Seeing SETUP ( s ) (′′) GIRAFFE Program 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0063 (A) 2009-09-02 02:48:43 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='455 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='88 H13-1 2009-09-01 01:03:00 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='87 H13-2 2009-09-01 01:50:54 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='72 H13-3 2009-09-13 23:32:32 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='91 H13-4 2009-09-14 00:31:12 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='82 H14-1 2009-09-14 01:17:51 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='78 H14-2 2009-09-14 02:04:21 2700 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='75 H14-3 Program 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0136 (A) 2017-07-14 06:21:39 2400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='75 H11-1 2017-07-14 04:34:44 2400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='67 H11-2 2017-09-02 01:50:12 2400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='61 H11-4 2017-09-07 02:53:12 2400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='54 H13-1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Spectral data reduction The UVES spectra were obtained using the FLAMES-UVES setup centred at 580 nm, covering the wavelength range 480 - 680 nm, from the ESO Programs 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0063 (A) (PI: S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Or- tolani) and 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0136 (A) (PI: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Valentini).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The latter ESO program was coordinated with the program GO11126 (PI: M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Valentini) for campaign 11 of the K2 satellite (K2 is the repur- posed Kepler mission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Howell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2014): the goal was to obtain asteroseismology for the giants in the sample GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' However, ob- taining reliable light curves for these stars was not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The log of observations is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We performed the FLAMES-UVES data reduction pro- cedure using the ESO-Reflex software with the UVES-Fibre pipeline (Ballester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Modigliani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The cor- Article number, page 3 of 20 N 个 E F438W F555WA&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Photometric data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Left panel: All stars within the FOV obtained from the HST Project (GO-11628, PI: Noyola).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Middle panel: Final differential-reddening-corrected CMD with selected stars (black) and discarded stars (grey).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The Gaia EDR3 member stars from the Vasiliev & Baumgardt (2021) catalogue matched with 2MASS to obtain the HST are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The green triangles represent the observed stars with HST magnitudes obtained from the isochrone calibration, and the sizes are from the S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The HB region is plotted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Right panel: Differential reddening map for stars within a half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The resolution of the map is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='024 arcminutes (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 arcseconds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' responding spectra of each star were corrected for the radial ve- locity computed using the Python library PyAstronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The radial velocities were obtained by cross-correlating the stellar spectra with the Arcturus spectrum (Hinklen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The values of the heliocentric radial velocity of each spectrum and their mean values are presented in Table 2 for the member stars, selected from the membership analysis (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The spectra of stars 1546 and 1239 from ESO Program 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0063, have a low signal-to-noise ratio (S/N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' < 15), which is significantly lower than those obtained from ESO Program 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The spectra of these two stars are therefore strongly affected by noise, which makes it very difficult to distinguish strong lines and prevents a satisfactory radial velocity deriva- tion from the cross-correlation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For consistency, they can therefore not be confirmed as members of NGC 6355 given the uncertainties in their radial velocity values even though these stars are considered members from the proper-motion member- ship check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Consequently, the final observed star sample is com- posed of the four stars of ESO Program 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Based on our final sample, we found a mean heliocentric ra- dial velocity for NGC 6355 of −193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 km s−1, which agrees well with the value of −194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 km s−1 obtained from the indi- vidual stars of Gaia DR21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, the normalized spectra were combined and were weighted by the median flux to obtain the final stellar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Membership selection The power of Gaia astrometry has been demonstrated in differ- ent ways, such as in the search for new open clusters and in the selection of the most probable members of a GC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In particular, regarding the latter, Gaia was not available until recent years, 1 https://people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='smp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='au/HolgerBaumgardt/ globular/appendix/ngc6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='txt Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Heliocentric radial velocity obtained for each extracted spec- trum and the average value for each star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Target Vhel r σVr Target Vhel r σVr km s−1 km s−1 km s−1 km s−1 1546_1 −227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 1239_1 −66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 1546_2 −29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='70 1239_2 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='56 1546_3 −216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 1239_3 −68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='31 1546_4 −318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='42 1239_4 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='90 1546_5 −166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='35 1239_5 −187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='22 1546 −216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='96 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='72 1239 −187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='28 133_1 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='47 1176_1 −196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='56 133_2 −191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='52 1176_2 −196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='65 133_3 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='48 1176_3 −193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='69 133_4 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='45 1176_4 −193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='68 133 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='53 1176 −195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='56 1539_1 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='41 1363_1 −193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='41 1539_2 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 1363_2 −194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 1539_3 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='41 1363_3 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 1539_4 −191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='41 1363_4 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 1539 −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 1363 −193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='79 and now the membership probabilities should be verified in all samples preceding the Gaia era, and in particular for our sample stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To remove bias from our sample, we performed a member- ship analysis to determine which stars observed in both ESO pro- grams are members of NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Considering both programs, we have a total of nine stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We selected the Gaia DR3 stars within 10′ from the cluster center, and we applied the Gaus- sian mixture models (GMM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2011) clustering method to separate the cluster members from the field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The derived mean proper-motion for NGC 6355 is < µ∗ α >= Article number, page 4 of 20 N star HB region 0 HST GO-11628 (PI:E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Noyola) P>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 O 0 80 0 Gaia RGBs 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Obs Stars 60 R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='024 arcmin +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 40 (arcmin) +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 DEC 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04 Q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='76 arcmin 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 ARA (arcmin) C O 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 F438W-F555W F438W-F555WSouza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' : Globular cluster NGC 6355 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Proper-motion density map from Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The stars show all the observed stars in both programs (members are plotted in white, and non-members are given in black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The red lines show the position of the mean proper motion of NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='06 mas yr−1 and < µδ >= −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 mas yr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This agrees very well with the new values computed by Vasiliev & Baumgardt (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The membership probabilities were computed considering cluster and field distributions, following the method presented in Bellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' When we had determined the membership probability, we cross-matched our sample stars with the Gaia data (Table 3), which are indicated with stars in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We found that six of nine stars from both programs have member- ship probabilities above 80%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Combining the information of ra- dial velocity and the proper-motion membership probability, we therefore disregard the non-member stars in the following anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fundamental parameters 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Atmospheric stellar parameters 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Stellar magnitudes The photometric effective temperature (Teff) and surface gravity (log g) were derived from the VIJHKS magnitudes given in Ta- ble 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For comparison purposes, we obtained the Teff from the Transiting Exoplanet Survey Satellite (TESS) input catalogue (TIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018) for our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The 2MASS J, H, and KS magnitudes were taken from Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To obtain the Teff from a wide wavelength range, we calculated the colour V − I employing the photometric systems relations G − V = f(GBP − GRP) and G − I = f(GBP − GRP) from Gaia EDR3 (Riello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Photometric effective temperatures Teff and gravities log g The Teff values were derived from V − I, V − KS , and J − KS colour-temperature calibrations of Casagrande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To use the calibrations, we must perform the reddening corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For NGC 6355, we assumed the metallicity [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='33, E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='77, and (m − M)V = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='21 from Harris (1996, 2010 edition) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Table 4 lists the derived photometric effective temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The < Teff > value given in the fifth column is the mean effective temperature without the TESS values (which are too hot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To derive the photometric log g value, we used the classical ratio log(g∗/g⊙), where log g⊙ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 is log g∗ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 + 4 log Teff∗ T⊙ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4(Mbol − Mbol⊙) + log M∗ M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (1) We adopted the values of from Table 4, M∗ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='85M⊙ and Mbol⊙ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The derived values of the photometric Teff and log g are given in the left columns of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Spectroscopic stellar parameters The final spectroscopic stellar parameters Teff, log g, and the mi- croturbulence velocity vt of NGC 6355 were derived together with [Fe/H] based on excitation and ionization equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Equiv- alent widths (EW) for a list of lines of Fe i and Fe ii lines were measured using DAOSPEC (Stetson & Pancino 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Using a visual inspection of the stellar spectrum, we remeasured some lines with the IRAF routine to evaluate the impact of blending lines, mainly for Fe ii, and some lines that were poorly fitted with DAOSPEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The employed lines are listed in the appendix (Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1) with the adopted oscillator strengths (log gf) for Fe i lines obtained from the VALD3 and NIST databases (Piskunov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Martín et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2002) and for Fe ii lines from Meléndez & Barbuy (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We extracted 1D photospheric models for our sample us- ing the MARCS grid of atmospheric models (Gustafsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The adopted CN-mild models consider [α/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='20 for [Fe/H]= −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 and [α/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 for [Fe/H]≤ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For the solar Fe abundance, we adopted ϵ(Fe) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 (Grevesse & Sauval 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The mean photometric and log g values calculated in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 were assumed as initial guesses to derive the spec- troscopic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The method consists of obtaining the exci- tation and ionization equilibrium of Fe i and Fe ii lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Figure 4 shows the excitation and ionization equilibrium for star 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The derived spectroscopic parameters Teff, log g, [Fe i/H], [Fe ii/H], [Fe/H], and vt are presented in the right columns of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To derive the final metallicity, we generated a Monte Carlo (MC) sample for each star to construct their [FeI/H] and [FeII/H] distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The distributions composed of the individual MC sample of each star are shown in Figure 5 as grey and red for [FeI/H] and [FeII/H], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, the cluster metal- licity distribution was obtained by combining the two distribu- tions (grey and red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The best metallicity value, the correspond- ing standard deviation, and the error of the mean are [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This metallicity agrees well with the Carretta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2009a) metallicity scale, which gives a value of [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 for NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Age and distance We employed the SIRIUS code (Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020) to perform the isochrone fitting to the CMD [F555W, F438W-F555W] of NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The code can provide a Bayesian view of the fun- damental parameters age, reddening (E(B − V)), d⊙, and metal- licity ([Fe/H]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We adopted the isochrones from the Dartmouth Stellar Evolutionary Database (Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2008) with a further linear interpolation in age and [Fe/H] with the random values given by the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' As a Gaussian prior for the metallicity, we employed the value derived in this work, while for the other parameters, we adopted uniform priors: 10 Gyr ≤ age ≤ 14 Gyr, Article number, page 5 of 20 NGC6355 4 Observed 2 0 μs [mas/yr] 6 10 10 2 0 2 4 6 μ*[mas/yr]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Identifications, coordinates, magnitudes from JHKs 2MASS survey, VI, HST/ACS, and matched Gaia DR3 information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The first two stars are from program 083.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0063 (A), and the four last stars are from 099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='D-0136 (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' ID ID RA DEC V V − I J H KS F438W F555W †µ∗ α µδ G BP−RP S/N 2MASS (deg) (deg) 2MASS HST/WFC3 (mas yr−1) 1546 17235883 − 2620183 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='996 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='338 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='24 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='359 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='45 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='42 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='747 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='523 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='33 1239 17240227 − 2621267 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='010 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='357 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='284 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='92 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='81 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='81 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='839 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='394 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='66 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 1539 17235356 − 2620223 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='973 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='339 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='82 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='942 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='73 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='780 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='659 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 1363 17240101 − 2620597 261.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='004 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='349 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='892 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='944 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='63 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='16 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='942 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='591 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='49 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='93 1176 17235712 − 2621441 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='988 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='362 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='684 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='90 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='59 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='66 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='69 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='041 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='609 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='26 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='33 133 17235528 − 2621088 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='980 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='352 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='24 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='435 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='62 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='21 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='64 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='64 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='572 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='635 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='36 †µ∗ α = µα cos δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Photometric parameters derived using calibrations by Casagrande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2010) for V − I, V − K, J − K colours are given in columns 2-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In columns 9-14 are given the spectroscopic stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Photometric parameters Spectroscopic parameters ID T(V−I) T(V−KS ) T(J−KS ) BCV Mbol log g Teff log g [Fe i/H] [Fe ii/H] [Fe/H] vt (K) (K) (K) (K) (K) (km s−1) 1539 4359 4330 4297 4330 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='615 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='74 4300 ± 65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='23 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 1363 4246 4315 4152 4246 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='702 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='64 4296 ± 76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='24 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 1176 4573 4642 4660 4642 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='481 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 4580 ± 69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='26 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='23 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 133 4373 4328 4250 4328 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='606 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='94 4378 ± 76 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ionization and excitation equilibria for NGC 6355 star 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The black dots and red squares correspond to the [FeI/H] and [FeII/H] lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The crosses are the FeI lines that were excluded through a 3σ clipping method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' E(B−V) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0, and d⊙ ≤ 20 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We used the CMD structure constraints similar to the procedure described by VB13 to im- prove the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Nevertheless, we kept the Bayesian nature of the code and used the structure pattern of the CMD as priors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The direct comparison between observational data and isochrones cannot give an accurate physical interpretation of the cluster (D’Antona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018) because the likelihood in this case is purely geometrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Therefore, the prior distributions are of great importance to improve the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In that sense, we adopted a more robust prior to the magnitude of the horizontal branch (HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This prior is crucial to give a more precise dis- tance derivation when it is very close to the magnitude level of RR Lyrae stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To constrain the HB magnitude, we employed the relation by Recio-Blanco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2005), MZAHB F555W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='981 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='410 × [M/H] + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='061 × [M/H]2, (2) where [M/H] = [Fe/H] + log � 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='638 × 10[α/Fe] + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='362 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We as- sumed [α/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 because this is the expected value for GCs with a similar metallicity (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Then, we re- calculated the magnitude level for each iteration of the Markov Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Metallicity distribution from sample stars of NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The final distribution (black step histogram) considers both [FeI/H] (grey) and [FeII/H] (red) for all lines of our sample member stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' chain Monte Carlo (McMC) sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For the apparent magni- tude of the HB, we assumed mZAHB F555W = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 by a visual inspection, which is very close to the value derived by Ortolani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2003) of VHB = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Another morphological parameter is the magnitude differ- ence between zero-age HB (ZAHB) and the turn-off point (TO), also known as vertical parameter (Vandenberg, Bolte, & Stet- son 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' However, this parameter is strongly dependent on the ZAHB level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Because of this, we de- cided to use the horizontal parameter (Vandenberg, Bolte, & Stetson 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Rosenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The horizontal parame- ter is the colour difference between the TO and the point at the RGB that is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 magnitude brighter than the TO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In order to implement the horizontal method in the observed CMD, we computed the fiducial or ridge line of NGC 6355 us- ing the method described in Marín-Franch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2009, hereafter MF09).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The procedure is briefly described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We first computed a simple fiducial line by binning the cluster magnitude and calculating the median colour for each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We applied a dif- Article number, page 6 of 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 I,II/HJ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 Fel 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 + α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00011 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' = - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00484 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' = - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 = - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 = - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 20 40 60 80 100 0 1 2 3 4 5 0 6 EW (mA) Xex [eV]: [FeI/H] [FeI,II/H] <[FeI,II/H]> 4 [Fe/H]= - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08) 3 density 2 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 [Fe/H]Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' : Globular cluster NGC 6355 ferential binning method to have more points around the TO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The second step was to derive the median colour perpendicular to each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This method is most important for the subgiant branch (SGB) because this sequence is almost horizontal for bluer fil- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, the algorithm computes the horizontal parameter for the cluster fiducial line and each McMC isochrone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The posterior distributions of the parameters are given by the 50th percentile as the best value, and the 16th and 84th per- centiles to provide the uncertainties (right corner plots of Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In Figure 6, the NGC 6355 CMD (left panel) is over-plotted by the best solution of the isochrone fitting composed of the me- dian value (solid line) and the 1σ region (shaded region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Because the expected extinction is relatively high, it is neces- sary to consider the Teff correction to the isochrones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is worth noting that the Teff correction effect increases with the tempera- ture and changes the isochrone morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The method is well described in Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020) and Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We found the following equations: AF438W/AV = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='688 − 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='606x + 325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='254x2 − 407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='219x3 (3) AF555W/AV = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='043 − 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='394x + 507.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='496x2 − 634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='233x3(4) AJ/AV = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='128 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='428x − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='309x2 + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='573x3 (5) AKS/AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='061 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='677x + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='522x2 − 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='134x3 (6) AG/AV = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='346 − 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='867x + 183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='277x2 − 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='296x3 (7) AGBP/AV = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='899 − 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='627x + 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='243x2 − 364.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='345x3 (8) AGRP/AV = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='154 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='695x − 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='175x2 + 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='449x3, (9) where x is log Teff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The immediate effect on the isochrone is an offset in the direction of the CMD blue-brighter region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' There- fore, the horizontal (E(438 − 555)) and vertical ((m − M)F555W) displacements should be different from those without a correc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In addition, the morphology is defined essentially by the age and metallicity when the helium mass fraction (Y) is fixed (see Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In our case, the metallicity was con- strained to the value derived here from high-resolution spec- troscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Therefore, only age changes the isochrone morphol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Because of this, the age considering the Teff correction tends to be older than the simple isochrone fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The result is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In this work, we derived the absolute age of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 Gyr for NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The considerable uncertainty on the age deriva- tion is due to the narrow colour baseline adopted in this work (F438W-F555W), which spread the TO region slightly more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Al- though we provide the first absolute age for NGC 6355 through isochrone fitting, KC16 derived an age of ∼ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Gyr using in- tegrated magnitudes, and C21 reported the age as 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 Gyr for NGC 6355 by comparing its CMD with that of NGC 6205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The age derived in this work assuming the Teff correction agrees very well with the age in C21+VB13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This illustrates the importance of this correction for highly reddened clusters in the central part of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Nataf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2016) discussed the extinction towards GCs lo- cated in the Galactic bulge, where the RV value can be as low as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Pallanca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021) reported a straightforward method for determining the best value of RV for highly reddened clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The method was also applied by Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020), who derived a value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 for Pal6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The method for deriving the RV consists of simultaneously fitting CMDs with different colour baselines with the same set of reddening and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Here we fitted (in addition to the HST CMD) the CMDs [J, J − KS ] from Valenti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2007) and [G, GBP − GRP] from Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' From the HST CMD, we found E(438 − 555) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 and (m − M)F555W = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' These values were con- verted into E(B − V) and (m − M)0 for different values of RV, as shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The best RV is the mean between the best values for Valenti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2007) and Gaia DR3 CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We find RV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='84±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Hence, for NGC 6355 with the derived RV, we find E(B − V) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 and d⊙ = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The distance value is crucial for deriving the orbital parame- ters of the clusters, as demonstrated by PV20 and illustrated by the case of Palomar 6, as discussed in Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To verify our distance derivation, we collected the RR Lyrae star members of NGC 6355 from the fourth data release of the Op- tical Gravitational Lensing Experiment (OGLE-IV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We adopted the calibrations from Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2017, G17) using the least-squares (LQS) and Bayesian (BA) methods (Muraveva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018, M18), and Oliveira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2022, O22)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' All distances are displayed in Figure 8, including the derivation by Baumgardt & Vasiliev (2021, B22) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The value of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='19 kpc derived in this work agrees well with the oth- ers, particularly the B22 value of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='22 kpc, which is the most recent value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Abundance analysis We carried out a detailed abundance analysis employing line-by- line spectrum synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We employed the spectrum synthesis code PFANT (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018c) to derive the abundances of the elements C, N, O, Na, Mg, Al, Si, Ca, Ti, V, Mn, Co, Cu, Zn, Y, Zr, Ba, La, Nd, and Eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The line list with the abundance ratios for each line are given in the appendix (Table A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The code PFANT is an update of the Meudon code by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Spite and adopts local thermodynamic equilibrium (LTE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The atomic line list is from VALD3 (Ryabchikova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The abundance values were derived through the χ2 mini- mization algorithm described in detail in Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Figure 9 gives a visual illustration of the method for star 1363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The observed spectrum around the lines Na i 5682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='633 Å and Al i 6698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='673 Å is shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The best-fit solution is the solid red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For completeness, we also compare the spectrum without the abundance contribution of the current element (solid green line), the best fit plus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 (solid magenta line), and the best fit minus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 (solid cyan line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, we adopted the so- lar abundances from Grevesse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' C, N, and O abundances The CNO abundances were derived through an iterative fitting of the C2(1,0) Swan bandhead at 5635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3 Å, and CN(6,2) at 6478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='48 Å of the A2ΠX2Σ system band heads and the forbidden oxygen line [OI] 6300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='31 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The algorithm fits the three lines simulta- neously and takes the interdependent continuum variation due to changes in C, O, and N values into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Table 5 lists the de- rived abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Because the region of the C2(1,0) bandhead is strongly affected by the S/N and the line is weak, we assumed the C abundances as upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, before fitting the [OI] line, we verified the contamination by telluric lines in this region and concluded that for our sample, none of the stars has telluric line contamination on the [OI] line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The spectral fitting for C, N, and O for star 1363 are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' As expected for most GCs (Piotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2017), NGC 6355 seems to host multiple stellar populations (MPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' see the reviews Gratton, Sneden, & Carretta 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Grat- ton, Carretta, & Bragaglia 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Bastian & Lardo 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Milone 2 https://people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='smp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='uq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='au/HolgerBaumgardt/ globular/fits/disfit/ngc6355_dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='pdf Article number, page 7 of 20 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Isochrone fitting for NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The best solution is composed of the median values of the posterior distributions (solid dark red line), and the 1σ extrapolation is constructed from the 16th and 84th percentiles (shaded dark red region).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The corner plot shows the correlations among the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Simultaneous isochrone fitting to derive the cluster RV using three CMDs: HST (left panel), 2MASS JKS from Valenti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2007) (top right panel), and Gaia DR3 (bottom right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The isochrones are coloured according to their RV value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In each panel, the best solution is represented by the solid dark red isochrone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For the two right panels, the χ2 analysis is plotted in the inset plot, and the dots are coloured by the same colour as the corresponding isochrone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' & Marino 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The relatively high nitrogen abundance of stars 1176 and 133 with relatively low values of carbon abundances indicates the presence of MPs in NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The other two stars have a relatively low N abundance and relatively normal (solar) C one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Because stellar evolution theory predicts an N-C anti- correlation, we must further investigate to confirm the presence of MPs in NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This is further analysed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Our distance derivation compared with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The violins show the distance distribution using RR Lyrae stars, the recent distance derivation by Baumgardt & Vasiliev (2021), and the distance found in this work through isochrone fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For the derived RR Lyrae distances, four calibrations were adopted that are represented by the first four vio- lins (see the text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Carbon, nitrogen, and oxygen abundances [X/Fe] from C2, CN bandhead, and [OI], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' [C/Fe] [N/Fe] [O/Fe] Star C2 CN(6,2) [OI] 5635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 Å 6478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='60 Å 6300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='31 Å 1539 ≤ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='21 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='43 1363 ≤ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='49 1176 ≤ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='87 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='37 133 ≤ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='70 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='24 Article number, page 8 of 20 6 Age(Gyr) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 E(438 - 555) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='78 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='01 (m - M)F555W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 00 O [Fe/H] = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39] +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 E(438 - 555) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='80 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' mF555W 20 2 [Fe/H] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 g0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 Age (Gyr) E(438 - 555) (m - M)F555w mF438W - mF555W10 17 12 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Rv 18 14 19 O 16 20 18 mF555W b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 J-Ks 21 14 22 0 O 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 G Rv 16 23 G O 24 18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 GBP - GRP mF438W - mF555W12 T 11 10 (ody) 9 8 7 6 M18 B22 G17 (BA) This workSouza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' : Globular cluster NGC 6355 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Example of line-profile fitting for star 1363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The upper panel shows the result for the Na i 5682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='633 Å, and the bottom panel shows the fit for the Al i 6698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='673 Åline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The black lines correspond to the observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line shows the best-fit solution as the median.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For comparison purposes, we also plot the best-fit solution with a variation of ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 (solid cyan and magenta lines) and the spectrum without the element abundance (green line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Spectral fitting of C, N, and O for star 1363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The observed spectrum is given in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line is the best fit, and the cyan and magenta lines show the best fit ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The yellow line shows the line region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For C2 (upper panel) we also show the bandhead lines in dotted silver lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' alpha-elements The α-elements O and Mg are the most reliable indicators of enrichment in α-elements from hydrostatic phases of massive stars (Woosley & Weaver 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Together with the explosive α- elements Si and Ca, they are good indicators of a fast early en- richment of the proto-cluster gas by supernovae type II (SNII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ti is classified as an iron-peak element (Woosley & Weaver 1995), but shows a similar α-element behaviour and is often included as another α-element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The spectral fitting results for Mg, Si, Ca, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as figure 10 for Mg, Si, Ca, and Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line is the best fit, and the cyan and magenta lines show the best fit ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' and Ti of star 1363 are shown in Figure 11, and the results are presented in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Odd-Z elements The sodium abundances were derived from Na i 5682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='633 Å, 5688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='194 Å, 6154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='23 Å, and 6160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='753 Å lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The Al abun- dances were derived from lines Al i 6696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='185 Å, 6698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='673 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The (anti-)correlations indicating the effect of MPs are shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We also calculated the Spearman corre- lation parameter for each combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For N-Al, we found a strong correlation, and the anti-correlation for N-O, Na-O, and Al-O is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Moreover, the main correlations come from the ni- trogen abundances (Fernández-Trincado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' However, [Al/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 is also a threshold for second-generation (2G) stars (Mészáros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The figure shows a visible separa- tion of our sample into two groups: two stars are moderately rich in N and Al, and two stars have low values of [Al/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This af- fects their mean abundances (Table 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This is further discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Iron-peak elements We derived the abundances of the iron-peak elements V, Mn, Co, Cu, and Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' While V and Mn are members of the lower iron-peak element group, Co, Cu, and Zn are considered to be- Article number, page 9 of 20 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 [Na/Fel= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 a = 5682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='633 A Norm flux 5682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 5682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 5683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 5683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 [A1/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 ^ = 6698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='673 A 6698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6699.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Wavelength (A)star13＇1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='95 [C2/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 [C2/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 [C2/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='33 5634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 5634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 5635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 5635.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 star13 1363 Norm flux 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 [CN(6,2)/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='95 [CN(6,2)/Fe1= + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 [CN(6,2)/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 6478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 6478.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 6479.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 star13 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 [OI/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 [Ol/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='49 [O]/Fe] = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 6299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 6300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 6301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 Wavelength (A) star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 [Mg/Fel= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='47 ^= 6318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='720A 6318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6319.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 [Si/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='14 A=6237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='328A flux 6236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Norm star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='75 TCa/Fel= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 =6161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='295A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 6160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 [Ti/Fel= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 2 = 6064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='623 6064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Wavelength (A)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (Anti-)Correlations indicating effects of multiple stellar popula- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The dotted orange line in both left panels represents the transition to the N-rich regime at [N/Fe]∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='7 for [Fe/H] around the NGC 6355 value (Fernández-Trincado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Additionally, the grey line in the two bottom panels shows the upper limit for first-generation stars (Mészáros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The colour bar shows the Mg abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' long to the upper iron-peak group (Woosley & Weaver 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The first group is mainly produced in type Ia supernovae (SNIa) with a contribution from core-collapse supernovae (Nomoto, Kobayashi, & Tominaga 2013, and refereces therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In con- trast, Co, Cu, and Zn are predominantly produced by core- collapse supernovae (Woosley, Heger, & Weaver 2002, and ref- erences therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The atomic lines were adopted from Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018) and Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020), together with their hy- perfine structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The spectral fitting results for V, Mn, and Co are shown in Figure 13 for star 1363, and Cu and Zn are given in Figure 14 for star 1539.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Heavy elements The abundances of the heavy neutron-capture s-elements Y, Zr, Ba, La, and Nd, and the r-element Eu also were derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For Y, we measured the Y i 6435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='004 Å and the Y ii 6613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='73 Å lines, and we assumed for the mean that the ionized species of Y con- tributes with 99% to the abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For the barium abundance, we used the Ba ii lines 5853.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='675 Å, 6141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='713 Å, and 6496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='897 Å, with hyperfine structure from Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The Zr i 6127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='47 Å, 6134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='58 Å, 6140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='535.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='58 Å, and 6143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 Å, La ii 6262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='287 Å, 6320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='376 Å, and 6390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='477 Å, Nd ii 6740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='078 Å, 6790.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='372 Å, and 6549.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='525 Å, and Eu ii 6437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 Å and 6645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 Å were used for Zr, La, Nd, and Eu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The spectral fitting results for Y, Zr, Ba, La, Eu, and Nd are shown in Figures 15 and 16 for star 1363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Errors The uncertainties in spectroscopic parameters are given in the last four columns of Table 6 for star 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For each stellar pa- Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as figure 10 for V, Mn, and Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line is the best fit, and the cyan and magenta lines show the best fit ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as figure 10 for Cu and Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line is the best fit, and the cyan and magenta lines show the best fit ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' rameter, we adopted the usual uncertainties for similar samples (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2014, 2016, 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The sensitivities were com- puted by employing models with these modified parameters and recomputing lines of different elements considering changes of ∆Teff = +100 K, ∆log g= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2, ∆vt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The given er- ror is the difference between the new and the adopted abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The uncertainties due to non-LTE effects are negligible for these stellar parameters, as discussed in Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The same error analysis and estimations can be applied to other stars in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is worth noting that star 133 has the lowest S/N of the four sample stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The uncertainties given in Table 6 can therefore be considered as upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The faint La lines appear to be more reliable than the strong Ba lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, it is impor- Article number, page 10 of 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 [O/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3 p= - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 p= + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 [Na/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='80 b= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 [Al/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 p= + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='97 p = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='78 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 [N/Fe] [O/Fe] [Na/Fe] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46 [Mg/Fe]star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 [V/Fel= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 ^=6119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='520A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 6118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Norm flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 [Mn/Fel= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 = 6013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='513A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 6013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 [Co/Fe] = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 = 5342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='708 A 5342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 5342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 5343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 5343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Wavelength (A)star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 [Cu/Fe]= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 a= 5105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='537A Norm flux 5104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 5105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 5105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 5106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 5106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 [Zn/Fe]= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='20 a= 6362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='339 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='95卜 6361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Wavelength (A)Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' : Globular cluster NGC 6355 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as figure 10 for Y, Zr, and Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line is the best fit, and the cyan and magenta lines show the best fit ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as figure 10 for La, Eu, and Nd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line is the best fit, and the cyan and magenta lines show the best fit ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' tant to note that the main uncertainties in stellar parameters are due to uncertainties in the Teff, as shown in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Density probability map for the x − y and R − z projections of the set of orbits for NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Orange corresponds to higher probabil- ities, and the black lines show the orbits using the main observational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Dynamical properties In order to obtain the orbital parameters of NGC 6355, we em- ployed an axisymmetric potential McMillan (2017) adopting the Python package galpy (Bovy 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We integrated a set of 1000 initial conditions forward for 10 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The set was generated by using a MC algorithm adopting the observational uncertainties of the cluster data on proper motions µ∗ α and µδ, heliocentric ra- dial velocity, and the heliocentric distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The McMillan (2017) Galactic potential was adopted to compare our results with those of Massari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2019) and to relate NGC 6355 with its plausible progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' A more realistic potential, including a contribution of the Galactic bar (Pérez-Villegas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018, 2020), could provide a farther inward orbit for the GC members of the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The orbital parameters are listed in Table 7, including the values of the IOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Figure 17 shows the density probability map of the orbits of NGC 6355 in the x−y and R−z projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The space region in which the orbits of NGC 6355 cross more frequently are shown in orange, and the black curves are the orbits considering the cen- tral values of the observational parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' NGC 6355 is mostly confined within ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 kpc and therefore has a high probability of belonging to the bulge component (> 95%) when we adopt the distance of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='22 kpc that we estimated in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Our new distance derivation indicates that the cluster NGC 6355 lies far inward based on its maximum height of |z| < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1 kpc and the high eccentric orbit > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It may well be that this perigalactic distance is one the closest distances to the Galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Discussion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Kinematic classification The orbital analysis shows that the orbit of NGC 6355 is com- patible with a location at the Galactic bulge volume according to the classification of PV20, who presented the probability dis- tribution of belonging to each Galactic component through the values of rapo and |z|max (Figure 17 and Table 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' It is essential to mention that their classification is based on a Galactic po- tential that includes the contribution of the Galactic bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Another Article number, page 11 of 20 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='95 [Y/Fel= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 =6795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='414A 6794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6795.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Norm flux [Zr/Fel= ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 2=6127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='475A 6127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 [Ba/Fe]= + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 = 6496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='897 A 6496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 Wavelength (A) star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 [La/Fel= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 Λ = 6390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='477 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 6390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6391.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Norm flux 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 [Eu/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='55 ^= 6437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='640A 6437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 star 1363 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='9 [Nd/Fe] =+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 a= 6740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='078 A 6739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 6740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 6741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Wavelength (A)2 2 1 [kpc] 0 0 y 1 1 2 2 0 2 0 2 [kpc] 0 0 Z 1 1 2 2 2 1 0 1 2 0 1 2 x [kpc] R [kpc]A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Abundances in the four UVES member stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The mean values were computed considering all four stars (< all >), considering only 1G stars (< 1G >), and only 2G stars (< 2G >).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The last four columns show the abundance sensitivity due to variation in atmospheric parameters for star 15 (133) considering uncertainties of ∆Teff = 100 K, ∆log g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2, and ∆vt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 km s−1, and the last column is the total error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' These errors were taken into account when we composed the final reported abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' [X/Fe] star 1539 star 1363 star 1176 star 133 < all > < 1G > < 2G > ∆T ∆ log g ∆vt ( 1 3 �x2)1/2 1G 2G K kms−1 C +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 N +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='29 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='12 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 O +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='20 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='12 Mn −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='02 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='06 Co +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='06 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 Cu −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='16 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='18 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='13 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 Zn −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='11 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='01 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='06 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04 [Fe/H] −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='15 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='07 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='17 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='13 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='13 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='10 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='04 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Orbital parameters, velocities, and membership probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Parameter Mean Unit E −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='03 ×105km2 s−2 LZ −31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='28 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='42 km s−1kpc rperi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 kpc rapo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='14 kpc |z|max 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='08 kpc ecc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='05 — vR −218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='27 ± 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 km s−1 vφ −192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='39 ± 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='54 km s−1 Pbulge 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 % Pdisk 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 % Pinner halo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 % Pouter halo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 % robust Galactic potential, taking into account the friction dynam- ics in addition to the contribution of the Galactic bar, was applied by Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Their orbital parameters are essentially compatible with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The values of E are precisely the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The LZ and rperi are compatible within 1σ, while our value of rapo is higher than that of Moreno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This indicates that a more realistic Galactic potential confines NGC 6355 even more within the Galactic bulge volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' With the results using the McMillan (2017) Galactic potential, NGC 6355 is a Galactic bulge GC with a probability of about 95%, and a 5% probability of belonging to the Galactic disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' After establishing that NGC 6355 currently is a member of the Galactic bulge, we investigated whether this cluster origi- nated from the primordial material of the Galaxy or if it is a remnant of the first mergers of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To do this, we stud- ied the chemodynamical and photometric information derived in previous sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Comparison with bulge field stars To study NGC 6355 in the context of the Galactic bulge, we compared the orbital parameters derived in this work with the field star population composed by the reduced proper motion (RPM) sample from Q21 and the bulge RR Lyrae from the OGLE Galaxy Variability Survey (Soszy´nski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We matched the OGLE sample with APOGEE DR17, which already provides the abundances, radial velocities, and proper motions (previously obtained from Gaia EDR3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' After this, the second sample was matched with Starhorse (Queiroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020) in order to obtain the distance values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, the resulting sam- ple consisted of 4132 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' NGC 6355 has a relatively high |Z|max and e, placing it in cell F of Figure 20 in Q21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This is reproduced here in the upper left panel of Figure 18 for the RPM sample and in the upper right panel for the RR Lyrae sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The normalized population density as a function of [Fe/H] (MDF), Rmean (mean between rapo and rperi), and vφ is shown in the three bottom panels of Figure 18, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Based on the MDF (lower left panel ), the RPM sample comprises the moderately metal-rich bulge MDF, while the RR Lyrae sample is the metal-poor tail one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' As expected, NGC 6355, an old GC, is located together with the peak of the RR Lyrae MDF and Rmean distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Nevertheless, the vφ of the cluster is in the tail of both samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The comparison with the bulge field populations shows that NGC 6355 is likely an in-situ GC compatible with the old and metal-poor RR Lyrae component of the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Comparison with chemodynamical models To investigate the chemical abundances in the context of nucle- osynthesis, we compared our results with chemical evolution models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The models for O, Mg, Si, Ca, V, Mn, Co, Cu, and Zn were computed with the code described in Friaça & Bar- buy (2017) (see also Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2020, 2022, for V, Mn, Co, Cu, and Zn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The star formation rate (SFR) was found to be best suited with ν = 1 Gyr−1 in order to fit the abundances of a selected sample of bulge stars in Razera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Article number, page 12 of 20 Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' : Globular cluster NGC 6355 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' NGC6355 compared with the RPM bulge sample of Q21 (left panel) and Galactic bulge RR Lyrae population (right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Upper panels: |Z|max as a function of the eccentricity plane divided into nine frames defined by the letter close to the horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The golden star represents the locus of NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The bottom panels show the population density of [Fe/H], Rmean, and vφ for cell F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The gold lines represent the position of NGC 6355 in each panel, and the shaded gold region shows the 1σ distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Therefore, we adopted this SFR for all elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The SFR is the rate at which the available gas mass is turned into stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Consequently, it measures the inverse of the system for- mation timescale: Our adopted ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Gyr−1 represents a rather fast star formation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The models assume a baryonic mass of 2 × 109 M⊙, a dark halo mass 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='3 × 1010 M⊙, and the cosmological parameters from Planck Collaboration (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The bulge is considered a classical spheroidal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Finally, the models project the chemical abundance distribution at dif- ferent radius ranges: r< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 kpc (dash-dotted line in Figure 19), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 < r < 1 kpc (dashed line), 1 < r < 2 kpc (dotted line), and 2 < r < 3 kpc (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For Na and Al, we used the Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020) models for the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' These models also assume an SFR ν ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Gyr−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In addition to the chemodynamical models, in the follow- ing analysis, we also compare the abundance ratios obtained in this work with the bulge GCs Palomar 6 (Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2021), HP 1 (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018a), NGC 6558 (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2018b), and NGC 6522 (Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Furthermore, we compare NGC 6355 with the RPM and RR Lyrae samples (for the case of α and odd-Z elements) inside cell F of Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Figure 19 shows the abundances of O, Mg, Si, and Ca as a function of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To better illustrate the comparison, the mean locus of the RPM sample is shown as a solid black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' We de- rived [α/Fe]= +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='09 considering all α elements O, Mg, Si, Ca, and Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Our mean [α/Fe] is compatible within 1σ with the assumed value for the isochrone fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This reinforces the consistency of the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In all cases, NGC 6355 is compatible with the RR Lyr locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Additionally, NGC 6355 is also compat- ible with the other GCs, execpt for Ca, in which the cluster is relatively richer than the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' As a result of the presence of MPs, the spread in [Na/Fe] is higher than for the other elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This effect can be observed in the top panel of Figure 20 with the discrepancy between the two models and the mean locus for the case of low metallici- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Souza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021) found that Pal 6 is not compatible with a bulge [Na/Fe].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' They argued that the reason is due to the pres- ence of a 2G star in their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The same effect is expected for [Al/Fe] because the Al abundance is a good indicator of the presence of 2G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The lower panel of the same figure shows the high error bars of [Al/Fe] for NGC 6355 that are due to the presence of two moderately Al-rich stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In Figure 21 we investigate the iron-peak elements V, Mn, Co, and Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' To increase the bulge sample, we also compared our results with bulge GC stars from Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018) and the bulge field stars from Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The chemical evolution model fits NGC 6355 perfectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' For Cu abundances, the selected bulge clusters have relatively lower values than NGC 6355, indicating a different possible scenario for its early evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In the case of V (top left panel), the evolution model is shifted to lower abundances for all metallicities than the mean locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The models suitably fit the abundances of NGC 6355, the selected clusters, and the bulge GC stars for Mn, Co, and Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The Zn abundances derived in this work are based only on the line Zn i 6362.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='339 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' In Figure 22, NGC 6355 is perfectly fitted by the models and is compatible with all reference clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Here it is worth noting that the models predict supersolar zinc abundances for metallicities above −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 and subsolar for values Article number, page 13 of 20 RPM Bulge sample Bulge RR Lyrae stars 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 (kpc) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 ecc ecc NGC6355 RPM RR Lyr 2 4 0 0 200 200 [Fe/H] (kpc) Vμ (km s-1)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' aanda Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' O, Mg, Si, and Ca abundance as a function of [Fe/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The KDE plot represents the RPM bulge selection from cell F, and the blue contours represent the RR Lyrae sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The stars are abundances of bulge GCs: NGC 6558 (cyan), NGC 6522 (pink), HP1 (green), and Pal 6 (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The golden star represents the mean abundance of NGC 6355.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The chemodynamical evolution models are shown in different radii ranges: r< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 kpc (dash-dotted line), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 < r < 1 kpc (dashed line), 1 < r < 2 kpc (dotted line), and 2 < r < 3 kpc (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' below −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The low Zn as an indicator of an ex-situ origin was suggested only for the case of near-solar metal-rich stars Minelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as Figure 19 for odd-Z elements Na (upper) and (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The solid red line is the chemical evolution model from Kobayashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as Figure 19 for V, Mn, Co, and Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The black squares are bulge GC stars from Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018), and black crosses show bulge field stars from Ernandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The chemodynamical evo- lution models are shown in different radius ranges: r< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 kpc (dash- dotted line), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 < r < 1 kpc (dashed line), 1 < r < 2 kpc (dotted line), and 2 < r < 3 kpc (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Same as Figure 19 for Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The Friaça & Barbuy (2017) evo- lution models are shown in different radius ranges: r< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 kpc (dash- dotted line), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 < r < 1 kpc (dashed line), 1 < r < 2 kpc (dotted line), and 2 < r < 3 kpc (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' The comparison of heavy-element abundances of NGC 6355 with literature GCs is shown in Figure 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' NGC 6355 abun- dances are compatible with HP 1 in almost all heavy elements except for Ba, for which NGC 6355 has higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' There is a rather large scatter in the abundances of n-capture elements, especially Y, Zr, and Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' This pattern is better explained in Chi- appini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2011), Cescutti & Chiappini (2014), and Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Analysis of abundance discriminators The [Mg/Mn]-[Al/Fe] plane is often used in the context of the Galactic halo to split the original MW population from merger remnants (Hawkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Limberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' 2022) because the accreted population shows lower [Al/Fe] abundances and high α abundances due to the abrupt evolution interruptions of the merger progenitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' Horta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2021) applied the same idea for a star sample located in the Galactic centre to find debris stars within the Galactic bulge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' They called this inner Galaxy struc- Article number, page 14 of 20 Mg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 [X/Fe] Si Ca 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='50 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='00 2 1 0 2 1 0 [Fe/H] Bulge RR Lyrae stars ★★★★木 NGC 6558 - Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018) v= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Gyr-1: r<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 kpc NGC 6522 - Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2020) HP 1 - Barbuy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content=' (2018) v= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='0 Gyr-1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mNE4T4oBgHgl3EQftw2O/content/2301.05227v1.pdf'} +page_content='5 k0 +(0, 0), +otherwise, +(14) +where αky denotes the rotation angle, ∆ky and ∆kx denote +the degree of motion along x and y direction, respectively, +and k0 is the delay time of the phase error due to the centric +k-space filling. In our simulation, k0 is fixed to π/10, αky is +randomly sampled from [−2◦, 2◦], ∆ky and ∆kx are sampled +from [−1cm, 1cm] and [−0.5cm, 0.5cm], respectively, at each +k-space line. +The second type of simulated motion is respiratory motion, +which appears as a sinusoidal function in k-space [16], [23]: +(α, Φ) = +� +(0, ky∆ky sin(mky + n)), +|ky| > k0 +(0, 0), +otherwise, +(15) +where ∆ky, m, and n denote the amplitude, period, and +phase shift of the sinusoidal function, respectively. Because +the respiratory motion appears in abdominal MR images, we +simulate it with the liver MR image dataset. Parameters for +the simulation are sampled as follows: k0 ∼ U[π/10, π/5], +∆ky ∼ U[1cm, 1.5cm], m ∼ U[0.1, 5.0], and n ∼ U[0, π/4], +where U[a, b] denotes the uniform distribution with the interval +[a, b] +C. Comparison Methods +We compared our method with three state-of-the-art meth- +ods to verify the performance of the method. The first com- +parison method is MARC [16], a method for reducing liver +MRI motion artifacts. Because it is a supervised method, we +train MARC models using simulated motion-corrupted images +with Eqs. (14) and (15). +The second comparison method is Cycle-MedGAN V2.0 +[20], an unpaired deep learning method based on CycleGAN +[22]. Cycle-MedGAN V2.0 can be trained with both simulated +or in vivo motion-corrupted data, but we train it with only +simulated motion-corrupted data because the training of Cycle- +MedGAN V2.0 was unstable when using in vivo data. +We also employed the bootstrap subsampling and aggre- +gation method in [23] as a comparison method. Because +this method requires only motion-free images during training, +simulated or in vivo motion-corrupted images were not used +during training. +D. Evaluation Methods +For the quantitative evaluation, we used the peak signal-to- +noise ratio (PSNR) and the structural similarity index metric +(SSIM). Because there is no ground truth matched with in +vivo motion-corrupted images, the quantitative evaluation was +performed with simulated motion-corrupted images. +In addition, we also conducted a clinical evaluation with +the results using in vivo motion-corrupted data. Specifically, +a radiologist with 13 years of experience in abdominal MR +imaging performed an analysis of the results of various +methods. The image analysis was conducted from various per- +spectives. First, the performance in reducing motion artifacts +is rated using a 5-point scoring system: 1 = non-diagnostic +(severe artifacts causing impaired diagnostic capability of the +readers); 2 = substantial artifacts with image quality decrease, +but diagnostic performance impairment; 3 = mild artifacts, no +significant (only mild) image quality disturbance; 4 = minimal +artifacts, sharp image; 5 = no artifacts. The image noise level +is also evaluated with the following scoring system: 1 = non- +diagnostic (severe noise causing impaired diagnostic capability +of the readers); 2 = substantial noise with image quality +decrease, but diagnostic performance impairment; 3 = mild +noise, no significant (only mild) image quality disturbance; +4 = minimal noise; 5 = no noise. Next, the blurring can be +induced when reducing the motion artifact, so the rating of +image blurring level is performed: 1 = non-diagnostic (severely +pixelated texture causing impaired diagnostic capability of the +readers); 2 = substantially pixelated, artificial sensation with +concerns about the loss of normal texture, without diagnostic +performance impairment; 3 = mildly pixelated, artificial sensa- +tion, without image quality decrease; 4 = minimal alteration of +image texture; 5 = no alteration of image texture. Furthermore, +because the hepatic artery (HA) on the arterial phase should +be visualized clearly, the vessel clarity is evaluated with a +scoring system: 1 = not delineated due to motion or low +signal-to-noise ratio (SNR); 2 = blur or decreased SNR; 3 = +clear common hepatic artery (CHA) and proper hepatic artery +(PHA), but blurred HA and gastroduodenal artery (GDA); 4 += entire HA is clearly visible, clear CHA, GDA, bilateral +HA; 5 = strong contrast-to-noise ratio with score 4. Last, the +overall image quality is assessed by following scoring system: +1 = non-diagnostic; 2 = not satisfactory image quality, but re- +examination is not needed; 3 = acceptable image quality (im- +age quality may not be very good, but clinically acceptable); 4 += good image quality without significant artifact; 5 = excellent +image quality without artifact and good spatial resolution. The +score is rated for each volume in all assessments. Also, the +results were presented to the radiologist in a random order +without any labeling for a fair comparison. +V. RESULTS +A. Results with Simulated Data +Fig. 3 shows the motion artifact reduction results of various +methods with random simulated motion-corrupted data. As +shown in Fig. 3(a), it is hard to recognize detailed structures of +brains due to motion artifacts. MARC [16] reduces the motion +artifact but the output images of MARC are too blurry or + +6 +Fig. 3: The simulated random motion artifact reduction results with the HCP brain dataset: (a) motion-corrupted input image, +(b) MARC [16], (c) Cycle-MedGAN V2.0 [20], (d) bootstrap subsampling and aggregation [23], (e) the proposed method, and +(f) motion-free label image. The difference maps show the difference between each image and the label image. PSNR and +SSIM values of each image are shown in the corner of the images. +Fig. 4: The simulated respiratory motion artifact reduction results with the CNUH liver dataset: (a) motion-corrupted input +image, (b) MARC [16], (c) Cycle-MedGAN V2.0 [20], (d) bootstrap subsampling and aggregation [23], (e) the proposed +method, and (f) motion-free label image. The difference maps show the difference between each image and the label image. +PSNR and SSIM values of each image are shown in the corner of the images. +smoothed (Fig. 3(b)). In the results of MARC, the boundary +between gray matter and white matter is not clear (the first +row in Fig. 3(b)), and the structure of the choroid plexus is not +properly restored (the second row in Fig. 3(b)). Next, in Fig. +3(c) and (d), Cycle-MedGAN V2.0 [20] and bootstrap sub- +sampling and aggregation [23] remove random motion artifacts +significantly and show increased quantitative results compared +to input images. However, there are some differences between +label images and outputs of Cycle-MedGAN V2.0 as shown in +difference maps, and bootstrap subsampling and aggregation + +(a) +29.85 /0.809 (b) +30.93 / 0.916 (c) +31.41 /0.916 (d) +32.31 /0.881 +(e) +33.65/0.942 +(f) +PSNR (dB)/ SSIM +口 +口 +C +29.55 / 0.785 +30.80 / 0.879 +30.25 /0.877 +32.14 / 0.878 +33.19 / 0.910 +PSNR (dB)/ SSIM +口 +口 +口 +口 +口38.28 /0.945 (f) +(a) +37.66 /0.931 (b) +38.61 /0.948 (c) +36.77 /0.932 (d) +37.74 /0.939 (e) +PSNR (dB) / SSIM +39.73 / 0.946 +PSNR (dB) /SSIM +41.74 / 0.968 +38.67 / 0.950 +39.95 / 0.957 +40.84 / 0.9637 +[23] shows blurrier edge details compared to label images. On +the other hand, as shown in Fig. 3(e), the proposed method +shows the best qualitative and quantitative results among all +methods. Especially, the proposed method shows the sharpest +boundary between gray and white matters among methods as +shown in the first row of Fig. 3. +Next, we compare motion artifact reduction methods using +simulated respiratory motion-corruption data. In Fig. 4(a), the +vasculature of the liver is damaged or blurred due to motion +artifacts. Especially, artifacts appear most severe around blood +vessels. MARC removes motion artifacts and achieves high +quantitative metric values, but the blood vessels still look +blurry as shown in Fig. 4(b). On the other hand, Cycle- +MedGAN V2.0 [20] sharp reconstructed results but the PSNR +of results of Cycle-MedGAN V2.0 is lower than that of +input images (4(c)). It is maybe because Cycle-MedGAN +V2.0 changes image intensity or details. Results of bootstrap +subsampling and aggregation [23] are shown in Fig. 4(d), +resulting in images with reduced motion artifacts and improved +quantitative metrics compared to input images. However, some +motion artifacts near the blood vessels remain (the first row +of Fig. 4(d)), and it is hard to recognize the vessel due +to blurring and remaining artifacts (the second row of Fig. +4(d)). Meanwhile, the proposed method shows the most similar +restoration results to the label images as shown in Fig. 4(e) +and (f). Specifically, the vascular structure is most clearly and +accurately restored by the proposed method. Furthermore, our +method significantly reduces motion artifacts around the blood +vessels compared to other methods. +TABLE I shows the quantitative metric values of motion +artifact reduction methods. In experiments using simulated +random motion-corrupted data, the proposed method achieves +the highest PSNR and SSIM, and it is consistent with the +qualitative results in Figs. 3 and 4. On the other hand, MARC +shows the highest quantitative results when using simulated +respiratory motion-corrupted data. However, as confirmed in +Figs. 3 and 4, reconstructed images by MARC are extremely +blurred, so the detailed structures are indistinguishable. Com- +pared to MARC, the proposed method removes the motion +artifacts without losing information on image details. Further- +more, the quantitative metric value of our method is the highest +among that of unpaired/unsupervised methods. +TABLE I: Quantitative results of various methods with sim- +ulated motion-corrupted data (Cycle: Cycle-MedGAN V2.0, +BSA: Bootstrap Subsampling and Aggregation). +Method +PSNR (dB) +SSIM +Brain +Random motion +Input +27.83 +0.751 +MARC [16] +29.29 +0.891 +Cycle [20] +28.79 +0.894 +BSA [23] +30.18 +0.839 +Proposed +31.40 +0.916 +Liver +Respiratory motion +Input +36.15 +0.912 +MARC [16] +37.87 +0.947 +Cycle [20] +35.54 +0.926 +BSA [23] +36.45 +0.932 +Proposed +37.01 +0.940 +Fig. 5: The in vivo motion artifact reduction results with the +CNUH liver dataset: (a) motion-corrupted input image, (b) +MARC [16], (c) Cycle-MedGAN V2.0 [20], (d) bootstrap sub- +sampling and aggregation [23], and (e) the proposed method. + +(a) +(b) +(c) +(d) +(e)8 +B. Results with In Vivo Data +Because the simulated motion artifacts only consider rigid +motion artifacts, it should be verified that the method can also +be applied to non-rigid in vivo motion artifact removal. In +Fig. 5(a), motion artifacts due to transient dyspnea degrade +the quality of liver MR image. We attempt to remove motion +artifacts in Fig. 5(a), and results are shown in Fig. 5(b) to +(e). Again, MARC removes not only motion artifacts but also +detailed structures of blood vessels, so the reconstructed image +is extremely blurry (Fig. 5(b)). Conversely, in Fig. 5(c), Cycle- +MedGAN V2.0 makes the image sharper, but it also amplifies +motion artifacts or noise in the input image. Next, the bootstrap +subsampling and aggregation method also fails to remove the +motion artifacts. Specifically, as shown in the yellow and green +boxes of Fig. 5(d), motion artifacts around the blood vessels +remain in the output image. Unlike comparison methods, the +proposed method successfully removes the motion artifacts +and reduces the noise level of the input image. Furthermore, +our method reconstructs detailed structures. For example, in +the yellow box of Fig. 5(e), the sharpness of the lesion +increased as the motion artifact disappeared. Also, the vascular +structure is recovered due to the reduction of motion artifacts +as shown in the green box of Fig. 5(e). Through the experiment +using in vivo motion-corrupted data, we confirmed that the +proposed method also removes in vivo motion artifacts that +contain the non-rigid motion of patients. +TABLE II: Clinical evaluation results of various methods with +in vivo motion-corrupted data (average ± standard deviation) +(Cycle: Cycle-MedGAN V2.0, BSA: Bootstrap Subsampling +and Aggregation). Higher scores indicate higher performance. +Method +Motion artifact +Noise +Blurring +Vessel clarity +Overall quality +Input +3.03 ± 0.91 +3.03 ± 0.68 +3.92 ± 0.71 +3.45 ± 1.20 +3.00 ± 1.09 +MARC [16] +3.37 ± 0.79 +3.34 ± 0.81 +2.29 ± 0.87 +2.97 ± 1.15 +2.50 ± 1.03 +Cycle [20] +3.42 ± 0.92 +3.13 ± 0.81 +3.97 ± 0.94 +3.47 ± 1.18 +3.21 ± 1.09 +BSA [23] +3.45 ± 1.22 +3.39 ± 0.75 +3.89 ± 1.06 +3.45 ± 1.29 +3.29 ± 1.18 +Proposed +3.63 ± 1.10 +3.58 ± 0.76 +3.97 ± 0.91 +3.71 ± 1.31 +3.45 ± 1.25 +C. Clinical Evaluation +Because it is impossible to quantitatively evaluate results +using in vivo motion-corrupted datasets due to the lack of +paired motion-free data, we evaluate motion artifact reduction +results by clinical evaluation. +TABLE II shows the scores by evaluating each method on +various criteria. MARC achieved scores of 3.37 and 3.34 in +terms of motion artifact and noise evaluation, respectively, +while input images score 3.03 in both evaluations. These +results indicate that MARC was good in motion artifact +improvement or noise reduction. However, MARC scored 2.29 +in the blurring evaluation, which is lower than the score of +input images (score: 3.92). The blurring effect of MARC +also can be confirmed in Fig. 5(b). Therefore, the overall +quality score of MARC (score: 2.50) is lower than that of +input images (score: 3.00). On the other hand, Cycle-MedGAN +V2.0 got the highest score in the blurring evaluation (score: +3.97). However, Cycle-MedGAN V2.0 scored 3.13 in noise +evaluation, which is lower than the scores of other methods. +This high level of noise affects the image quality drop of +Cycle-MedGAN V2.0, so Cycle-MedGAN V2.0 gets only 3.29 +points in terms of the overall image quality. As shown in Fig. +5 and TABLE II, the bootstrap subsampling and aggregation +method shows higher scores than the other existing methods +in most assessments. However, the outputs of the bootstrap +subsampling and aggregation method were slightly blurred, +so its score was lower than the input images in the blurring +evaluation. +While the other methods each showed drawbacks, the pro- +posed method achieved the highest performance in all evalua- +tions. First, in terms of motion artifact removal, the proposed +method achieves the highest score (score: 3.63) while other +methods get similar lower scores (score: 3.37-3.45). Next, +our method scored 3.58 and 3.97 in the noise and blurring +evaluations, respectively. From these results, we confirm that +our method does not amplify image noise level or blur output +images through the clinical evaluation. Moreover, the motion- +corrupted input images scored 3.45 in terms of vessel clarity. +The proposed method shows a significant improvement in +vessel clarity score (score: 3.71) while the vessel clarity of the +other three methods is similar to or lower than that of motion- +corrupted input images (score: 2.97-3.47). Finally, our method +gets the best score (score: 3.45) for overall image quality. To +sum up, the proposed method achieves the highest score in all +clinical evaluations, and this result indicates that our method +is useful in clinical practice. +VI. DISCUSSION +A. Comparison with Other Methods +In Section V, it was verified that MARC [16] generates +blurry outputs in both simulation and in vivo study. The +blurring results may be a limitation of methods based on su- +pervised learning. Because the supervised learning minimizes +the loss (e.g. L1, mean squared error (MSE)) between output +and label, it achieves high quantitative results as shown in +TABLE I. However, it can also lead to the loss of information +on image details because L1 or MSE losses do not assure the +perceptual quality of output images. +Unlike MARC, Cycle-MedGAN V2.0 [20] is an unpaired +method that does not require paired input and label im- +ages. Instead of using losses between input and label, it +translates an image from one domain to another domain by +utilizing cycle consistency loss and adversarial loss. Because +the discriminators of Cycle-MedGAN V2.0 distinguish real +and fake generated images, the generators of Cycle-MedGAN +V2.0 provide realistic images with sharp details. However, we +have confirmed that Cycle-MedGAN V2.0 also magnifies the +artifacts or noise of images. We conjecture that it is because the +networks of Cycle-MedGAN V2.0 consider resolution degra- +dation due to the motion artifacts to be the main difference +between the two image domains. Therefore, the networks of +Cycle-MedGAN V2.0 try to improve resolution rather than +eliminate motion artifacts. +Compared to the previous two methods, bootstrap sub- +sampling and aggregation [23] showed stable qualitative and +quantitative results. Nevertheless, because [23] works under +the assumption that the motion artifact appears as sparse + +9 +TABLE III: Ablation studies on hyperparameters with simu- +lated respiratory motion-corrupted data. The gray rows indi- +cate the hyperparameters that are selected in our experiments. +Hyperparameters +PSNR (dB) +SSIM +Time/image (sec) +λN′ +0 +36.36 +0.935 +19.30 +0.01 +37.01 +0.940 +19.30 +0.1 +36.58 +0.927 +19.30 +N′ +1 +36.45 +0.935 +1.834 +10 +37.01 +0.940 +19.30 +100 +36.88 +0.938 +195.6 +M +1 +36.43 +0.934 +6.358 +3 +37.01 +0.940 +19.30 +5 +37.28 +0.942 +32.48 +N′ × M +10 × 3 +37.01 +0.940 +19.30 +30 × 1 +36.90 +0.938 +19.30 +outliers in k-space, the performance of this method is degraded +if the assumption is not satisfied. For example, we simulated +the respiratory motion with Eq. (15), so the respiratory motion +appears as a continuous sinusoidal form in k-space. Because +the motion did not appear as sparse outliers, the performance +of [23] was dropped compared to when it works with simulated +random motion-corrupted data. +On the other hand, our proposed method presented out- +standing results compared to other comparison methods. The +proposed method successfully removes motion artifacts and +retrieves high-frequency image details in both simulation and +in vivo studies. +Nevertheless, our method is not free of limitations. Because +the score-based diffusion models require several steps of +reverse diffusion, it takes a long time to generate outputs. Al- +though we utilized the CCDF algorithm to reduce the inference +time, our method also requires several seconds as shown in +TABLE III. Therefore, the acceleration of the proposed method +should be done for clinical use. +B. Effects of Annealing Hyperparameters +In our method, we injected high-frequency components of +measurements (k-space of motion-corrupted images) with the +hyperparameter λN ′ to preserve detailed structures of MR +images. To confirm the effect of high-frequency component +injection, we conduct our method for simulated liver motion- +corrupted images with various λN ′. As shown in TABLE +III, the proposed method with λN ′ = 0 shows lower quan- +titative results than the proposed method with λN ′ = 0.01. +It is because detailed structures such as vessels cannot be +reconstructed perfectly without high-frequency component in- +jection. When λN ′ = 0.1, the quantitative results drop again +compared to results with λN ′ = 0.01. We conjecture that it is +because the high-frequency component of measurements also +contains motion artifacts, and the remaining artifacts degrade +the quality of reconstructed images. Therefore, we choose to +inject high-frequency components with λN ′ = 0.01 in our +experiments. +Next, we also confirm the effect of the selection of N ′. +When N ′ = 1, the motion artifacts remain in output images, +so the quantitative results deteriorate. On the other hand, our +method also shows the degraded performance when N ′ = 100. +It may be because the structures that cannot be seen in the +input image were generated during the iterations of the reverse +diffusion process. Moreover, the required inference time of +the proposed method with N ′ = 100 is quite long as shown +in TABLE III, so we choose N ′ = 10 that shows the best +qualitative and quantitative performance. +Finally, the number of iterations of the reverse diffusion +process M is also one of the important hyperparameters of +our method. Through the experiments on M, we find that the +proposed method cannot completely remove motion artifacts +when M = 1. On the other hand, when M = 5, the required +inference time for one image is too long while the performance +gain is negligible compared to when M = 3. Therefore, M = +3 is selected in our experiments. +In addition, we also verify the effect of the combination +of N ′ and M. The proposed method shows different results +depending on the combination of N ′ and M as shown in +TABLE III, even if it takes the same inference time. The +proposed method with N ′ = 30, M ′ = 1 shows lower quan- +titative performance compared to the method with N ′ = 10, +M ′ = 3. It is because the motion artifacts cannot be removed +perfectly with only one iteration of the diffusion process even +though N ′ is large. Through the experiment, we verify that +the combination of N ′ = 10, M ′ = 3 is better than N ′ = 30, +M ′ = 1 for the performance of our proposed method. +VII. CONCLUSION +In this paper, we proposed a novel MRI motion artifact +reduction method using the annealed score-based diffusion +model. By applying the diffusion process iteratively and +gradually imposing data consistency with high-frequency in- +jection, the proposed method successfully reduced simulated +and in vivo motion artifacts in MR images. Furthermore, we +verified that our method provides higher-quality images and +more clinical meaning compared to other state-of-the-art deep +learning methods. We believe that our algorithm can be a +useful framework for MRI motion artifact reduction. +VIII. ACKNOWLEDGEMENT +This work was supported by Institute of Information & com- +munications Technology Planning & Evaluation (IITP) grant +funded by the Korea government (MSIT) (No.2019-0-00075, +Artificial Intelligence Graduate School Program(KAIST)), +National Research Foundation(NRF) of Korea grant NRF- +2020R1A2B5B03001980, and by the KAIST Key Research +Institute (Interdisciplinary Research Group) Project. +REFERENCES +[1] A. Nishie, D. Kakihara, Y. Asayama, Y. Ushijima, Y. Takayama, N. Fu- +jita, D. Shimamoto, K. 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Vincent, “A connection between score matching and denoising au- +toencoders,” Neural computation, vol. 23, no. 7, pp. 1661–1674, 2011. + diff --git a/qdE1T4oBgHgl3EQfPgMa/content/tmp_files/load_file.txt b/qdE1T4oBgHgl3EQfPgMa/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c75ac821067c77128dd7af17d03ec5a3f2a217f3 --- /dev/null +++ b/qdE1T4oBgHgl3EQfPgMa/content/tmp_files/load_file.txt @@ -0,0 +1,937 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf,len=936 +page_content='1 Annealed Score-Based Diffusion Model for MR Motion Artifact Reduction Gyutaek Oh, Jeong Eun Lee, and Jong Chul Ye, Fellow, IEEE Abstract—Motion artifact reduction is one of the important research topics in MR imaging, as the motion artifact degrades image quality and makes diagnosis difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Recently, many deep learning approaches have been studied for motion artifact reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Unfortunately, most existing models are trained in a supervised manner, requiring paired motion-corrupted and motion-free images, or are based on a strict motion-corruption model, which limits their use for real-world situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' To address this issue, here we present an annealed score-based diffusion model for MRI motion artifact reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Specifically, we train a score-based model using only motion-free images, and then motion artifacts are removed by applying forward and reverse diffusion processes repeatedly to gradually impose a low- frequency data consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Experimental results verify that the proposed method successfully reduces both simulated and in vivo motion artifacts, outperforming the state-of-the-art deep learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Index Terms—MRI, motion artifact, score-based models, dif- fusion models I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' INTRODUCTION M AGNETIC resonance imaging (MRI) is an imaging technique that provides various types of contrast im- ages without radiation exposure or invasive procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Despite many advantages, MRI requires a long acquisition time due to its imaging physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, the long acquisition time leads to motion artifacts due to the movement of the patient, so the motion artifact is considered one of the main problems of MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In addition, the contrast agent injection may cause motion artifacts in MRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' For example, gadoxetic acid (Gd-EOB- DTPA) is one of the liver-specific MRI contrast agents that can help the diagnosis of diseases such as hepatocellular carcinoma, liver metastases [1], [2] by providing hepatobiliary phase (HBP) imaging [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, the administration of Gd- EOB-DTPA can occur acute transient dyspnea, resulting in transient severe motion (TSM) [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' If TSM occurs, the image quality of the arterial phase is degraded, and the accuracy of diagnosis can be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' So, an algorithm to correct the motion artifact due to TSM is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' There have been several attempts to reduce the motion artifact of MRI by tracking the motion [5], [6], or changing G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Oh is with the Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology (KAIST), Daejeon 34141, Republic of Korea (e-mail: okt0711@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Ye is with the Kim Jaechul Graduate School of AI, Korea Advanced Institute of Sci- ence and Technology (KAIST), Daejeon 34141, Republic of Korea (e-mail: jong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='ye@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='kr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Lee is with the Department of Radiology, Chung- nam National University Hospital, Chungnam National University College of Medicine, 282 Munhwa-ro, Jung-gu, Daejeon 35015, Republic of Korea (e-mail: leeje290@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='com).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' sampling trajectory or imaging sequence [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, they require additional devices or scan time, and the types of motion artifacts corrected by these methods are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Motion artifact correction algorithms based on compressed sensing (CS) [10]–[12] also have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' CS-based algorithms have shown high-quality results, but they have limitations such as the difficulty of hyperparameter tuning and high computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, many CS algorithms require raw k-space data, which are rarely obtained in clinical environments due to storage limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Recent studies for MRI motion artifact reduction are based on deep learning methods [13]–[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Deep learning methods have shown improved performance and reduced run time compared to previous methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, most of the deep learning methods are based on supervised learning approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Since paired motion-free and corrupted data are difficult to obtain, these methods usually utilize simulated motion artifact images to train the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, it is difficult to apply them to real motion-corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' To overcome the limitation of simulation-based deep learn- ing methods, deep learning methods using unpaired data have also been explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Some methods interpret the motion artifact reduction problem as image-to-image translation [20], [21], and address them based on CycleGAN architecture [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Although they utilize real motion artifact data, the performance of these algorithms is often limited because there is no explicit motion artifact rejection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Recently, we proposed an algorithm for MR motion artifact reduction using bootstrap subsampling and aggregation [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Under the assumption that the motion artifact appears as k- space outliers, the method removes the motion artifact by re- jecting k-space outliers in a probabilistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Although our prior method outperforms other simulation-based or unpaired deep learning methods, there exist limitations if the motion artifact does not appear as sparse outliers in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Recently, score-based diffusion models [24]–[26] have shown remarkable performance in the field of image gener- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In score-based diffusion models, a network that esti- mates the score, the gradient of the log probability density function, is first trained, and then images can be generated by solving the reverse-time stochastic differential equation (SDE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, it has been verified that unconditionally trained diffusion models can be applied to solve various inverse problems by adjusting sampling procedure using constraints [27]–[31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Importantly, the unconditionally trained diffusion models do not require paired data, so it is possible to solve inverse problems in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Inspired by this, here we propose a novel MRI motion arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='03027v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='IV] 8 Jan 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 1: The overall procedure of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' xN ′ is generated from the motion-corrupted image x0 by the forward diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Then the MR image with reduced motion artifact x0 is sampled by solving the reverse-time SDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' y denotes the measurement (k-space of motion-corrupted image), and it is used in the data consistency step to prevent the severe deformation of the output image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The output goes through forward and reverse diffusion iteratively to obtain the final reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' artifact reduction method using score-based diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 1 shows the overall procedure of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' During reverse diffusion, the low-frequency data consistency is gradually imposed in an iterative manner so that the overall structure of the original image is maintained and helps to remove only motion artifact components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In particular, our constraint is designed based on the ob- servation that the motion artifacts in MRI usually occurred in the high-frequency region of k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' This is because k- space acquisition is usually performed first in the center region and motion occurs after a certain period after the start of acquisition, so k-space samples that include motion artifacts generally appear in high-frequency regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, during the reverse diffusion, the low-frequency region needs to be maintained and only the high-frequency region should be corrected by diffusion sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, because the high- frequency region of k-space also contains the information of details, the detailed structures of images can be altered or van- ished if the data consistency step in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (9) is applied directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' To address this issue, we propose an annealed reverse sampling procedure where the data consistency step is gradually applied in a repeated manner to maintain high-frequency details of measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The remaining parts of the paper are constructed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Section II introduces backgrounds of score-based diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Section III contains the key idea of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The experimental setting is explained in Section IV, and qualitative and quantitative results are shown in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Section VI and VII contains the discussion and conclusion of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' BACKGROUNDS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Score-Based Diffusion Models A continuous diffusion process can be represented as {x(t)}1 t=0, where t ∈ [0, 1] denotes the time variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Here, x(0) ∼ pdata where pdata is the data distribution, and x(1) ∼ p1 where p1 refers to the noise distribution which is commonly set to Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Then, the diffusion process can be modeled by the solution of the following stochastic differential equation: dx = f(x, t)dt + g(t)dw, (1) where f : Rd → Rd is a drift coefficient, g : R → R is a diffusion coefficient, and w denotes a standard Wiener process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (1), it is possible to transmit a sample from the data distribution to that of the noise distribution through the forward diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' If it is possible to reverse the diffusion process in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (1), then we can obtain samples of the data distribution from samples of the noise distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In [32], it was shown that the reverse process is also a diffusion process that can be modeled by following reverse SDE: dx = [f(x, t) − g(t)2∇x log pt(x)]dt + g(t)d ¯w (2) where ¯w is also a standard Wiener process from time 1 to 0, and ∇x log pt(x) denotes the score function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, if the score function can be estimated, it is possible to derive the reverse diffusion process and generate samples of data distribution from random Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Among the many possible choices of f and g, we choose variance exploding SDE (VE-SDE) [26], where f and g are defined by f = 0, g = � d[σ2(t)] dt , (3) and σ(t) = σmin �σmax σmin �t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (4) Then, the reverse SDE in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (2) can be rewritten as: dx = −d[σ2(t)] dt ∇x log pt(x)dt + � d[σ2(t)] dt d ¯w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (5) Data consistency Data consistency Data consistency Data consistency Predictor Corrector Predictor Corrector Forward diffusion Reverse diffusion Data consistency3 Algorithm 1 CCDF with PC sampler Require: x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' N ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' {σi}N ′ i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' {ϵi}N ′ i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' sθ 1: z ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' I) 2: xN ′ ← x0 + σN ′z ▷ Forward diffusion 3: for i = N ′ to 1 do 4: x′ i−1 ← xi + (σ2 i − σ2 i−1)sθ(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' σi) 5: z ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' I) 6: xi−1 ← x′ i−1 + � σ2 i − σ2 i−1z ▷ Predictor 7: xi−1 ← Data consistency(xi−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' y) 8: ▷ Data consistency 9: z ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' I) 10: xi−1 ← xi−1 + ϵisθ(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' σi) + √2ϵiz ▷ Corrector 11: xi−1 ← Data consistency(xi−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' y) 12: ▷ Data consistency 13: end for Here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' the time index t is usually discretized uniformly into N intervals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' and xi and σi can be defined as xi := x(t)|t= i−1 N−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' σi := σmin �σmax σmin � i−1 N−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (6) The score function ∇x log pt(x) is generally estimated by training a neural network sθ(x(t), t) with denoising score matching [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The training of the score-based model with denoising score matching can be done by minimizing the following objective function: min θ Et � λ(t)Ex(0)Ex(t)|x(0) � ��sθ(x(t), t) − ∇x(t) log p0t(x(t)|x(0)) ��2 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (7) After training the neural network and plugging it into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (5), the reverse SDE can be solved by numerical SDE solvers or predictor-corrector (PC) samplers [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Come-Closer-Diffuse-Faster (CCDF) The main drawback of score-based diffusion models is their slow sampling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because the sampling starts from the random Gaussian noise and usually requires thousands of steps, the sampling time of score-based diffusion models is too long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In the prior work [28], the authors proposed a method called Come-Closer-Diffuse-Faster (CCDF) to reduce the sampling time of diffusion models when solving inverse problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Specifically, instead of starting sampling from ran- dom Gaussian noise, the forward diffusion is first applied from the initial reconstruction, leading to only few steps of reverse diffusion to get the final reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' More specifically, Algorithm 1 shows the CCDF sampling procedure using the PC sampler, where y denotes the initial measurement, and N ′ = Nt′ is the number of reverse diffusion steps where t′ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Here, the data consistency step should be non-expansive to maintain the stochastic contraction mapping nature of reverse diffusion sampling [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' With a better initialization followed by one-step forward diffusion, CCDF largely reduces the reverse sampling time for solving inverse problems [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 2: The data consistency step of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' MAIN CONTRIBUTION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Motivation In our prior work [23], we solved the motion artifact reduction problem by regarding motion artifacts as sparse outliers in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Specifically,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' if the motion is occurred by translation or rotation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' it is assumed to result in k-space phase shift or rotation at the specific phase encoding lines: ˆy(kx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' ky) = � F {Rαx}e−jΦ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' ky ∈ K F {x},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (8) where ˆy denotes the motion-corrupted k-space with the indices along the frequency encoding direction kx and phase encoding direction ky,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' and x is the motion-free image,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' F denotes the Fourier transform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Rα denotes the rotation operation with the angle α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Φ is the displacement in radian,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' and K is the phase encoding indices where the rotation or translation occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Based on this assumption, the network is trained to recon- struct fully sampled motion-free images from randomly sub- sampled images along the phase encoding direction in which the corrupted k-space data can be removed in a probabilistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Although this method does not require simulated motion artifact images and shows improved performance, it has a limitation in that it is difficult to apply when the motion artifacts cannot be considered as sparse outliers in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, because the index of outliers is not known, some outliers that are not removed by subsampling can remain in the reconstructed image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Proposed Method Rather than using the sparse outlier assumption in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (8), our method is based on a more relaxed assumption that the motion artifacts in MRI mainly occur in the high-frequency region of k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' This is because k-space acquisition is usually performed first in the center region and motion occurs after a certain period after the start of acquisition so that k- space samples with motion artifacts generally appear in high- frequency regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, the high-frequency region of k- space should be corrected to remove the motion artifact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The application of CCDF in Algorithm 1 starts from the one-step forward diffusion from the initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' a n¨aive way of using data consistency for reverse diffusion would be to impose the low-frequency region consistency: xi−1 = (I − F −1PΩF )x′ i−1 + F −1PΩy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (9) Fourier transform Inverse Fourier transform4 Algorithm 2 MR Motion Artifact Reduction Require: x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' N ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' {σi}N ′ i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' {ϵi}N ′ i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' {λi}N ′ i=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' sθ 1: for j = 1 to M do 2: z ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' I) 3: xN ′ ← x0 + σN ′z ▷ Forward diffusion 4: for i = N ′ to 1 do 5: x′ i−1 ← xi + (σ2 i − σ2 i−1)sθ(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' σi) 6: z ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' I) 7: xi−1 ← x′ i−1 + � σ2 i − σ2 i−1z ▷ Predictor 8: xi−1 ← (1 − λi)(I − F −1PΩF )xi−1 9: +λiF −1(I − PΩ)y + F −1PΩy 10: ▷ Data consistency 11: z ∼ N(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' I) 12: xi−1 ← xi−1 + ϵisθ(xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' σi) + √2ϵiz ▷ Corrector 13: xi−1 ← (1 − λi)(I − F −1PΩF )xi−1 14: +λiF −1(I − PΩ)y + F −1PΩy 15: ▷ Data consistency 16: end for 17: end for where PΩ is the operator that samples only the low-frequency region of k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In other words, during reverse diffusion, the low-frequency region is maintained so that only the high- frequency region is corrected by the diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, because the high-frequency region of k-space also contains the information of details, the detailed structures of images can be altered or vanished if the data consistency step in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (9) is applied directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' To address this issue, we propose an annealed data consistency step to maintain high-frequency details of measurements as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 2: xi−1 = (1 − λi)(I − F −1PΩF )x′ i−1 + λiF −1(I − PΩ)y + F −1PΩy, (10) where λi ∈ [0, 1] is the annealing hyperparameter to control the weight of high-frequency components of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, as shown in Algorithm 2, we choose relatively small N ′, and repeat forward and reverse processes M times so that the high-frequency components of the measurement are gradually added at each data consistency step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Here, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (10) can be written as xi−1 = T (xi−1) := Ax′ i−1 + b, where A = (1 − λi)(I − F −1PΩF ), b = λiF −1(I − PΩ) + F −1PΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Since ��I − F −1PΩF �� ≤ 1 [28], it is also true that ∥A∥ = ��(1 − λi)(I − F −1PΩF ) �� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, T is a non- expansive mapping, so it can accelerate the reverse diffusion process through the CCDF principle [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Implementation Details In our implementation, we choose VE-SDE, which results in the following one-step forward sampling: x(t) = x(0) + σ(t)z (11) where z ∼ N(0, I) and x(0) is the clean training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' By plugging this in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (7), we have the following cost function [26]: min θ EtEx(0)Ex(t)|x(0) � ����σ(t)sθ(x(t), t) − x(t) − x(0) σ(t) ���� 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (12) Here, we choose the number of discretized steps N = 1000, and σmin and σmax in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (6) are set to σmin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 and σmax = 50, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' We train the score model for 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='3M iterations and follow [30] for the setting of other hyperparameters such as optimization, batch size, learning rate, gradient clipping, or exponential moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In addition, for N ′, M and λi in Algorithm 2, we choose N ′ = 10, M = 3, and λi = λN ′ N ′ − 1(i − 1), (13) where λN ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In other words, λi linearly decreases to 0 as i goes to 1, so the weight of high-frequency components of the measurement decreases as reverse diffusion proceeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In CCDF [28], it was shown that a better initialization pro- vides faster reverse sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Accordingly, the neural network (NN) initialization could be utilized if available as it is better than the original artifact-corrupted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Accordingly, we also employed NN initialization with [20] for the brain dataset, and [23] for the liver dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' METHODS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Experimental Data In our experiments, we use two MR datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The first dataset is the human connectome project (HCP) dataset which is the public dataset that contains human brain MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' This dataset is acquired by Siemens 3T system with 3D spin echo sequence, and the scan parameters are as follows: TR = 3200 ms, TE = 565 ms, echo train duration = 1105, matrix size = 320×320, voxel size = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='7×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='7×0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='7 mm3, and phase encoding direction = anterior-posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because the HCP dataset does not contain motion-corrupted images, it is used for quantitative evaluation with motion artifact simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The score model is trained with 3000 motion-free MR images from 150 subjects, and other 800 images from 40 subjects are used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The second dataset is collected from Chungnam National University Hospital (CNUH), and it includes Gd-EOB-DTPA- enhanced liver MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' It is obtained by a 3T Philips Achieva MR system with the following scan parameters: TR = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='1 ms, TE = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='5 ms, flip angle = 10◦, field of view = 256×256 mm2, slice thickness/intersection gap = 2/0 mm, acquisition matrix = 320×192, and phase encoding direction = anterior- posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Also, dynamic imaging including various phases was obtained, but only arterial phase images are used for experiments because TSM usually occurs during the arterial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The liver dataset consists of two groups, motion-free images, and motion-corrupted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' For the training of the score model, 3097 motion-free images from 18 subjects are 5 used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' After training, 444 simulated motion-corrupted images from 5 subjects are selected for the quantitative evaluation, and 38 MR volumes with in vivo motion-corrupted images are used for qualitative and radiologist evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Artifact Simulation For the quantitative evaluation, we used simulated motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The simulation was performed similarly to prior works [16], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The first type of motion artifact that we simulate is random translation and rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' We simulate the first type of motion artifact with the HCP brain dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The motion artifact with random translation and rotation can be simulated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (8) with (α, Φ) = � (αky, ky∆ky + kx∆kx), |ky| > k0 (0, 0), otherwise, (14) where αky denotes the rotation angle, ∆ky and ∆kx denote the degree of motion along x and y direction, respectively, and k0 is the delay time of the phase error due to the centric k-space filling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In our simulation, k0 is fixed to π/10, αky is randomly sampled from [−2◦, 2◦], ∆ky and ∆kx are sampled from [−1cm, 1cm] and [−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='5cm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='5cm], respectively, at each k-space line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The second type of simulated motion is respiratory motion, which appears as a sinusoidal function in k-space [16], [23]: (α, Φ) = � (0, ky∆ky sin(mky + n)), |ky| > k0 (0, 0), otherwise, (15) where ∆ky, m, and n denote the amplitude, period, and phase shift of the sinusoidal function, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because the respiratory motion appears in abdominal MR images, we simulate it with the liver MR image dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Parameters for the simulation are sampled as follows: k0 ∼ U[π/10, π/5], ∆ky ∼ U[1cm, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='5cm], m ∼ U[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0], and n ∼ U[0, π/4], where U[a, b] denotes the uniform distribution with the interval [a, b] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Comparison Methods We compared our method with three state-of-the-art meth- ods to verify the performance of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The first com- parison method is MARC [16], a method for reducing liver MRI motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because it is a supervised method, we train MARC models using simulated motion-corrupted images with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (14) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The second comparison method is Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 [20], an unpaired deep learning method based on CycleGAN [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 can be trained with both simulated or in vivo motion-corrupted data, but we train it with only simulated motion-corrupted data because the training of Cycle- MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 was unstable when using in vivo data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' We also employed the bootstrap subsampling and aggre- gation method in [23] as a comparison method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because this method requires only motion-free images during training, simulated or in vivo motion-corrupted images were not used during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Evaluation Methods For the quantitative evaluation, we used the peak signal-to- noise ratio (PSNR) and the structural similarity index metric (SSIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because there is no ground truth matched with in vivo motion-corrupted images, the quantitative evaluation was performed with simulated motion-corrupted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In addition, we also conducted a clinical evaluation with the results using in vivo motion-corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Specifically, a radiologist with 13 years of experience in abdominal MR imaging performed an analysis of the results of various methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The image analysis was conducted from various per- spectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' First, the performance in reducing motion artifacts is rated using a 5-point scoring system: 1 = non-diagnostic (severe artifacts causing impaired diagnostic capability of the readers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 2 = substantial artifacts with image quality decrease, but diagnostic performance impairment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 = mild artifacts, no significant (only mild) image quality disturbance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4 = minimal artifacts, sharp image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5 = no artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The image noise level is also evaluated with the following scoring system: 1 = non- diagnostic (severe noise causing impaired diagnostic capability of the readers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 2 = substantial noise with image quality decrease, but diagnostic performance impairment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 = mild noise, no significant (only mild) image quality disturbance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4 = minimal noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5 = no noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Next, the blurring can be induced when reducing the motion artifact, so the rating of image blurring level is performed: 1 = non-diagnostic (severely pixelated texture causing impaired diagnostic capability of the readers);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 2 = substantially pixelated, artificial sensation with concerns about the loss of normal texture, without diagnostic performance impairment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 = mildly pixelated, artificial sensa- tion, without image quality decrease;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4 = minimal alteration of image texture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5 = no alteration of image texture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, because the hepatic artery (HA) on the arterial phase should be visualized clearly, the vessel clarity is evaluated with a scoring system: 1 = not delineated due to motion or low signal-to-noise ratio (SNR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 2 = blur or decreased SNR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 = clear common hepatic artery (CHA) and proper hepatic artery (PHA), but blurred HA and gastroduodenal artery (GDA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4 = entire HA is clearly visible, clear CHA, GDA, bilateral HA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5 = strong contrast-to-noise ratio with score 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Last, the overall image quality is assessed by following scoring system: 1 = non-diagnostic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 2 = not satisfactory image quality, but re- examination is not needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 = acceptable image quality (im- age quality may not be very good, but clinically acceptable);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4 = good image quality without significant artifact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5 = excellent image quality without artifact and good spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The score is rated for each volume in all assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Also, the results were presented to the radiologist in a random order without any labeling for a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Results with Simulated Data Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 shows the motion artifact reduction results of various methods with random simulated motion-corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3(a), it is hard to recognize detailed structures of brains due to motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' MARC [16] reduces the motion artifact but the output images of MARC are too blurry or 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3: The simulated random motion artifact reduction results with the HCP brain dataset: (a) motion-corrupted input image, (b) MARC [16], (c) Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 [20], (d) bootstrap subsampling and aggregation [23], (e) the proposed method, and (f) motion-free label image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The difference maps show the difference between each image and the label image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' PSNR and SSIM values of each image are shown in the corner of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4: The simulated respiratory motion artifact reduction results with the CNUH liver dataset: (a) motion-corrupted input image, (b) MARC [16], (c) Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 [20], (d) bootstrap subsampling and aggregation [23], (e) the proposed method, and (f) motion-free label image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The difference maps show the difference between each image and the label image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' PSNR and SSIM values of each image are shown in the corner of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' smoothed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In the results of MARC, the boundary between gray matter and white matter is not clear (the first row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3(b)), and the structure of the choroid plexus is not properly restored (the second row in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Next, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3(c) and (d), Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 [20] and bootstrap sub- sampling and aggregation [23] remove random motion artifacts significantly and show increased quantitative results compared to input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, there are some differences between label images and outputs of Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 as shown in difference maps, and bootstrap subsampling and aggregation (a) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='85 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='809 (b) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='93 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='916 (c) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='41 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='916 (d) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='31 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='881 (e) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='65/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='942 (f) PSNR (dB)/ SSIM 口 口 C 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='55 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='785 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='80 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='879 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='25 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='877 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='14 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='878 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='19 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='910 PSNR (dB)/ SSIM 口 口 口 口 口38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='28 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='945 (f) (a) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='66 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='931 (b) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='61 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='948 (c) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='77 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='932 (d) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='74 /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='939 (e) PSNR (dB) / SSIM 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='73 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='946 PSNR (dB) /SSIM 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='74 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='968 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='67 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='950 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='95 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='957 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='84 / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='9637 [23] shows blurrier edge details compared to label images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' On the other hand, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3(e), the proposed method shows the best qualitative and quantitative results among all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Especially, the proposed method shows the sharpest boundary between gray and white matters among methods as shown in the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Next, we compare motion artifact reduction methods using simulated respiratory motion-corruption data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4(a), the vasculature of the liver is damaged or blurred due to motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Especially, artifacts appear most severe around blood vessels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' MARC removes motion artifacts and achieves high quantitative metric values, but the blood vessels still look blurry as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' On the other hand, Cycle- MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 [20] sharp reconstructed results but the PSNR of results of Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 is lower than that of input images (4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' It is maybe because Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 changes image intensity or details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Results of bootstrap subsampling and aggregation [23] are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4(d), resulting in images with reduced motion artifacts and improved quantitative metrics compared to input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, some motion artifacts near the blood vessels remain (the first row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4(d)), and it is hard to recognize the vessel due to blurring and remaining artifacts (the second row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4(d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Meanwhile, the proposed method shows the most similar restoration results to the label images as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 4(e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Specifically, the vascular structure is most clearly and accurately restored by the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, our method significantly reduces motion artifacts around the blood vessels compared to other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' TABLE I shows the quantitative metric values of motion artifact reduction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In experiments using simulated random motion-corrupted data, the proposed method achieves the highest PSNR and SSIM, and it is consistent with the qualitative results in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' On the other hand, MARC shows the highest quantitative results when using simulated respiratory motion-corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, as confirmed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 3 and 4, reconstructed images by MARC are extremely blurred, so the detailed structures are indistinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Com- pared to MARC, the proposed method removes the motion artifacts without losing information on image details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Further- more, the quantitative metric value of our method is the highest among that of unpaired/unsupervised methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' TABLE I: Quantitative results of various methods with sim- ulated motion-corrupted data (Cycle: Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0, BSA: Bootstrap Subsampling and Aggregation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Method PSNR (dB) SSIM Brain Random motion Input 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='751 MARC [16] 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='891 Cycle [20] 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='894 BSA [23] 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='839 Proposed 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='916 Liver Respiratory motion Input 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='912 MARC [16] 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='947 Cycle [20] 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='926 BSA [23] 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='932 Proposed 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='940 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5: The in vivo motion artifact reduction results with the CNUH liver dataset: (a) motion-corrupted input image, (b) MARC [16], (c) Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 [20], (d) bootstrap sub- sampling and aggregation [23], and (e) the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (a) (b) (c) (d) (e)8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Results with In Vivo Data Because the simulated motion artifacts only consider rigid motion artifacts, it should be verified that the method can also be applied to non-rigid in vivo motion artifact removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(a), motion artifacts due to transient dyspnea degrade the quality of liver MR image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' We attempt to remove motion artifacts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(a), and results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(b) to (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Again, MARC removes not only motion artifacts but also detailed structures of blood vessels, so the reconstructed image is extremely blurry (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Conversely, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(c), Cycle- MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 makes the image sharper, but it also amplifies motion artifacts or noise in the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Next, the bootstrap subsampling and aggregation method also fails to remove the motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Specifically, as shown in the yellow and green boxes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(d), motion artifacts around the blood vessels remain in the output image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Unlike comparison methods, the proposed method successfully removes the motion artifacts and reduces the noise level of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, our method reconstructs detailed structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' For example, in the yellow box of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(e), the sharpness of the lesion increased as the motion artifact disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Also, the vascular structure is recovered due to the reduction of motion artifacts as shown in the green box of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Through the experiment using in vivo motion-corrupted data, we confirmed that the proposed method also removes in vivo motion artifacts that contain the non-rigid motion of patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' TABLE II: Clinical evaluation results of various methods with in vivo motion-corrupted data (average ± standard deviation) (Cycle: Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0, BSA: Bootstrap Subsampling and Aggregation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Higher scores indicate higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Method Motion artifact Noise Blurring Vessel clarity Overall quality Input 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='00 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='09 MARC [16] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='79 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='97 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='50 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='03 Cycle [20] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='92 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='47 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='21 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='09 BSA [23] 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='89 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='29 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='18 Proposed 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='63 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='76 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='71 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='25 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Clinical Evaluation Because it is impossible to quantitatively evaluate results using in vivo motion-corrupted datasets due to the lack of paired motion-free data, we evaluate motion artifact reduction results by clinical evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' TABLE II shows the scores by evaluating each method on various criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' MARC achieved scores of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='37 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='34 in terms of motion artifact and noise evaluation, respectively, while input images score 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='03 in both evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' These results indicate that MARC was good in motion artifact improvement or noise reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, MARC scored 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='29 in the blurring evaluation, which is lower than the score of input images (score: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='92).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The blurring effect of MARC also can be confirmed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, the overall quality score of MARC (score: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='50) is lower than that of input images (score: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' On the other hand, Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 got the highest score in the blurring evaluation (score: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='97).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 scored 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='13 in noise evaluation, which is lower than the scores of other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' This high level of noise affects the image quality drop of Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0, so Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 gets only 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='29 points in terms of the overall image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 5 and TABLE II, the bootstrap subsampling and aggregation method shows higher scores than the other existing methods in most assessments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, the outputs of the bootstrap subsampling and aggregation method were slightly blurred, so its score was lower than the input images in the blurring evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' While the other methods each showed drawbacks, the pro- posed method achieved the highest performance in all evalua- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' First, in terms of motion artifact removal, the proposed method achieves the highest score (score: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='63) while other methods get similar lower scores (score: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='37-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Next, our method scored 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='58 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='97 in the noise and blurring evaluations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' From these results, we confirm that our method does not amplify image noise level or blur output images through the clinical evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Moreover, the motion- corrupted input images scored 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45 in terms of vessel clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The proposed method shows a significant improvement in vessel clarity score (score: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='71) while the vessel clarity of the other three methods is similar to or lower than that of motion- corrupted input images (score: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='97-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Finally, our method gets the best score (score: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45) for overall image quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' To sum up, the proposed method achieves the highest score in all clinical evaluations, and this result indicates that our method is useful in clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' DISCUSSION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Comparison with Other Methods In Section V, it was verified that MARC [16] generates blurry outputs in both simulation and in vivo study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The blurring results may be a limitation of methods based on su- pervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because the supervised learning minimizes the loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' L1, mean squared error (MSE)) between output and label, it achieves high quantitative results as shown in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, it can also lead to the loss of information on image details because L1 or MSE losses do not assure the perceptual quality of output images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Unlike MARC, Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 [20] is an unpaired method that does not require paired input and label im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Instead of using losses between input and label, it translates an image from one domain to another domain by utilizing cycle consistency loss and adversarial loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because the discriminators of Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 distinguish real and fake generated images, the generators of Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 provide realistic images with sharp details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' However, we have confirmed that Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 also magnifies the artifacts or noise of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' We conjecture that it is because the networks of Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 consider resolution degra- dation due to the motion artifacts to be the main difference between the two image domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, the networks of Cycle-MedGAN V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='0 try to improve resolution rather than eliminate motion artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Compared to the previous two methods, bootstrap sub- sampling and aggregation [23] showed stable qualitative and quantitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Nevertheless, because [23] works under the assumption that the motion artifact appears as sparse 9 TABLE III: Ablation studies on hyperparameters with simu- lated respiratory motion-corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The gray rows indi- cate the hyperparameters that are selected in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Hyperparameters PSNR (dB) SSIM Time/image (sec) λN′ 0 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='935 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='940 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='927 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='30 N′ 1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='935 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='834 10 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='940 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='30 100 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='938 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='6 M 1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='934 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='358 3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='940 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='30 5 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='942 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='48 N′ × M 10 × 3 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='940 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='30 30 × 1 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='938 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='30 outliers in k-space, the performance of this method is degraded if the assumption is not satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' For example, we simulated the respiratory motion with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' (15), so the respiratory motion appears as a continuous sinusoidal form in k-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because the motion did not appear as sparse outliers, the performance of [23] was dropped compared to when it works with simulated random motion-corrupted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' On the other hand, our proposed method presented out- standing results compared to other comparison methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The proposed method successfully removes motion artifacts and retrieves high-frequency image details in both simulation and in vivo studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Nevertheless, our method is not free of limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Because the score-based diffusion models require several steps of reverse diffusion, it takes a long time to generate outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Al- though we utilized the CCDF algorithm to reduce the inference time, our method also requires several seconds as shown in TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, the acceleration of the proposed method should be done for clinical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Effects of Annealing Hyperparameters In our method, we injected high-frequency components of measurements (k-space of motion-corrupted images) with the hyperparameter λN ′ to preserve detailed structures of MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' To confirm the effect of high-frequency component injection, we conduct our method for simulated liver motion- corrupted images with various λN ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' As shown in TABLE III, the proposed method with λN ′ = 0 shows lower quan- titative results than the proposed method with λN ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' It is because detailed structures such as vessels cannot be reconstructed perfectly without high-frequency component in- jection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' When λN ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='1, the quantitative results drop again compared to results with λN ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' We conjecture that it is because the high-frequency component of measurements also contains motion artifacts, and the remaining artifacts degrade the quality of reconstructed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, we choose to inject high-frequency components with λN ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='01 in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Next, we also confirm the effect of the selection of N ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' When N ′ = 1, the motion artifacts remain in output images, so the quantitative results deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' On the other hand, our method also shows the degraded performance when N ′ = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' It may be because the structures that cannot be seen in the input image were generated during the iterations of the reverse diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Moreover, the required inference time of the proposed method with N ′ = 100 is quite long as shown in TABLE III, so we choose N ′ = 10 that shows the best qualitative and quantitative performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Finally, the number of iterations of the reverse diffusion process M is also one of the important hyperparameters of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Through the experiments on M, we find that the proposed method cannot completely remove motion artifacts when M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' On the other hand, when M = 5, the required inference time for one image is too long while the performance gain is negligible compared to when M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Therefore, M = 3 is selected in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' In addition, we also verify the effect of the combination of N ′ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The proposed method shows different results depending on the combination of N ′ and M as shown in TABLE III, even if it takes the same inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' The proposed method with N ′ = 30, M ′ = 1 shows lower quan- titative performance compared to the method with N ′ = 10, M ′ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' It is because the motion artifacts cannot be removed perfectly with only one iteration of the diffusion process even though N ′ is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Through the experiment, we verify that the combination of N ′ = 10, M ′ = 3 is better than N ′ = 30, M ′ = 1 for the performance of our proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' CONCLUSION In this paper, we proposed a novel MRI motion artifact reduction method using the annealed score-based diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' By applying the diffusion process iteratively and gradually imposing data consistency with high-frequency in- jection, the proposed method successfully reduced simulated and in vivo motion artifacts in MR images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' Furthermore, we verified that our method provides higher-quality images and more clinical meaning compared to other state-of-the-art deep learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' We believe that our algorithm can be a useful framework for MRI motion artifact reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' ACKNOWLEDGEMENT This work was supported by Institute of Information & com- munications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content='2019-0-00075, Artificial Intelligence Graduate School Program(KAIST)), National Research Foundation(NRF) of Korea grant NRF- 2020R1A2B5B03001980, and by the KAIST Key Research Institute (Interdisciplinary Research Group) Project.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} +page_content=' 1661–1674, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE1T4oBgHgl3EQfPgMa/content/2301.03027v1.pdf'} diff --git a/rdAzT4oBgHgl3EQfcfxL/content/tmp_files/2301.01403v1.pdf.txt b/rdAzT4oBgHgl3EQfcfxL/content/tmp_files/2301.01403v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8462fc39b3eb9576d1636c8eda21b12c89ecf9c --- /dev/null +++ b/rdAzT4oBgHgl3EQfcfxL/content/tmp_files/2301.01403v1.pdf.txt @@ -0,0 +1,1113 @@ + + +1 + +Atomic-scale Modulation of Synthetic Magnetic Order in Oxide Superlattices + +Seung Gyo Jeong, Sehwan Song, Sungkyun Park, Valeria Lauter, and Woo Seok Choi* + +S. G. Jeong, W. S. Choi +Department of Physics, Sungkyunkwan University, Suwon 16419, Korea +E-mail: choiws@skku.edu + +S. Song, S. Park +Department of Physics, Pusan National University, Busan 46241, Korea + +V. Lauter +Neutron Scattering Division, Neutron Sciences Directorate, Oak Ridge National Laboratory, +Oak Ridge 37831, USA + +Keywords: atomic-scale modulation, synthetic magnetic order, tunable magnetic +noncollinearity, magnetic oxide superlattices, polarized neutron reflectometry + +Atomic-scale precision control of magnetic interactions facilitates a synthetic spin order +useful for spintronics, including advanced memory and quantum logic devices. Conventional +modulation of synthetic spin order has been limited to metallic heterostructures that exploit +RKKY interaction through a nonmagnetic metallic spacer; however, they face problems +arising from Joule heating and/or electric breakdown. The practical realization and +observation of a synthetic spin order across a nonmagnetic insulating spacer would lead to the +development of spin-related devices with a completely different concept. Herein, we report +the atomic-scale modulation of the synthetic spiral spin order in oxide superlattices composed +of ferromagnetic metal and nonmagnetic insulator layers. The atomically controlled +superlattice exhibit an oscillatory magnetic behavior, representing the existence of a spiral +spin structure. Depth-sensitive polarized neutron reflectometry evidences modulated spiral +spin structures as a function of the nonmagnetic insulator layer thickness. Atomic-scale +customization of the spin state could lead the field one step further to actual spintronic +applications. + + + + + +2 + + +1. Introduction +Synthetic spin order in magnetic heterostructures promotes novel spintronic functionalities[1] +including colossal magnetoresistance,[2, 3] tunneling magnetoresistance,[4] topological Hall +effect,[5] spin Hall effect,[6, 7] spin-wave propagation,[8] and terahertz spin-transfer torque.[9] In +typical ferromagnetic (FM)/nonmagnetic-metal (NM-M)/FM heterostructures, the relative spin +orientation between two FM layers can be modulated by the thickness of the NM-M layer, +thereby realizing a synthetic magnetic order, which is useful for designing magnetic storage +and logic devices.[1] This is generally understood by the Ruderman-Kittel-Kasuya-Yosida +(RKKY) interlayer exchange interaction between the FM layers, which is mediated by the +conduction electrons in the NM-M layer. In this scheme, the interaction strength oscillates as +a function of the NM-M layer thickness.[1] In contrast, FM/nonmagnetic-insulator (NM-I)/FM +heterostructures foster synthetic spin order, with unfamiliar exchange mechanisms other than +the RKKY interaction.[10, 11] Recently, chiral phonon has been suggested to carry the spin +information across the NM-I layer via strong spin-phonon and spin-orbit coupling,[11] +similarly acting as the conduction electron across the NM-M layer for the RKKY +interaction.[10, 11] It has been shown that the chiral phonons could mediate the interlayer +exchange interaction leading to an oscillatory magnetization as a function the NM-I layer +thickness, evidenced by confocal Raman spectroscopy.[10] If the synthetic spin order is +atomically controllable in NM-I based heterostructures, the inherent limitations such as Joule +heating and electric breakdown in NM-M based heterostructures can be resolved. + +Polarized neutron reflectometry (PNR) is a favorable technique for investigating the atomic +layer-resolved distribution of spins, particularly when combined with atomic-scale epitaxy of +synthetic magnetic heterostructures.[12-15] Frist, the in-plane spin orientation in the magnetic +layers can be obtained by comparing the non-spin-flip and spin-flip components of neutrons +in the PNR spectra. Second, the depth profile of the spin orientation can be obtained by +assessing the out-of-plane component of the wavevector transfer (Qz) in the specular PNR. +Third, application of the PNR to magnetic superlattices is further advantageous because the +superlattice Bragg peak results in an enhanced reflectivity signal for the analyses. + +In this study, we present an atomic-scale thickness control of the synthetic spin structures in +oxide superlattices. We epitaxially deposited SrRuO3 (SRO, FM layer)/SrTiO3 (STO, NM-I +layer) superlattices on (001)-oriented single crystal STO substrates using pulsed laser epitaxy. + + + +3 + +As schematically shown in Figure 1a,b, the spiral spin structure of six-unit-cell-thick (1 u.c. +~0.4 nm) FM SRO layers was modulated by the y-u.c.-thick NM-I STO layers with ten +repetitions, within the [6|y] superlattice. Consequently, y-dependent oscillatory net +magnetization was obtained consistently from both the magnetization and the PNR +measurements. We note that the magnetic easy axis of SRO thin films usually point to the out- +of-plane direction. However, the y-dependent oscillatory behavior is only observed for the in- +plane magnetization measurements. Since PNR also identifies the in-plane magnetization of +the thin films, it is an ideal tool to characterize the important magnetic features of the +superlattices in microscopic scale. + +2. Results and Discussion +Figure 1c,d show X-ray reflectivity (XRR) and diffraction (XRD) results, respectively, +validating the atomically defined periodicities of the SRO/STO superlattices. The XRR curves +exhibit distinct Bragg peaks (SL+n) and Kiessig fringes corresponding to the total thickness of +the superlattice thin film, thereby indicating a well-defined periodic supercell structure with +atomically sharp interfaces (Figure 1c). Figure S1 shows the XRR fitting results of the [6|4], +[6|6], and [6|8] superlattices. Small decay slopes of the XRR indicate that the surface +roughness of the superlattices is < 1 u.c., which is consistent with the topographic images +obtained by atomic force microscopy (Figure S2). XRD θ-2θ scans show coherent diffraction +peaks (SL±n) of the superlattices on the (001)-oriented single crystal STO substrates (substrate +diffraction peaks marked by asterisks) (Figure 1d). With increasing y, the separation between +the superlattice peaks decreases systematically, which indicates the atomically controlled +periodicities (𝛬SL). When thickness of the x = 6 u.c. of SRO layer is assumed to be fixed at +2.358 nm (obtained from epitaxially strained single SRO thin films on STO substrates), the +estimated thicknesses of the y u.c. of STO layer are 0.768, 1.573, 2.318, 3.078, and 6.957 nm +for the [6|2], [6|4], [6|6], [6|8], and [6|18] superlattices, respectively. These values are +obtained from the Bragg’s law, 𝛬SL = 2π(Qn – Qn – 1)–1, where n and Qn denote the superlattice +peak order and the Qz position of the nth-order superlattice peak, respectively. The deviation +between the target and measured thicknesses is smaller than half a u.c. (< 0.2 nm), thus +manifesting structurally well-controlled superlattices. Finally, Figure 1e representatively +shows the X-ray reciprocal space mapping of [6|8] superlattice around the (103) Bragg +diffraction peak of the STO substrate, confirming a fully strained state. + + + + +4 + +The in-plane magnetization of the [6|y] superlattices shows an unexpected oscillation as a +function of y at low-temperature (-T) and low-magnetic- (H-) fields. Field-cooled M (T) +curves of the [6|y] superlattices were measured at 0.01 T of H-field along the in-plane +direction (Figure 2a). In addition to a robust FM transition at approximately 130 K +(consistent with 6 u.c. SRO single layer on the STO substrate[16, 17]), the in-plane M shows a +peak as T is further reduced, indicating that the FM order is disturbed. The M (H) curves at +low T exhibit a double hysteresis with a large coercive field (Hc) of approximately 1.8 T (see +the inset of Figure 2a), which supports the disturbed magnetic order in the ground state. +Moreover, the M (H) curve of y = 18 u.c. superlattice shows a typical single hysteresis loop +with a small value of Hc, but with a saturation M (Ms) of ~0.3 μB/Ru similar to those of y = 6 +and 8 u.c. superlattices, which indicates the suppression of the interlayer exchange interaction +at a sufficiently thick NM-I layer. We note that the enhancement of Ms for y = 4 u.c. +superlattice might originate from the structural modulation (orthorhombic to tetragonal) in the +SRO layers with decreasing y.[16] The oscillation of the in-plane M at 5 K and 0.01 T is shown +as a function of y in Figure 2f.[10] The magnetization results suggest the existence of an +interlayer exchange interaction across the NM-I STO layer and the possible unconventional +synthetic magnetic order in the SRO/STO superlattices. The interlayer exchange interaction +strength is estimated as J = tFMHcMs, where tFM is the thickness of the FM SRO layer. Figure +S3 shows the y-dependent oscillatory behavior of J, which depends on both Hc and Ms. This +confirms the unconventional character of the interlayer exchange interaction between FM +SRO layer across the NM-I STO layer, which is supposed to originate from the chiral +phonon.[10,11] The J values in SRO/STO superlattices are approximately two orders of +magnitude less than those for the RKKY interaction at the same thickness of the NM-M +layer.[18] Although conventional analyses of the M (H) curve would be useful to examine the +macroscopic strength of magnetic interaction, we note that the observed magnetic order is +rather fragile and is easily destroyed even in a moderate H-field. Therefore, we will focus on +the low H-field region of 0.01 T. + +To understand the unexpected y-dependent oscillatory magnetization, we examine a simple +model, wherein spins in SRO layers have a rotation angle ϕ with respect to the spins in the +adjacent SRO layer. Let us first assume that each SRO layer has the same uniform in-plane Mi +value (i is the layer index). Although it has been reported that spin ordering may differ +depending on the position away from the interface within magnetic heterostructures,[19,20] the +modulation is generally small in atomically defined heterostructures with symmetric + + + +5 + +interfaces. Second, we assume that ϕ depends linearly on y, that is, with an increase in the +NM-I layer thickness, the rotation of Mi increases, as in the case of the RKKY with the NM- +M layer. Figure 2b–d shows a schematic representation of the model. We estimate the sum of +the projections of Mi (y, ϕ) along a certain in-plane direction of the H-field for each SRO layer +to simulate the results of y-dependent M because the magnetization measurement gives only +the average scalar M value of the entire superlattice along the H-field direction. The average +M value is determined by the relation M = ∑ Mi (ϕ) +i += ∑ [Ma + Mb +i +cos((i − 1)ϕ)], where +Ma and Mb are constants. Next, we compare the result with that obtained from the experiment +and calculate the sum of the squared errors (SSE) as a function of ϕ, defined as [(Simulated +M) – (Measured M)]2 (Fig. 2(e)). In particular, for y = 2 u.c. (~0.8 nm), ϕ = ~80 and ~100º +results in the lowest SSE values within the spiral spin models with Ma = 0.070 μB/Ru and Mb += 0.629 μB/Ru. Figure 2(f) shows that the spiral spin models with ϕ = ~200º, 300º, and 400º +(40º) for the [6|4], [6|6], and [6|8] superlattices consistently describe the experimental y- +dependent oscillatory behavior of M at 5 K. Note that the underestimation of M for the [6|4] +superlattice might be due to the absence of the decaying term with increasing y in our model, +which would further complicate the model. + +PNR is employed to clarify the complex synthetic spin structure suggested by the +magnetization measurement discussed earlier (Figure 3 and Figure 4). Figure 3a +schematically shows that a collimated polychromatic neutron beam is incident on the film +surface at a grazing angle (α). The PNR signal was measured as a function of Qz = 4πsin(α)/λ +along the out-of-plane direction (z-direction in Figure 3a), where λ is the neutron wavelength. +We assume that a homogeneous in-plane M vector (M +⃗⃗⃗ ) is rotated by angle ϕ within the xy- +plane, defining the in-plane Mx and My components. Then the PNR signal would include both +non-spin-flip (R++ and R––) and spin-flip (R+– and R–+) contributions.21 R++ and R— are +defined as 1/4|(r+ + r–) + (r+ – r–) cosϕ|2 and 1/4|(r+ + r–) – (r+ – r–) cosϕ|2, respectively, and R+– +and R–+ are described as 1/4|r+ – r–|2 sin2ϕ, respectively, where r± are the complex reflection +amplitudes for the up and down spins.[13] Because the simulated experimental signal with +spin-flip polarization (R+– and R–+) is rather small for our samples, and hence, required a long +measuring time to obtain high statistics, we did not employ spin-flip polarization analyses. +Therefore, the spin-flip reflection was taken into account only in the data analyses as follows. +Our results show spin-up and spin-down polarizations of the PNR spectra, R+ = R++ + R+– and +R– = R–– + R–+, which describe depth-sensitive spin-vector rotation in the SRO/STO +superlattices. + + + +6 + + +To minimize the number of parameters in PNR fitting, we confirm the prerequisite structural +parameters and Ms values of the SRO/STO superlattices. First, the structural parameters, +including the interface roughness, thickness, and density of each layer, are obtained from non- +polarized neutron reflectivity data at 300 K when the sample is nonmagnetic. The top panels +of Figure 3c,e,g show the unpolarized neutron reflectivity and fitting results for the y = 4, 6, +and 8 superlattices, respectively, which are consistent with the XRR results shown in Figure +1c and Figure S1. The structural parameters of PNR analyses at 300 K have been confirmed +with those of XRR fitting. We summarize the structural fitting parameters of the XRR and +PNR results with the measured thickness consistently obtained by using XRR, PNR, and +scanning transmission electron microscopy[10] in Table S1. These results manifest the +atomically well-controlled SRO/STO superlattices with the minimized interdiffusion. Second, +to confirm the Ms values of the superlattices, we measured the PNR spectra at 85 K with a 1 T +(> Hc at 85 K) of in-plane H-field. The M (H) curves at 85 K exhibit a single hysteresis loop +as for a typical FM, as shown in Figure 3b. At this temperature with 1 T of H-field, the net M +is almost saturated (net M ~ Ms), which indicates that the in-plane rotation is relatively +suppressed (ϕ ~ 0º). The bottom panels of Figure 3c,e,g show the PNR spectra at 85 K in a 1 +T field. The insets in Figure 3c,e,g highlight the differences in PNR intensity between R+ and +R– near the Bragg peaks of each superlattice. This clear separation is consistently observed in +both the experimental result and fit, thereby indicating a robust FM order of the SRO layers at +a high H-field at 85 K. Figure 3d,f,h show the nuclear and magnetic scattering length densities +(SLD) fitted using GenX for [6|4], [6|6], and [6|8] superlattices, respectively.22 They show an +atomically well-defined periodic structure and ferromagnetically aligned Mi vectors in each +SRO layer at 85 K with a 1 T field. The obtained Ms values are estimated to be 0.4 μB/Ru for +[6|4] superlattice and 0.3 μB/Ru for [6|6] and [6|8] superlattices, respectively, which are highly +consistent with the M (H) curves in Figure 3b. + +In the ground state (experiments at 5 K and 0.01 T of H-field), the PNR spectra reveal a +modulated synthetic spiral spin structure suggested by magnetization measurements. Figure +4a shows the simulation result of a synthetic spiral spin structure with the best fit, which is +consistent with the experimental PNR result. For a single magnetic film, the PNR spectrum at +small Qz is influenced by the direct beam, whereas with increasing Qz, the reflectivity decays +exponentially such that the experimental data may be affected by the background signal.[12] In +contrast, as briefly discussed previously, superlattice Bragg peaks provide a better signal-to- + + + +7 + +noise ratio even at finite Qz values due to the coherent periodic structure.[23] Therefore, we +primarily compare the PNR data and simulation results near the Qz values associated with the +superlattice Bragg peaks. Figure S4 summarizes PNR simulations of the SRO/STO +superlattices for a typical FM model (left panels), a synthetic collinear antiferromagnetic +(sAFM) model with ϕ = 180º (middle panels), and a synthetic spiral spin structure (right +panels). The simulated spectra of collinear FM and sAFM models have two distinctions from +the experimental PNR spectra: (1) The collinear sAFM structure leads to doubling of the +magnetic unit cell of the superlattices. This doubling causes a significant discrepancy between +R+ and R– at half Qz of the superlattice Bragg peak.[24] As highlighted by the red rectangle in +Figure S4a, this model produces spectra that are inconsistent with the experimental PNR +spectra. (2) The collinear FM model results in a strong intensity contrast between the R+ and +R– in the superlattice Bragg peak due to the difference in the scattering of spin up and spin +down neutrons of periodic superlattice structures.[24] Note the similarity to the PNR spectra at +85 K with robust FM ordering. As highlighted by the blue rectangle in Figure S4b, the +experimental PNR spectra exhibit negligible separation in the superlattice Bragg peaks. +Hence, the PNR experimental results show that a collinear spin configuration is unlikely in +SRO/STO superlattices. Instead, the spiral spin structure model best describes the +experimental PNR spectra among the considered simple models, confirming the in-plane spin +rotation in the SRO/STO superlattice as a function of y. + +3. Conclusion +In summary, we presented the modulation of synthetic magnetic order in FM/NM-I/FM +superlattice via atomic-scale precision thickness control. To customize the in-plane rotation of +spin vectors within ferromagnetic SRO layers, we atomically controlled the thickness y of the +NM-I (STO) layer within the SRO/STO superlattices. The oscillatory in-plane magnetic +behavior as a function of y, determined by magnetization measurements, indicates the +presence of the synthetic magnetic order in the superlattices. The PNR result manifests the +depth profile of spin vectors within SRO layers with varying rotation angle ϕ and its +controllability via the precise control of y. Both magnetization and PNR results consistently +confirm that the atomically controlled thickness of the NM-I layers effectively changes the +rotation angle of the spiral spin structures in the FM layers within the FM/NM-I superlattices. +Our approach suggests an atomic-scale control knob for modulating synthetic spin spiral +structures for future spintronics. + + + + + +8 + + +Figure 1. Atomically controlled SRO/STO superlattices for controlling synthetic magnetic +order. a) and b) Schematic illustrations of modulation of synthetic magnetic order in FM +(red)/NM-I (gray)/FM (red) heterostructures via atomic-scale precision thickness control of +NM-I layer. c) X-ray reflectivity and d) θ-2θ scan results of the atomically well-defined +superlattices. e) X-ray reciprocal space mapping of [6|8] superlattice representatively shows +the fully strained state. + + + +a +C +STO(103) +[6|18] +7.8 +Intensitiy (arb. units) +units) +[6]8] +(arb. +wu) +[6]6] +SRO +Intensitiy +7.6 +b +[6|4] +SL+1 +[6|2] +SL +SL +SI +2 +7.4 +2 +3 +10 +20 +30 +2.4 +2.6 +2.8 +@ low-T, low-H-field +Q, (nm-1) +Q, (nm-1) +Qx (nm-1) + +9 + + +Figure 2. Synthetic in-plane magnetic behavior of SRO/STO superlattices. a) Field-cooled M +(T) curves of superlattices with 0.01 T of in-plane H-field. The inset shows the M (H) curves +of superlattices at 5 K. Schematic representation of in-plane Mi vector of the SRO layer for b) +[6|4], c) [6|6], and d) [6|8] superlattices. External H-field applied along the y-direction. The +red arrows in the bottom panels denote the summation of Mi vectors (Msum). e) Sum of square +error values as a function of ϕ for y = 2 u.c. superlattice, determined by [(Simulated M) – +(Measured M)]2. f) Summary of experimentally measured M and the simulated M values as a +function of y. The error bars indicate the experimental deviation from two different datasets. + + +a +0.4 +e +1.0 +f +0.6 +y = 2 u.c. (~0.8 nm) +0.3 +Experiment +Error +Simulation +0.3 +0.0 +M (ug/Ru) +M (ug/Ru) +M +Squre +0.2 +@5K +0.2 +0.6 +0.5 +80°100° +-5 +0 +5 +(L) H +0.1 +0.1 +[6|2] +[6|8] +[6|18] +wns +[6|4] +[9]9] +S +0.0 +0.0 +0.0 +0 +100 +200 +300 +0 +60 +120 +180 +2 +4 +6 +8 +T (K) +Φ(°) +y (u.c.) +[6|4] +Mo +[6|6] +d [6]8] +Mg +M4 +M3,Mg +M2, Mg +M- +Mg +Mg +M6 +Ma, +M4, +M1, +M1 +M10 +M10 +M10 +M +M5 +M3 +M2 +M +M5 +M6 +M +M. +M3 + +10 + + +Figure 3. PNR results of the FM state of SRO/STO superlattices at 85 K and 1 T of H-field. +a) Schematic representation of a PNR configuration with oxide thin films. b) M (H) curves of +[6|y] superlattices with different y at 85 K. PNR spectra at 300 K and 85 K with 1 T H-field +for c) [6|4], e) [6|6], and g) [6|8] superlattices. The insets show the extended PNR spectra near +the Bragg peaks of each superlattice. Nuclear and magnetic SLD of d) [6|4], f) [6|6], and h) +[6|8] superlattices. The vertical dashed lines in SLD are eye guides. + + +b +0.6 +a +C +d +(wu) +Polarized Neutron +[6]4], 300 K +Nuclear +Magnetic +[6]4] +R +Reflectivity (PNR) +50 +STO +Mx +0.4 +[6]6] +PNR (arb. units) +Fit + substrate +: SRO +My +[6]8] +0.2 +40 +[6]4], 85 K, 1 T +My +0.0 +R* +30 +Fit +Distance from s +R- +-0.2 +Fit +20 +A +1.6 +1.8 +in-plane H +Qz (nm-1) +-0.4 +@ 85 K +10 +R+ +R +-0.6 +L +0 +-5.0-2.5 +0.0 +2.5 +5.0 +0.5 +1.0 +1.5 +2.0 +0.0 0.2 0.4 +0.00 0.02 +Spin-polarized neutrons +(I) H +Qz (nm-1) +SLD (10-4nm-2) +e +(wu) +Nuclear +Magnetic +g +h +substrate (nm) +[6|6] +[6|8] +Nuclear +Magnetic +50 +50 +. units) +substrate ( +. units) +40 +40 +PNR (arb. +[6]6] +PNR (arb. units) +(arb. +[6]8] +PNR (arb. units) +30 +30 +from +from s +PNR +20 +20 +1.2 +1.4 +1.2 +Qz (nm-1) +Distance +Q, (nm-1) +istance +10 +10 +D +0 +0 +0.5 +1.0 +1.5 +2.0 +0.0 0.2 0.4 +0.00 0.02 +0.5 +1.0 +1.5 +2.0 +0.0 0.2 0.4 +0.00 0.02 +Qz (nm-1) +SLD (10-4nm-2) +Qz (nm-1) +SLD (10-4nm-2) + +11 + + +Figure 4. y-dependent in-plane rotation of synthetic spiral spin structures in SRO/STO +superlattices at 5 K and 0.01 T of H-field. a) PNR spectra at 5 K and 0.01 T of H-field for +[6|4], [6|6], and [6|8] superlattices. b) Magnetic SLD and c) schematic representation of the +synthetic spin order of [6|4], [6|6], and [6|8] superlattices. The result evidences that the +atomic-scale modulation of the synthetic spiral spin structure in SRO/STO superlattices as a +function of y consistent with magnetization measurements. + + + + +Distance from substrate (nm) +a +Magnetic +c [6]4] +PNR (arb. units) +[6|41,5K,0.01T +50 +M, +R+ +Fit +M +40 +R +Fit +30 +20 +10 +0.5 +1.0 +1.5 +2.0 +-0.02 +0.00 +0.02 +Qz (nm-1) +SLD (10-4nm-2) +Distance from substrate (nm) +[6]6] +Magnetic +[6]6] +PNR (arb.units) +50 +40 +30 +1!++++ +20 +10 +0.5 +1.0 +1.5 +2.0 +-0.02 +0.00 +0.02 +Qz (nm-1) +SLD (10-4nm-2) +: from substrate (nm) +Magnetic +[6]8] +[6]8] +PNR (arb.units) +50 +40 +30 +20 +Distance f +10 +0.5 +1.0 +1.5 +2.0 +-0.02 +0.00 +0.02 +Q, (nm-1) +SLD (10-4nm-2) + +12 + +4. Methods +Atomic-scale epitaxy: To validate the atomically designed FM/NM-I/FM heterostructures, we +deposited epitaxial SRO/STO superlattices on (001)-oriented single crystal STO substrates by +using pulsed laser epitaxy.[10, 17, 25–28] We fixed the number of laser pulses to grow the SRO +layers and systematically changed the number of pulses to modulate y. We controlled the six +u.c. of the SRO layers and y u.c. of the STO layers with ten repetitions, i.e., the [6|y] +superlattice. Before deposition, we treated the surfaces of the STO substrates using buffered +HF and annealed them at 1000 °C for 6 h under atmospheric conditions. We used +stoichiometric ceramic SRO and STO targets and an excimer (KrF) laser (248 nm; IPEX 868, +LightMachinery) with a 1.5 J/cm2 of fluence and a repetition rate of 5 Hz. The temperature +and oxygen partial pressure employed for the stoichiometric growth of both SRO and STO +layers were 750 °C and 100 mTorr, respectively. To characterize the atomically controlled +periodicity of superlattice, we performed XRR and XRD θ-2θ measurements using a high- +resolution PANalytical X'Pert and Rigaku SmartLab X-ray diffractometer with Cu K-α1. + +Magnetization measurement: The field-cooled M (T) curves of the SRO/STO superlattice +along the in-plane direction were measured using a magnetic property measurement system +(MPMS, Quantum Design). The in-plane M (H) curves of the SRO/STO superlattice were +measured at 5 and 85 K. We did not observe any exchange bias effect in M (H) curves of the +SRO/STO superlattice. When we estimated the orientation of the spin vectors to describe the +oscillatory M behavior, we included the layer-repetition-dependent M of superlattices as well +(Figure S5).[10] + +Polarized neutron reflectometry: The PNR experiments were performed on the time-of-flight +Magnetism Reflectometer (BL-4A) at the Spallation Neutron Source at Oak Ridge National +Laboratory (SNS, ORNL).[29] We used the closed-cycle refrigerator systems with a Bruker +electromagnet to induce an in-plane H-field. We used polarized neutron beam with +polarization efficiencies from 99 to 97.5% and λ within a band of 2 to 8 Å. The PNR spectra +with nuclear and spin structures were fitted using GenX.[22] + +Supporting Information +Supporting Information is available from the Wiley Online Library. + + + + + +13 + +Acknowledgements +This research used resources at the Spallation Neutron Source, a Department of Energy +(DOE) Office of Science User Facility operated by the Oak Ridge National Laboratory. We +gratefully acknowledge Dr. Haile Ambaye (Spallation Neutron Source) for technical support +with PNR experiments and data processing. We also thank Core Research Facilities, Pusan +National University for MPMS. This research was supported by the Basic Science Research +Programs through the National Research Foundation of Korea (NRF- +2020K1A3A7A09077715, 2021R1A2C2011340, and 2022R1C1C2006723). + + +References +[1] +Duine, R. A.; Lee, K.-J.; Parkin, S. S. P.; Stiles, M. D., Nat. Phys. 2018, 14 (3), 217-219. +[2] +Dieny, B., J. Magn. Magn. Mater. 1994, 136 (3), 335-359. +[3] +Kim, H. H.; Yang, B.; Patel, T.; Sfigakis, F.; Li, C.; Tian, S.; Lei, H.; Tsen, A. W., Nano +Lett. 2018, 18 (8), 4885-4890. +[4] +Yuasa, S.; Nagahama, T.; Fukushima, A.; Suzuki, Y.; Ando, K., Nat. Mater. 2004, 3 (12), +868-871. +[5] +Jiang, W.; Chen, G.; Liu, K.; Zang, J.; te Velthuis, S. G. E.; Hoffmann, A., Phys. Rep. +2017, 704, 1-49. +[6] +Sinova, J.; Valenzuela, S. O.; Wunderlich, J.; Back, C. 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Lett. 2019, 122 (18), 187202. +[20] +Skoropata, E.; Mazza, A. R.; Herklotz, A.; Ok, J. M.; Eres, G.; Brahlek, M.; Charlton, +T. R.; Lee, H. N.; Ward, T. Z., Phys. Rev. B 2021, 103 (8), 085121. +[21] +Blundell, S. J.; Gester, M.; Bland, J. A. C.; Lauter, H. J.; Pasyuk, V. V.; Petrenko, A. V., +Phys. Rev. B 1995, 51 (14), 9395-9398. +[22] +Bjorck, M.; Andersson, G., J. Appl. Crystallogr. 2007, 40 (6), 1174-1178. +[23] +Lauter-Pasyuk, V.; Lauter, H. J.; Toperverg, B. P.; Romashev, L.; Ustinov, V., Phys. Rev. +Lett. 2002, 89 (16), 167203. +[24] +Zabel, H., Phys. B: Condens. Matter 1994, 198 (1), 156-162. +[25] +Jeong, S. G.; Min, T.; Woo, S.; Kim, J.; Zhang, Y.-Q.; Cho, S. W.; Son, J.; Kim, Y.-M.; +Han, J. H.; Park, S.; Jeong, H. Y.; Ohta, H.; Lee, S.; Noh, T. W.; Lee, J.; Choi, W. S., Phys. Rev. +Lett. 2020, 124 (2), 026401. +[26] +Jeong, S. G.; Kim, H.; Hong, S. J.; Suh, D.; Choi, W. S., ACS Appl. Nano Mater. 2021, +4 (2), 2160-2166. +[27] +Jeong, S. G.; Seo, A.; Choi, W. S., Adv. Sci. 2022, 9 (7), 2103403. +[28] +Cho, S. W.; Jeong, S. G.; Kwon, H. Y.; Song, S.; Han, S.; Han, J. H.; Park, S.; Choi, W. +S.; Lee, S.; Choi, J. W., Acta Mater. 2021, 216, 117153. +[29] +Lauter, V.; Ambaye, H.; Goyette, R.; Hal Lee, W.-T.; Parizzi, A., Physica B: Condensed +Matter 2009, 404 (17), 2543-2546. + + + +15 + +Atomic-scale precision epitaxy and microscopic observation let us customize the synthetic +magnetic order useful for spintronic applications. However, conventional approaches have +been limited to metallic heterostructures, which have Joule heating and/or electric breakdown. +Here, we report atomic-scale modulation of synthetic magnetic order across the insulating +spacer, observed by a polarized neutron reflectometer. This approach can yield novel +controllability of magnetic order across insulating spacers. + +Seung Gyo Jeong, Sehwan Song, Sungkyun Park, Valeria Lauter, and Woo Seok Choi* + +Atomic-scale Modulation of Synthetic Magnetic Order in Oxide Superlattices + +ToC figure + + + + + +FM + +16 + +Supporting Information + +Atomic-scale Modulation of Synthetic Magnetic Order in Oxide Superlattices + +Seung Gyo Jeong, Sehwan Song, Sungkyun Park, Valeria Lauter, and Woo Seok Choi* + + + +Figure S1. XRR fitting results of [6|y] superlattices with different y. The symbols (solid lines) +are the experimental data (fit) of the XRR data. + +Figure S2. Atomic force microscopy images of [6|y] superlattices with different y shows +typical step and terrace structures indicating atomically flat surfaces of the samples. + + +Figure S3. a) In-plane M (H) curves of [6|y] superlattices with different y at 5 K. b) Estimated +J of superlattices as a function of y. + + +XRR (arb. units) +[6|4] +0 +R +Fit +XRR (arb. units) +[6|6] +XRR (arb. units) +[6|8] +2 +3 +Qz (nm-1)[6|2] +[6]4] +[6]6] +800nma +0.6 +b +[6|4] +0.4 +[6|6] +0.03 +[6]8] +0.2 +J (erg/cm²) +/Ru) +[6]18] +0.02 +M (μB) +0.0 +-0.2 +0.01 +-0.4 +[6]y] +-0.6 +0.00 +-5.0 +-2.5 +0.0 +2.5 +5.0 +4 +6 +8 +18 +H (T) +y (u.c.) + +17 + + + + + +Figure S4. Extended PNR spectra with three different spin models (FM, sAFM, and spiral model, see Results and Discussion) at +5 K and 0.01 T of in-plane H-field near a) the half of the Bragg peaks and b) Bragg peaks of each superlattice. The red and blue +rectangles highlight distinctions between collinear FM and sAFM models and PNR spectra results, respectively. + +a +b +PNR (arb. units) +[6]4], 5 K, 0.01 +[6]4], 5 K, 0. +[6]4], 5 K, 0.01 T +[6]4], 5 K, 0.01 T +[6]4], 5 K, 0.01 T +[6]4] 5 K, 0.01 T +FM, R +SAFM, I +Spiral, R* +FM +sAFM, R* +Spiral, R* +FM, R +sAFM. +Spiral, +FM +sAFM, R +Spiral, R' +0.6 +0.9 +1.0 +0.6 +0.7 +0.8 +0.9 +1.0 +0.6 +0.7 +0.8 +0.9 +1.0 +.4 +1.8 +1.4 +1.5 +1.6 +1.7 +1.8 1.4 +1.5 +1.6 +1.7 +1.8 +PNR (arb. units) +PNR (arb. units) +[6]6], 5 K, 0.01 +[6|6], 5 K, 0.0 +[6]6], 5 K, 0.01 +[6]6], 5 K, 0.01 T +[6]6], 5 K, 0.01 T +[6]6], 5 K, 0.01 T +FM,R +sAFM, R +Spiral, R+ +R+ +FM, R* +sAFM, R* +Spiral, R* +sAFM, R +Spiral, R +FM, R +sAFM, R' +Spiral, R' +0.5 +0.6 +0.7 +0.8 +0.5 +0.6 +0.7 +0.8 +0.5 +0.6 +0.7 +0.8 +1.2 +1.3 +1.4 +1.5 +1.2 +1.3 +1.4 +1.5 +1.2 +1.3 +1.4 +1.5 +(arb. units) +PNR (arb. units) +[6]8], 5 K, 0.01 T +[6]8], 5 K, 0.01 T +[6]8], 5 K, 0.01 T +R +SAFM, R* +Spiral, R* +FM +sAFM, R +Spiral, R' +{6|8], 5 K, 0.01 T +[6]8], 5 K, 0.01 T +[6]8], 5 K, 0.01 T +PNR +一 +R +SAFM, R* +Spiral, R* +FM +SAFM, R' +Spiral, R' +0.4 +0.5 +0.6 +0.0 +0.4 +0.5 +0.6 +0.7 +0.4 +0.5 +0.6 +0.7 +1.0 +1.1 +1.2 +1.3 +1.0 +1.1 +1.2 +1.3 +1.0 +1.1 +1.2 +1.3 +Q, (nm-1 +Q, (nm-1) +Q, (nm- +Q, (nm-1) +Q, (nm-1) +Qz (nm-1) + +18 + + +Figure S5. Comparison between the measured M and the simulated M values at 5 K for layer- +repetition- (z-) dependent [6|4]z heterostructures. + + + +Table S1. Summary of fitting parameters for XRR and PNR analyses and comparison +measured thickness of SRO/STO superlattices + + + + + + + + + + + + + + + +Samples +XRR +PNR +Roughness (nm) +Density (g/cm3) +Roughness +(nm) +Density (f.u./Å3) +SRO +STO +SRO +STO +SRO +STO +[6|4] +0.14 +0.23 +5.9 +4.81 +0.2 +0.0153 +0.017 +[6|6] +0.38 +0.19 +6 +4.12 +Samples +Target thickness (nm) +Measured thickness (nm) +XRR +PNR +STEM +[6|2] +31.366 +31.256 +- +32.590 +[6|4] +39.176 +39.306 +39.134 +- +[6|6] +46.986 +46.756 +47.430 +- +[6|8] +54.796 +54.356 +53.930 +54.930 + +0.3 +Experiment +Simulation +Ms5 k (μg/Ru) +0.2 +0.1 +0.0 +2 +3 +5 +Repetition (z) \ No newline at end of file diff --git a/rdAzT4oBgHgl3EQfcfxL/content/tmp_files/load_file.txt b/rdAzT4oBgHgl3EQfcfxL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f3f6cd8cd23ad217018fe309ea3ea11d6c5c10cb --- /dev/null +++ b/rdAzT4oBgHgl3EQfcfxL/content/tmp_files/load_file.txt @@ -0,0 +1,1022 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf,len=1021 +page_content='1 Atomic-scale Modulation of Synthetic Magnetic Order in Oxide Superlattices Seung Gyo Jeong, Sehwan Song, Sungkyun Park, Valeria Lauter, and Woo Seok Choi* S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Jeong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Choi Department of Physics, Sungkyunkwan University, Suwon 16419, Korea E-mail: choiws@skku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='edu S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Park Department of Physics, Pusan National University, Busan 46241, Korea V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Lauter Neutron Scattering Division, Neutron Sciences Directorate, Oak Ridge National Laboratory, Oak Ridge 37831, USA Keywords: atomic-scale modulation, synthetic magnetic order, tunable magnetic noncollinearity, magnetic oxide superlattices, polarized neutron reflectometry Atomic-scale precision control of magnetic interactions facilitates a synthetic spin order useful for spintronics, including advanced memory and quantum logic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Conventional modulation of synthetic spin order has been limited to metallic heterostructures that exploit RKKY interaction through a nonmagnetic metallic spacer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' however, they face problems arising from Joule heating and/or electric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The practical realization and observation of a synthetic spin order across a nonmagnetic insulating spacer would lead to the development of spin-related devices with a completely different concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Herein, we report the atomic-scale modulation of the synthetic spiral spin order in oxide superlattices composed of ferromagnetic metal and nonmagnetic insulator layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The atomically controlled superlattice exhibit an oscillatory magnetic behavior, representing the existence of a spiral spin structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Depth-sensitive polarized neutron reflectometry evidences modulated spiral spin structures as a function of the nonmagnetic insulator layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Atomic-scale customization of the spin state could lead the field one step further to actual spintronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Introduction Synthetic spin order in magnetic heterostructures promotes novel spintronic functionalities[1] including colossal magnetoresistance,[2, 3] tunneling magnetoresistance,[4] topological Hall effect,[5] spin Hall effect,[6, 7] spin-wave propagation,[8] and terahertz spin-transfer torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [9] In typical ferromagnetic (FM)/nonmagnetic-metal (NM-M)/FM heterostructures, the relative spin orientation between two FM layers can be modulated by the thickness of the NM-M layer, thereby realizing a synthetic magnetic order, which is useful for designing magnetic storage and logic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [1] This is generally understood by the Ruderman-Kittel-Kasuya-Yosida (RKKY) interlayer exchange interaction between the FM layers, which is mediated by the conduction electrons in the NM-M layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' In this scheme, the interaction strength oscillates as a function of the NM-M layer thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [1] In contrast, FM/nonmagnetic-insulator (NM-I)/FM heterostructures foster synthetic spin order, with unfamiliar exchange mechanisms other than the RKKY interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10, 11] Recently, chiral phonon has been suggested to carry the spin information across the NM-I layer via strong spin-phonon and spin-orbit coupling,[11] similarly acting as the conduction electron across the NM-M layer for the RKKY interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10, 11] It has been shown that the chiral phonons could mediate the interlayer exchange interaction leading to an oscillatory magnetization as a function the NM-I layer thickness, evidenced by confocal Raman spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10] If the synthetic spin order is atomically controllable in NM-I based heterostructures, the inherent limitations such as Joule heating and electric breakdown in NM-M based heterostructures can be resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Polarized neutron reflectometry (PNR) is a favorable technique for investigating the atomic layer-resolved distribution of spins, particularly when combined with atomic-scale epitaxy of synthetic magnetic heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [12-15] Frist, the in-plane spin orientation in the magnetic layers can be obtained by comparing the non-spin-flip and spin-flip components of neutrons in the PNR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Second, the depth profile of the spin orientation can be obtained by assessing the out-of-plane component of the wavevector transfer (Qz) in the specular PNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Third, application of the PNR to magnetic superlattices is further advantageous because the superlattice Bragg peak results in an enhanced reflectivity signal for the analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' In this study, we present an atomic-scale thickness control of the synthetic spin structures in oxide superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We epitaxially deposited SrRuO3 (SRO, FM layer)/SrTiO3 (STO, NM-I layer) superlattices on (001)-oriented single crystal STO substrates using pulsed laser epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 3 As schematically shown in Figure 1a,b, the spiral spin structure of six-unit-cell-thick (1 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 nm) FM SRO layers was modulated by the y-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='-thick NM-I STO layers with ten repetitions, within the [6|y] superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Consequently, y-dependent oscillatory net magnetization was obtained consistently from both the magnetization and the PNR measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We note that the magnetic easy axis of SRO thin films usually point to the out- of-plane direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' However, the y-dependent oscillatory behavior is only observed for the in- plane magnetization measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Since PNR also identifies the in-plane magnetization of the thin films, it is an ideal tool to characterize the important magnetic features of the superlattices in microscopic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Results and Discussion Figure 1c,d show X-ray reflectivity (XRR) and diffraction (XRD) results, respectively, validating the atomically defined periodicities of the SRO/STO superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The XRR curves exhibit distinct Bragg peaks (SL+n) and Kiessig fringes corresponding to the total thickness of the superlattice thin film, thereby indicating a well-defined periodic supercell structure with atomically sharp interfaces (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure S1 shows the XRR fitting results of the [6|4], [6|6], and [6|8] superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Small decay slopes of the XRR indicate that the surface roughness of the superlattices is < 1 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=', which is consistent with the topographic images obtained by atomic force microscopy (Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' XRD θ-2θ scans show coherent diffraction peaks (SL±n) of the superlattices on the (001)-oriented single crystal STO substrates (substrate diffraction peaks marked by asterisks) (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' With increasing y, the separation between the superlattice peaks decreases systematically, which indicates the atomically controlled periodicities (𝛬SL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' When thickness of the x = 6 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' of SRO layer is assumed to be fixed at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='358 nm (obtained from epitaxially strained single SRO thin films on STO substrates), the estimated thicknesses of the y u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' of STO layer are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='768, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='573, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='318, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='078, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='957 nm for the [6|2], [6|4], [6|6], [6|8], and [6|18] superlattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' These values are obtained from the Bragg’s law, 𝛬SL = 2π(Qn – Qn – 1)–1, where n and Qn denote the superlattice peak order and the Qz position of the nth-order superlattice peak, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The deviation between the target and measured thicknesses is smaller than half a u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 nm), thus manifesting structurally well-controlled superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Finally, Figure 1e representatively shows the X-ray reciprocal space mapping of [6|8] superlattice around the (103) Bragg diffraction peak of the STO substrate, confirming a fully strained state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 4 The in-plane magnetization of the [6|y] superlattices shows an unexpected oscillation as a function of y at low-temperature (-T) and low-magnetic- (H-) fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Field-cooled M (T) curves of the [6|y] superlattices were measured at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T of H-field along the in-plane direction (Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' In addition to a robust FM transition at approximately 130 K (consistent with 6 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' SRO single layer on the STO substrate[16, 17]), the in-plane M shows a peak as T is further reduced, indicating that the FM order is disturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The M (H) curves at low T exhibit a double hysteresis with a large coercive field (Hc) of approximately 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='8 T (see the inset of Figure 2a), which supports the disturbed magnetic order in the ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Moreover, the M (H) curve of y = 18 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' superlattice shows a typical single hysteresis loop with a small value of Hc, but with a saturation M (Ms) of ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='3 μB/Ru similar to those of y = 6 and 8 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' superlattices, which indicates the suppression of the interlayer exchange interaction at a sufficiently thick NM-I layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We note that the enhancement of Ms for y = 4 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' superlattice might originate from the structural modulation (orthorhombic to tetragonal) in the SRO layers with decreasing y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [16] The oscillation of the in-plane M at 5 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T is shown as a function of y in Figure 2f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10] The magnetization results suggest the existence of an interlayer exchange interaction across the NM-I STO layer and the possible unconventional synthetic magnetic order in the SRO/STO superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The interlayer exchange interaction strength is estimated as J = tFMHcMs, where tFM is the thickness of the FM SRO layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure S3 shows the y-dependent oscillatory behavior of J, which depends on both Hc and Ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' This confirms the unconventional character of the interlayer exchange interaction between FM SRO layer across the NM-I STO layer, which is supposed to originate from the chiral phonon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10,11] The J values in SRO/STO superlattices are approximately two orders of magnitude less than those for the RKKY interaction at the same thickness of the NM-M layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [18] Although conventional analyses of the M (H) curve would be useful to examine the macroscopic strength of magnetic interaction, we note that the observed magnetic order is rather fragile and is easily destroyed even in a moderate H-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Therefore, we will focus on the low H-field region of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' To understand the unexpected y-dependent oscillatory magnetization, we examine a simple model, wherein spins in SRO layers have a rotation angle ϕ with respect to the spins in the adjacent SRO layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Let us first assume that each SRO layer has the same uniform in-plane Mi value (i is the layer index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Although it has been reported that spin ordering may differ depending on the position away from the interface within magnetic heterostructures,[19,20] the modulation is generally small in atomically defined heterostructures with symmetric 5 interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Second, we assume that ϕ depends linearly on y, that is, with an increase in the NM-I layer thickness, the rotation of Mi increases, as in the case of the RKKY with the NM- M layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure 2b–d shows a schematic representation of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We estimate the sum of the projections of Mi (y, ϕ) along a certain in-plane direction of the H-field for each SRO layer to simulate the results of y-dependent M because the magnetization measurement gives only the average scalar M value of the entire superlattice along the H-field direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The average M value is determined by the relation M = ∑ Mi (ϕ) i = ∑ [Ma + Mb i cos((i − 1)ϕ)], where Ma and Mb are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Next, we compare the result with that obtained from the experiment and calculate the sum of the squared errors (SSE) as a function of ϕ, defined as [(Simulated M) – (Measured M)]2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 2(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' In particular, for y = 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='8 nm), ϕ = ~80 and ~100º results in the lowest SSE values within the spiral spin models with Ma = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='070 μB/Ru and Mb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='629 μB/Ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure 2(f) shows that the spiral spin models with ϕ = ~200º, 300º, and 400º (40º) for the [6|4], [6|6], and [6|8] superlattices consistently describe the experimental y- dependent oscillatory behavior of M at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Note that the underestimation of M for the [6|4] superlattice might be due to the absence of the decaying term with increasing y in our model, which would further complicate the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' PNR is employed to clarify the complex synthetic spin structure suggested by the magnetization measurement discussed earlier (Figure 3 and Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure 3a schematically shows that a collimated polychromatic neutron beam is incident on the film surface at a grazing angle (α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The PNR signal was measured as a function of Qz = 4πsin(α)/λ along the out-of-plane direction (z-direction in Figure 3a), where λ is the neutron wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We assume that a homogeneous in-plane M vector (M ⃗⃗⃗ ) is rotated by angle ϕ within the xy- plane, defining the in-plane Mx and My components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Then the PNR signal would include both non-spin-flip (R++ and R––) and spin-flip (R+– and R–+) contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='21 R++ and R— are defined as 1/4|(r+ + r–) + (r+ – r–) cosϕ|2 and 1/4|(r+ + r–) – (r+ – r–) cosϕ|2, respectively, and R+– and R–+ are described as 1/4|r+ – r–|2 sin2ϕ, respectively, where r± are the complex reflection amplitudes for the up and down spins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [13] Because the simulated experimental signal with spin-flip polarization (R+– and R–+) is rather small for our samples, and hence, required a long measuring time to obtain high statistics, we did not employ spin-flip polarization analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Therefore, the spin-flip reflection was taken into account only in the data analyses as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Our results show spin-up and spin-down polarizations of the PNR spectra, R+ = R++ + R+– and R– = R–– + R–+, which describe depth-sensitive spin-vector rotation in the SRO/STO superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 6 To minimize the number of parameters in PNR fitting, we confirm the prerequisite structural parameters and Ms values of the SRO/STO superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' First, the structural parameters, including the interface roughness, thickness, and density of each layer, are obtained from non- polarized neutron reflectivity data at 300 K when the sample is nonmagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The top panels of Figure 3c,e,g show the unpolarized neutron reflectivity and fitting results for the y = 4, 6, and 8 superlattices, respectively, which are consistent with the XRR results shown in Figure 1c and Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The structural parameters of PNR analyses at 300 K have been confirmed with those of XRR fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We summarize the structural fitting parameters of the XRR and PNR results with the measured thickness consistently obtained by using XRR, PNR, and scanning transmission electron microscopy[10] in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' These results manifest the atomically well-controlled SRO/STO superlattices with the minimized interdiffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Second, to confirm the Ms values of the superlattices, we measured the PNR spectra at 85 K with a 1 T (> Hc at 85 K) of in-plane H-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The M (H) curves at 85 K exhibit a single hysteresis loop as for a typical FM, as shown in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' At this temperature with 1 T of H-field, the net M is almost saturated (net M ~ Ms), which indicates that the in-plane rotation is relatively suppressed (ϕ ~ 0º).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The bottom panels of Figure 3c,e,g show the PNR spectra at 85 K in a 1 T field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The insets in Figure 3c,e,g highlight the differences in PNR intensity between R+ and R– near the Bragg peaks of each superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' This clear separation is consistently observed in both the experimental result and fit, thereby indicating a robust FM order of the SRO layers at a high H-field at 85 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure 3d,f,h show the nuclear and magnetic scattering length densities (SLD) fitted using GenX for [6|4], [6|6], and [6|8] superlattices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='22 They show an atomically well-defined periodic structure and ferromagnetically aligned Mi vectors in each SRO layer at 85 K with a 1 T field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The obtained Ms values are estimated to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 μB/Ru for [6|4] superlattice and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='3 μB/Ru for [6|6] and [6|8] superlattices, respectively, which are highly consistent with the M (H) curves in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' In the ground state (experiments at 5 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T of H-field), the PNR spectra reveal a modulated synthetic spiral spin structure suggested by magnetization measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure 4a shows the simulation result of a synthetic spiral spin structure with the best fit, which is consistent with the experimental PNR result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' For a single magnetic film, the PNR spectrum at small Qz is influenced by the direct beam, whereas with increasing Qz, the reflectivity decays exponentially such that the experimental data may be affected by the background signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [12] In contrast, as briefly discussed previously, superlattice Bragg peaks provide a better signal-to- 7 noise ratio even at finite Qz values due to the coherent periodic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [23] Therefore, we primarily compare the PNR data and simulation results near the Qz values associated with the superlattice Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure S4 summarizes PNR simulations of the SRO/STO superlattices for a typical FM model (left panels), a synthetic collinear antiferromagnetic (sAFM) model with ϕ = 180º (middle panels), and a synthetic spiral spin structure (right panels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The simulated spectra of collinear FM and sAFM models have two distinctions from the experimental PNR spectra: (1) The collinear sAFM structure leads to doubling of the magnetic unit cell of the superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' This doubling causes a significant discrepancy between R+ and R– at half Qz of the superlattice Bragg peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [24] As highlighted by the red rectangle in Figure S4a, this model produces spectra that are inconsistent with the experimental PNR spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' (2) The collinear FM model results in a strong intensity contrast between the R+ and R– in the superlattice Bragg peak due to the difference in the scattering of spin up and spin down neutrons of periodic superlattice structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [24] Note the similarity to the PNR spectra at 85 K with robust FM ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' As highlighted by the blue rectangle in Figure S4b, the experimental PNR spectra exhibit negligible separation in the superlattice Bragg peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Hence, the PNR experimental results show that a collinear spin configuration is unlikely in SRO/STO superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Instead, the spiral spin structure model best describes the experimental PNR spectra among the considered simple models, confirming the in-plane spin rotation in the SRO/STO superlattice as a function of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Conclusion In summary, we presented the modulation of synthetic magnetic order in FM/NM-I/FM superlattice via atomic-scale precision thickness control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' To customize the in-plane rotation of spin vectors within ferromagnetic SRO layers, we atomically controlled the thickness y of the NM-I (STO) layer within the SRO/STO superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The oscillatory in-plane magnetic behavior as a function of y, determined by magnetization measurements, indicates the presence of the synthetic magnetic order in the superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The PNR result manifests the depth profile of spin vectors within SRO layers with varying rotation angle ϕ and its controllability via the precise control of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Both magnetization and PNR results consistently confirm that the atomically controlled thickness of the NM-I layers effectively changes the rotation angle of the spiral spin structures in the FM layers within the FM/NM-I superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Our approach suggests an atomic-scale control knob for modulating synthetic spin spiral structures for future spintronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 8 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Atomically controlled SRO/STO superlattices for controlling synthetic magnetic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a) and b) Schematic illustrations of modulation of synthetic magnetic order in FM (red)/NM-I (gray)/FM (red) heterostructures via atomic-scale precision thickness control of NM-I layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' c) X-ray reflectivity and d) θ-2θ scan results of the atomically well-defined superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' e) X-ray reciprocal space mapping of [6|8] superlattice representatively shows the fully strained state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a C STO(103) [6|18] 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='8 Intensitiy (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) units) [6]8] (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' wu) [6]6] SRO Intensitiy 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 b [6|4] SL+1 [6|2] SL SL SI +2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 2 3 10 20 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='8 @ low T, low H field Q, (nm 1) Q, (nm 1) Qx (nm 1) 9 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Synthetic in-plane magnetic behavior of SRO/STO superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a) Field-cooled M (T) curves of superlattices with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T of in-plane H-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The inset shows the M (H) curves of superlattices at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Schematic representation of in-plane Mi vector of the SRO layer for b) [6|4], c) [6|6], and d) [6|8] superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' External H-field applied along the y-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The red arrows in the bottom panels denote the summation of Mi vectors (Msum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' e) Sum of square error values as a function of ϕ for y = 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' superlattice, determined by [(Simulated M) – (Measured M)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' f) Summary of experimentally measured M and the simulated M values as a function of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The error bars indicate the experimental deviation from two different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 e 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 y = 2 u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' (~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='8 nm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='3 Experiment Error Simulation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 M (ug/Ru) M (ug/Ru) M Squre 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 @5K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 80°100° -5 0 5 (L) H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='1 [6|2] [6|8] [6|18] wns [6|4] [9]9] S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0 100 200 300 0 60 120 180 2 4 6 8 T (K) Φ(°) y (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=') [6|4] Mo [6|6] d [6]8] Mg M4 M3,Mg M2, Mg M- Mg Mg M6 Ma, M4, M1, M1 M10 M10 M10 M M5 M3 M2 M M5 M6 M M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' M3 10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' PNR results of the FM state of SRO/STO superlattices at 85 K and 1 T of H-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a) Schematic representation of a PNR configuration with oxide thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' b) M (H) curves of [6|y] superlattices with different y at 85 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' PNR spectra at 300 K and 85 K with 1 T H-field for c) [6|4], e) [6|6], and g) [6|8] superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The insets show the extended PNR spectra near the Bragg peaks of each superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Nuclear and magnetic SLD of d) [6|4], f) [6|6], and h) [6|8] superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The vertical dashed lines in SLD are eye guides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 a C d (wu) Polarized Neutron [6]4], 300 K Nuclear Magnetic [6]4] R Reflectivity (PNR) 50 STO Mx 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 [6]6] PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) Fit substrate : SRO My [6]8] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 40 [6]4], 85 K, 1 T My 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 R* 30 Fit Distance from s R- -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 Fit 20 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='8 in-plane H Qz (nm-1) -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 @ 85 K 10 R+ R -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 L 0 -5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 Spin-polarized neutrons (I) H Qz (nm-1) SLD (10-4nm-2) e (wu) Nuclear Magnetic g h substrate (nm) [6|6] [6|8] Nuclear Magnetic 50 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) substrate ( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) 40 40 PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [6]6] PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [6]8] PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) 30 30 from from s PNR 20 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 Qz (nm-1) Distance Q, (nm-1) istance 10 10 D 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 Qz (nm-1) SLD (10-4nm-2) Qz (nm-1) SLD (10-4nm-2) 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' y-dependent in-plane rotation of synthetic spiral spin structures in SRO/STO superlattices at 5 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T of H-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a) PNR spectra at 5 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T of H-field for [6|4], [6|6], and [6|8] superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' b) Magnetic SLD and c) schematic representation of the synthetic spin order of [6|4], [6|6], and [6|8] superlattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The result evidences that the atomic-scale modulation of the synthetic spiral spin structure in SRO/STO superlattices as a function of y consistent with magnetization measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Distance from substrate (nm) a Magnetic c [6]4] PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) [6|41,5K,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01T 50 M, R+ Fit M 40 R Fit 30 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 Qz (nm 1) SLD (10 4nm 2) Distance from substrate (nm) [6]6] Magnetic [6]6] PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='units) 50 40 30 1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='++++ 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 Qz (nm 1) SLD (10 4nm 2) : from substrate (nm) Magnetic [6]8] [6]8] PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='units) 50 40 30 20 Distance f 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 Q, (nm 1) SLD (10 4nm 2) 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Methods Atomic-scale epitaxy: To validate the atomically designed FM/NM-I/FM heterostructures, we deposited epitaxial SRO/STO superlattices on (001)-oriented single crystal STO substrates by using pulsed laser epitaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10, 17, 25–28] We fixed the number of laser pulses to grow the SRO layers and systematically changed the number of pulses to modulate y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We controlled the six u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' of the SRO layers and y u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' of the STO layers with ten repetitions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=', the [6|y] superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Before deposition, we treated the surfaces of the STO substrates using buffered HF and annealed them at 1000 °C for 6 h under atmospheric conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We used stoichiometric ceramic SRO and STO targets and an excimer (KrF) laser (248 nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' IPEX 868, LightMachinery) with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 J/cm2 of fluence and a repetition rate of 5 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The temperature and oxygen partial pressure employed for the stoichiometric growth of both SRO and STO layers were 750 °C and 100 mTorr, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=" To characterize the atomically controlled periodicity of superlattice, we performed XRR and XRD θ-2θ measurements using a high- resolution PANalytical X'Pert and Rigaku SmartLab X-ray diffractometer with Cu K-α1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Magnetization measurement: The field-cooled M (T) curves of the SRO/STO superlattice along the in-plane direction were measured using a magnetic property measurement system (MPMS, Quantum Design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The in-plane M (H) curves of the SRO/STO superlattice were measured at 5 and 85 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We did not observe any exchange bias effect in M (H) curves of the SRO/STO superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' When we estimated the orientation of the spin vectors to describe the oscillatory M behavior, we included the layer-repetition-dependent M of superlattices as well (Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10] Polarized neutron reflectometry: The PNR experiments were performed on the time-of-flight Magnetism Reflectometer (BL-4A) at the Spallation Neutron Source at Oak Ridge National Laboratory (SNS, ORNL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [29] We used the closed-cycle refrigerator systems with a Bruker electromagnet to induce an in-plane H-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We used polarized neutron beam with polarization efficiencies from 99 to 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5% and λ within a band of 2 to 8 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The PNR spectra with nuclear and spin structures were fitted using GenX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [22] Supporting Information Supporting Information is available from the Wiley Online Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 13 Acknowledgements This research used resources at the Spallation Neutron Source, a Department of Energy (DOE) Office of Science User Facility operated by the Oak Ridge National Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We gratefully acknowledge Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Haile Ambaye (Spallation Neutron Source) for technical support with PNR experiments and data processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' We also thank Core Research Facilities, Pusan National University for MPMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' This research was supported by the Basic Science Research Programs through the National Research Foundation of Korea (NRF- 2020K1A3A7A09077715, 2021R1A2C2011340, and 2022R1C1C2006723).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' References [1] Duine, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' ;' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Valenzuela, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Wunderlich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Back, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [7] Safeer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Ingla-Aynés, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Herling, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Garcia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Vila, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Ontoso, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Calvo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Roche, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Hueso, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Casanova, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=', Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 2019, 19 (2), 1074-1082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [8] Ishibashi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Shiota, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Funada, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Moriyama, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Ono, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=', Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 2020, 6 (17), eaaz6931.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [9] Zhong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Qiao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Liang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Zhao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Kang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Magn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 2020, 497, 166070.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [10] Jeong, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Seo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Park, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Lauter, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Egami, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Han, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Müller, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Schröder, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Hövel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Karczewski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Wiater, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Wojtowicz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Kusrayev, Y.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Giebultowicz, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' B: Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Matter 2003, 335 (1), 44-49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 14 [13] Lauter-Pasyuk, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Lim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Park, S.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Charlton, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Ward, T.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Ambaye, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Goyette, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Hal Lee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Parizzi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=', Physica B: Condensed Matter 2009, 404 (17), 2543-2546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' 15 Atomic-scale precision epitaxy and microscopic observation let us customize the synthetic magnetic order useful for spintronic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' However, conventional approaches have been limited to metallic heterostructures, which have Joule heating and/or electric breakdown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Here, we report atomic-scale modulation of synthetic magnetic order across the insulating spacer, observed by a polarized neutron reflectometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' This approach can yield novel controllability of magnetic order across insulating spacers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Seung Gyo Jeong, Sehwan Song, Sungkyun Park, Valeria Lauter, and Woo Seok Choi* Atomic-scale Modulation of Synthetic Magnetic Order in Oxide Superlattices ToC figure FM 16 Supporting Information Atomic-scale Modulation of Synthetic Magnetic Order in Oxide Superlattices Seung Gyo Jeong, Sehwan Song, Sungkyun Park, Valeria Lauter, and Woo Seok Choi* Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' XRR fitting results of [6|y] superlattices with different y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The symbols (solid lines) are the experimental data (fit) of the XRR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Atomic force microscopy images of [6|y] superlattices with different y shows typical step and terrace structures indicating atomically flat surfaces of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a) In-plane M (H) curves of [6|y] superlattices with different y at 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' b) Estimated J of superlattices as a function of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' XRR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) [6|4] 0 R Fit XRR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) [6|6] XRR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) [6|8] 2 3 Qz (nm 1)[6|2] [6]4] [6]6] 800nma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 b [6|4] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 [6|6] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='03 [6]8] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 J (erg/cm²) /Ru) [6]18] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='02 M (μB) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0 4 6 8 18 H (T) y (u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=') 17 Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Extended PNR spectra with three different spin models (FM, sAFM, and spiral model, see Results and Discussion) at 5 K and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T of in-plane H-field near a) the half of the Bragg peaks and b) Bragg peaks of each superlattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' The red and blue rectangles highlight distinctions between collinear FM and sAFM models and PNR spectra results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' a b PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) [6]4], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 [6]4], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' [6]4], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T [6]4], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T [6]4], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T [6]4] 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T FM, R SAFM, I Spiral, R* FM sAFM, R* Spiral, R* FM, R sAFM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=" Spiral, FM sAFM, R Spiral, R' 0." metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='8 PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) PNR (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' units) [6]6], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 [6|6], 5 K, 0.' 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+page_content="01 T R SAFM, R* Spiral, R* FM sAFM, R Spiral, R' {6|8], 5 K, 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T [6]8], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='01 T [6]8], 5 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content="01 T PNR 一 R SAFM, R* Spiral, R* FM SAFM, R' Spiral, R' 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='6 0.' 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18 Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Comparison between the measured M and the simulated M values at 5 K for layer- repetition- (z-) dependent [6|4]z heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content=' Summary of fitting parameters for XRR and PNR analyses and comparison measured thickness of SRO/STO superlattices Samples XRR PNR Roughness (nm) Density (g/cm3) Roughness (nm) Density (f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='/Å3) SRO STO SRO STO SRO STO [6|4] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='0153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='017 [6|6] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='19 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='12 Samples Target thickness (nm) Measured thickness (nm) XRR PNR STEM [6|2] 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='366 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='256 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='590 [6|4] 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='176 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='306 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='134 [6|6] 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='986 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='756 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='430 [6|8] 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='796 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='356 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='930 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='930 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='3 Experiment Simulation Ms5 k (μg/Ru) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rdAzT4oBgHgl3EQfcfxL/content/2301.01403v1.pdf'} 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Shamasundar, Negar Daryanavardan, and Aria Nosratinia +Abstract +Reconfigurable intelligent surfaces (RIS) are passive controllable arrays of small reflectors that direct elec- +tromagnetic energy towards or away from the target nodes, thereby allowing better management of signals and +interference in a wireless network. The RIS has the potential for significantly improving the performance of wireless +networks. Unfortunately, RIS also multiplies the number of Channel State Information (CSI) coefficients between +the transmitter and receiver, which magnifies the challenges in estimating and communicating the channel state +information. Furthermore, the simplicity and cost-effectiveness of the passive RIS also implies that the incoming +links are not locally estimated at the RIS, and fresh pilots are not inserted into outgoing RIS links. This introduces +new challenges for training and estimation of channel state information. The rapid growth of the literature on CSI +acquisition in RIS-aided systems has been accompanied by variations in the underlying assumptions, models, and +notation, which can obscure the similarities and differences of various techniques, and their relative merits. This +paper presents a comprehensive exposition of principles and approaches in RIS channel estimation. The basic ideas +underlying each class of techniques are reduced to their simplest form under a unified model and notation, and various +approaches within each class are discussed. Several open problems in this area are identified and highlighted. +I. INTRODUCTION +The fifth generation (5G) wireless systems are being successfully deployed across the globe, achieving high data +rates using large antenna arrays [1], [2]. 6G systems will aim for even higher data rates, lower latency, and better +reliability, at low cost/complexity and high energy efficiency [3], [4]. This requires innovations in the physical layer +technologies. The continuing migration to the millimeter wave (mmWave) spectrum, while improving data rates, +has challenges including channel blockage and intermittent availability. Increasing the network density can solve +some of these problems, but it injects more power into the network, increases the interference levels, escalates cost +of deployment and operation, and raises scalability concerns. +Reconfigurable intelligent surfaces (RIS) are passive, controllable arrays of small reflectors that direct radio +waves toward or away from a target node, enabling better management of signals and interference in wireless +networks [5]–[9]. They are often interpreted as a mechanism that achieves software-defined control of the wireless +This work was supported in part by the National Science Foundation under Grants 1956213 and 2148211. +January 13, 2023 +DRAFT +arXiv:2301.05026v1 [cs.IT] 12 Jan 2023 + +2 +Controller +0 +2π +RIS +RIS Control + Link +Transmitter +Receiver +Fig. 1. Schematic representation of reconfigurable intelligent surface aided communication. +propagation environment (Fig. 1). This approach has the potential to address several of the challenges mentioned +above. Judiciously altering the channel characteristics in real-time as a part of the system operation, to achieve +favorable propagation environment, is an attractive idea to handle challenging channel conditions. Unlike a relay, +RIS does not inject more transmit power into the network, and its operation is independent of the details of PHY +signaling other than operating frequency and bandwidth. The combination of lower power consumption and simpler +construction makes it cheaper to build and operate, thus helping scalability. +RIS-induced channel coefficients must be estimated at the receiver for coherent communication, and shared +with the transmitter (and RIS) for beamforming. The RIS channel estimation problem is distinct from the MIMO +(multiple-input multiple-output) case because: (1) RIS is a two-hop channel observable only end-to-end, due to +passivity of RIS, (2) the number of channel gain coefficients is multiplied by the RIS size, making for a larger +vector to be estimated, and (3) in RIS channel estimation, training occurs through pilots plus passive RIS training +states, and the latter is without a direct counterpart in traditional channel training and estimation. +A. Contributions and Distinctions of the Present Work +The present work provides a thorough exposition of the ideas underlying the rapidly expanding literature on +RIS channel estimation, in a way that is both comprehensive and accessible to a wider audience. An up-to-date +discussion of the available estimation techniques and related issues, ranging from classical least squares to the most +recent artificial intelligence and machine learning (AI/ML)-based methods, is provided. +One of the contributions of the present work is to bring a unified notation and system model to the mathematical +expression of various RIS channel estimation problems and algorithms. This goes beyond the enumeration of +works in a conventional survey, and is greatly helpful in the interpretation and comparison of results in a rapidly +expanding literature. Among other features, this work illuminates the commonality and differences between RIS +channel estimation results, thus exposing synergies and facilitating the generation of new ideas. +January 13, 2023 +DRAFT + +3 +In addition to the exposition of various estimation approaches, this paper also explores critical assumptions, +explicit or implicit in RIS channel estimation, that are in need of further investigation and validation, e.g., channel +reciprocity or perfect deactivation of RIS elements. Other important issues, such as the near field effect in very +large RIS, or the dependence of reflection gains on phases, are also discussed. +To put this paper in context and highlight its distinctions, we briefly review related works. In a concise letter, +Wei et al. [10] explored sparsity, user correlation, and time scales for RIS channel estimation. Swindlehurst et +al. [10] consider the identifiability of the models as a function of the pilots and RIS training states, and further +consider special cases such as single-input single-output (SISO) and MIMO, availability or unavailability of a direct +link, and narrowband vs. wideband. Noh et al. [11] concentrates on characteristics of RIS channels in terahertz +(THz) and mmWave channels. Zheng et al. [12] provides a survey of RIS channel estimation that enumerates the +problems and outcomes, but does not dwell on methodology or characterization of the methods. In comparison, the +present work provides a simple yet sufficiently detailed exposition of ideas for a wider audience. Compared with +the works mentioned above, the present work also addresses new dimensions, including: the impact of RIS channel +estimation overhead on spectral efficiency, and the implications of optimizing spectral efficiency (through channel +estimation constraints) on the size of RIS array. The present work also includes coverage of machine learning +methods for RIS channel estimation. Also, as mentioned earlier, underlying assumptions with potential impact on +practical implementations are also examined in the present work. +This paper also presents several new interpretations and connections that have been unavailable in the literature +thus far. Among them, (a) Section VII-A provides an elegant explanation for available savings in multiuser RIS +systems, leading to suggestions for future work. (b) Section VII-C discusses the differences and similarities of +techniques that infrequently update the RIS coefficients, compared with traditional MIMO statistical CSI methods, +opening venues for future work, and (c) Section VII-D clarifies for the first time that the so-called “codebook +methods” for RIS actually have deep connections with opportunistic communication. +The organization of this paper is as follows. Section II presents RIS system and channel models. Section III +formulates RIS channel estimation problem and presents discussions on pilots, training states, and linear estimation +methods. Section IV presents the effect of training overhead and accuracy on the spectral efficiency of RIS, and dis- +cusses its impact on the choice of RIS dimensionality. Section V presents the formulation of RIS channel estimation +as a sparse recovery problem when operating in high frequency (mmWave/THz) channels, and discusses the solutions +proposed in the literature. Section VI presents estimation techniques for RIS-aided orthogonal frequency-division +multiplexing (OFDM) in wideband channels. Section VII presents various approaches for reducing estimation +overhead. Section VIII presents estimation approaches based on machine learning. Section IX outlines practical +issues of contemporary interest and investigation in RIS channel modeling. +II. SYSTEM AND CHANNEL MODELS +For easy reference, the mathematical terms and the key variables used in this paper are summarized in Table I. +January 13, 2023 +DRAFT + +4 +⊛ +circular convolution +⊗ +Kronecker product +⊙ +Hadamard product (element-wise product of two matrices) +⋄ +Khatri-Rao product (column-wise Kronecker product) +1 +vector of all-ones +T +matrix transpose +∗ +complex conjugate +H +conjugate transpose (Hermitian) +† +psuedo-inverse +diag(·) +diagonal matrix built from the argument +vec(·) +concatenation of columns of an M × N matrix into a vector of size MN +|| · || +vector norm +E +Expected value +Φ +RIS reflection coefficient matrix +H, G +incoming & outgoing RIS channel +Hc +end-to-end RIS-induced channel +Hd +direct (not through RIS) channel +Z +Training matrix, includes pilots & RIS training states +TABLE I: Nomenclature +A. Narrowband/Frequency-Flat Channels +Consider a narrowband point-to-point communication between a transmitter with Mt antennas and a receiver with +Mr antennas, assisted by an RIS with N elements. Let G ∈ CN×Mt denote the channel between the transmitter and +RIS, H ∈ CMr×N the channel between the RIS and the receiver, and Hd ∈ CMr×Mt the direct channel between +the transmitter and the receiver. The RIS reflection is represented with a matrix Φ = diag(ψ1, ψ2, . . . , ψN) whose +passive elements ψi = βiejφi have amplitude βi ∈ [0, 1]. Then, the received signal is given by +y = √ρ(Hd + HΦG)x + w, +(1) +where the transmit signal x ∈ CMt×1 satisfies E∥x∥2 = 1, the transmit power is ρ, and the receiver Gaussian noise +is w ∈ CMr×1. +In the microwave L and S bands, and parts of the C band,1 a suitable model of rich scattering involves many +multipath components, resulting in independent and identically distributed (i.i.d.) Gaussian gains for the direct and +1The frequency boundaries of the rich scattering model are subject to a number of variables, which are extensively covered in the channel +modeling literature [13]–[15]. +January 13, 2023 +DRAFT + +5 +RIS-aided channels. When a direct line-of-sight (LoS) path exists between any two nodes, a Rician fading model +applies. In that case, the Tx-RIS channel can be decomposed as follows: +G = +� +Kg +1 + Kg +¯G + +� +1 +1 + Kg +�G, +(2) +where Kg is the Rician factor, ¯G is the deterministic LoS component, and �G is the random non-LoS component. +Recently, RISs have been studied in higher frequencies (mmWave and THz) where the channels are represented +in parametric form; for example, a transmitter-RIS channel with L resolvable paths is given by +G = +L−1 +� +ℓ=0 +αℓa2(φ2,ℓ, θ2,ℓ)aH +1 (φ1,ℓ, θ1,ℓ), +(3) +where αℓ is the complex gain of path ℓ, a1(·) and a2(·) are steering vectors at the transmitter and RIS, respectively, +with φ1,ℓ(θ1,ℓ) and φ2,ℓ(θ2,ℓ) being azimuth (elevation) angles of departure and arrival for the path ℓ. In matrix +form: +G = A2 diag(α) AH +1 +(4) +This matrix representation will be revisited in Section V for channel estimation. +B. Wideband/Frequency-Selective Channels +Consider an RIS-aided communication between a single antenna transmitter and receiver (for ease of exposition) +in frequency selective channels using OFDM modulation with Mc orthogonal sub-carriers. Let L denote the number +of channel taps for both the direct and cascaded channels2, with hd = [hd(0), . . . , hd(L−1), 01×(Mc−L)]T ∈ CMc×1 +being the zero-padded (time-domain) direct channel vector, gn = [gn(0), . . . , gn(L − 1), 01×(Mc−L)]T the channel +between transmit antenna and the RIS element n, and hn = [hn(0), . . . , hn(L − 1), 01×(Mc−L)]T the channel +between RIS element n and receive antenna. Let qx be the Mc × 1 transmit frequency-domain OFDM symbol and +x be its Mc-point inverse discrete Fourier transform (IDFT). At the OFDM transmitter, qx is first transformed to x +and transmitted over the frequency-selective channel. The signal reaching the receiver due to direct path is given by +x⊛hd, where ⊛ denotes the circular convolution. The signal reaching the receiver via reflection from RIS element +n is ψn(x ⊛ gn) ⊛ hn, where ψn is the induced reflection coefficient. The overall received signal due to the direct +path and the RIS reflections is +y = √ρ +� N +� +n=1 +ψn(x ⊛ gn) ⊛ hn + x ⊛ hd +� ++ w. +(5) +The OFDM receiver performs DFT operation on y to obtain qy = FMcy, where FMc is the Mc × Mc DFT matrix. +Let qhd = FMchd denote the frequency domain Tx-Rx channel. Similarly, let qgn and qhn denote the frequency +domain RIS-aided channels due to element n. Then, the received signal in the frequency domain is given by +qy = √ρ +� N +� +n=1 +ψn(qx ⊙ qgn) ⊙ qhn + qx ⊙ qhd +� ++ w += √ρX +� N +� +n=1 +ψn(qgn ⊙ qhn) + qhd +� ++ w, +(6) +2Assumed for ease of exposition. Extensions for the case where direct and cascaded channels have a different number of paths is straightforward. +January 13, 2023 +DRAFT + +6 +where X = diag(qx) and ⊙ denotes the Hadamard (elementwise) product. The channel gains can follow either rich +or sparse scattering models depending on the propagation environment and frequency of operation. +III. PILOTS, TRAINING STATES, AND LINEAR ESTIMATION +A narrowband RIS-aided system is modeled by Eq. (1), or alternatively, +y = √ρ(xT ⊗ IMr)vec(Hd + HΦG) + w. +(7) +Recall ⊗ and vec(·) denote Kronecker product and vectorization, respectively. Define +Hc ≜ [vec(Hd) GT⋄ H] +(8) +where ⋄ is a columnwise Kronecker product, which is a special case of the Khatri-Rao product. We further define +hc ≜ vec(Hc). We organize the individual reflection coefficients ψ into a vector ψ, which carries the same +information as the diagonal matrix Φ = diag(ψ). Further, define ˜ψ ≜ +� +�1 +ψ +� +�. Then, Eq. (7) is expressed as follows +y = √ρ(xT ⊗ IMr)Hc ˜ψ + w += √ρ(˜ψ +T ⊗ xT ⊗ IMr)hc + w +≜ √ρZhc + w, +(9) +where Z represents a matrix that includes the transmit vector as well as the RIS training state, hc represents the +overall channel (including both cascaded and direct channels), whose estimation is necessary for coherent detection +at the receiver and beamforming at the transmitter and RIS. The transmission frame includes a training phase and +a data transmission phase. During the training phase, j = 1, . . . , J, the pilot signals xj and RIS training states ψj +give rise to the overall training matrix Zj = ˜ψ +T +j ⊗ xT +j ⊗ IMr, at the receiver resulting in yj = √ρZjhc + w. +A. Linear Estimation +Since yj is Mr-dimensional and hc is MrMt(N + 1)-dimensional, the least squares and minimum mean-square +error (MMSE) estimation requires at least J = Mt(N + 1) pilots. Define: +˜y ≜ +� +���� +y1 +... +yJ +� +���� +�Z ≜ +� +���� +Z1 +... +ZJ +� +���� +Then, the least squares estimate of hc is given by +ˆhc = +1 +√ρ +�Z†˜y, +(10) +where �Z† is the pseudo-inverse of �Z. Let Rhc denote the covariance matrix of hc. Then, the LMMSE estimate of +hc is given by +ˆhc = √ρRhc �ZH(ρ�ZRhc �ZH + IMrJ)−1˜y. +(11) +January 13, 2023 +DRAFT + +7 +Start +Training +i = 1 +Set RIS training state i + Transmit +orthonormal pilots +M t +i=N +1 ? +Yes +LS/LMMSE + Estimation +i=i+1 +No +Fig. 2. Flowchart representing the channel training process in RIS-aided MIMO. +Training Process: As illustrated in Fig. 2, the channel training occurs by the RIS assuming a training state ψi +that remains fixed over Mt consecutive slots, and the transmitter emitting Mt linearly independent (preferably +orthonormal) pilots. This process repeats N + 1 times with different RIS training states3 that generate linearly +independent extended training vectors �ψ. +B. RIS Training States +The accuracy of channel estimates depends on the choice of RIS training states. We consider the canonical, DFT, +and Hadamard training states. For simplicity of exposition, it is assumed that the direct path is absent, and the +channel gains follow i.i.d. CN(0, 1) resulting in Rhc = IMtMrN. The following identity is used in the derivation +of least squares and MMSE estimators. +�ZH �Z = Mt(ΨΨH)∗ ⊗ IMrMt, +(12) +where Ψ ≜ [ψ1 · · · ψN]. +1) Canonical Training: Canonical training activates one RIS element in each training state, deactivating the +remaining N −1 elements.4 Thus, the RIS training vectors constitute a so-called standard basis or canonical basis, +as follows: +ψi = ei ≜ [δi,1 . . . δi,N] +i = 1, . . . , N +where δi,j is the Kronecker delta function. This results in ΨΨH = IN, and hence, by the identity in Eq. (12), +˜ZH ˜Z = MtIMtMrN. Therefore, from (10), the least squares estimate with canonical training states is +ˆhc = +1 +Mt√ρ +�ZH ˜y. +(13) +3because of N RIS elements and one direct path, constituting N + 1 degrees of freedom. +4Nulling the reflection of an RIS element with respect to all angles has not been adequately addressed in the literature and remains an open +issue. See Section IX for further details. +January 13, 2023 +DRAFT + +8 +From (11), the MMSE estimate with canonical training is +ˆhc = +1 +Mt√ρ +� +1 + +1 +ρMt +� �ZH ˜y. +(14) +2) DFT Training: In DFT training, each RIS training state is a column of the standard N × N DFT matrix. +The orthogonality ΨΨH = NIN combined with the identity (12) gives �ZH �Z = MtNIMtMrN. The least squares +estimate with DFT training states is therefore +ˆhc = +1 +MtN√ρ +�ZH ˜y. +(15) +The MMSE estimate is +ˆhc = +1 +MtN√ρ +� +1 + +1 +ρMtN +� �ZH ˜y. +(16) +3) Hadamard Training: Another choice of RIS training states is to use the columns of N ×N Hadamard matrix, +which again results in ΨΨH = NIN and therefore �ZH �Z = MtNIMtMrN via the identity (12). The least squares +and MMSE estimators with Hadamard training are the same as with DFT training and hence are omitted for brevity. +Remark 1. (Ambiguity Problem) +For any diagonal matrix D ∈ CN×N, G′ ≜ D−1G, and H′ ≜ HD, +HΦG = H′ΦG′. +(17) +Therefore, G and H cannot be uniquely resolved by observing pilots via the channel HΦG. However, as noted +from Eq. (9), the knowledge of cascaded channel hc is sufficient for RIS beamforming, while also being more +efficient to estimate compared with estimating G and H separately. +Remark 2. (Channel Training under Finite Precision Reflection Coefficients) +The precision of RIS coefficients may be limited in practical implementations, with only a finite number of quantized +phase shifts being available, affecting both training and beamforming. Quantization of phase shifts has no effect on +channel estimation under canonical training states. Hadamard training requires only 1-bit phase precision, without +any compromise in estimation accuracy. DFT-training, however, requires N phase shifts, which may not be available +in practice for a large RIS array. When L phase shifts (L < N) are available at the RIS, a quantized-DFT training +can be achieved by mapping the phases of the DFT matrix FN to the nearest phases in the quantized phase set P +to obtain quantized-DFT matrix Fq,N as follows: +∠Fq,N(k, ℓ) = argminφ∈P|e−jφ − e−j 2π(k−1)(ℓ−1) +N +|. +(18) +Unfortunately, the quantized-DFT matrix is non-orthogonal, resulting in degraded channel estimates. This is espe- +cially an issue for low-precision RIS implementations. +Remark 3. (Grouping the RIS Elements) +Under some scenarios, the estimation of all channel gain parameters induced by the RIS may be prohibitive in +terms of time, power, or both. A remedy has been proposed [16]–[23] that constrains groups of RIS elements to +have the same reflection coefficient. Then, it is not difficult to see that the relevant estimation parameter is an +January 13, 2023 +DRAFT + +9 +aggregate channel gain corresponding to the total reflection produced by the RIS elements in each group (that have +the identical reflection coefficient). This scenario is effectively similar to an RIS with fewer (virtual) reflective +elements. Each of these virtual elements, representing multiple physical elements, will have a stronger reflection +and therefore a stronger channel gain. An example of this kind is discussed in Section VI. +IV. ESTIMATION VS. SPECTRAL EFFICIENCY +Because the RIS channel is not known ahead of time, channel resources must be spent to train and acquire +the channel state information. Pilots require transmit power, and transmission time is occupied for generating +independent channel observations, commensurate with the number of channel parameters being estimated. Any +channel resource used for training becomes unavailable for data transmission. A larger RIS can improve beamforming +gain that is beneficial for capacity, but also requires more training resources, which is detrimental for capacity. This +gives rise to an interesting and important tradeoff in the size of RIS and its effects on spectral efficiency, studied +in this section. +This section presents the training-based spectral efficiency results for RIS-aided single-antenna transmitters and +receivers without a direct path, whose insights carry over to multiple antenna systems as well. The developments +in this section follow [24]. Let T be the coherence interval of all channels, also used as block length. Td channel +symbols are dedicated to data transmission. In the absence of a direct path, a minimum of N temporal degrees of +freedom are needed for training, +T = N + Td. +Let ρτ and ρd denote the training and data powers, respectively, and let ρ denote the average power. Then, by +conservation of energy: +ρT = ρτN + ρdTd. +With these conditions, the following rate is achievable under canonical training [24]: +Rτ = +� +1 − N +T +� +Eˆhc log2 +� +1 + ρd +� �N +i=1 |ˆhc(i)| +�2 +1 + Nρd +1+ρτ +� +. +(19) +Under DFT training and Hadamard training, the following rate is achievable: +Rτ = +� +1 − N +T +� +Eˆhc log2 +� +1 + ρd +� �N +i=1 |ˆhc(i)| +�2 +1 + +Nρd +1+Nρτ +� +. +(20) +The spectral efficiency is a function of ρτ and ρd; the optimizer of Eq. (19) is ρdTd = β∗ +1ρT and ρτN = (1−β∗ +1)ρT, +with +β∗ +1 = +�� +1 + ρT +N +�� +1 + NρT +T −N +� +− +� +1 + ρT +N +� +� NρT +T −N − ρT +N +� +. +(21) +Similarly, the optimizer of Eq. (20) is ρdTd = β∗ +2ρT and ρτN = (1 − β∗ +2)ρT, with +β∗ +2 = +� +(1 + ρT) +� +1 + NρT +T −N +� +− (1 + ρT) +� NρT +T −N − ρT +� +. +(22) +January 13, 2023 +DRAFT + +10 +-10 +-5 +0 +5 +10 +15 +20 +25 +30 +2 +4 +6 +8 +10 +12 +14 +Fig. 3. Training-based bounds on capacity with canonical and DFT training. +-10 +-5 +0 +5 +10 +15 +20 +25 +30 +15 +20 +25 +30 +35 +40 +Fig. 4. Optimum RIS size that maximizes spectral efficiency. +Figure 3 shows the training-based bounds on the capacity of RIS-assisted system with 32 RIS elements when the +channel coherence interval is T = 150. The spectral efficiency with DFT training is higher compared with canonical +training.5 Specifically, with equal power allocation between training and data, DFT training achieves a gain of 3.5 +bits/s/Hz compared with canonical training. With optimal power allocation for both, DFT training achieves a gain +of 2 bits/s/Hz over canonical training. The reason for under-performance of canonical training is that the magnitude +of RIS training states multiplies the pilot power, therefore the zero coefficients in canonical training reduce the +received signal-to-noise ratio for pilots, and induce a penalty. On the other hand, DFT training activates all the RIS +elements in each training time slot, thereby efficiently utilizing the available pilot power. +5Hadamard training capacity expressions and numerical results are the same as DFT training, and are omitted for brevity. +January 13, 2023 +DRAFT + +11 +Figure 4 shows the RIS array size that maximizes the spectral efficiency, at each signal-to-noise ratio (SNR). +At low-SNR, power is at a premium, while degrees of freedom are less important. Therefore, it is beneficial to +estimate the channel induced by the entire (available) RIS array, even though the training requires degrees of +freedom. Conversely, at high-SNR, degrees of freedom are more important, therefore from a capacity perspective +it may be beneficial to utilize only part of an available RIS, so that fewer time slots are utilized for training. +V. SPARSE CHANNEL ESTIMATION +Whenever the channel has a sparse multipath structure, fewer channel parameters need to be estimated, and +the overhead incurred in transmission of pilots and feedback of channel coefficients is reduced. This is especially +relevant for higher frequencies (mmWave/THz) wherein the RIS has the most impact. To capture the efficiencies +arising from sparse channel structure, the RIS channel estimation is cast in the form of sparse vector recovery and +solved with compressive sensing algorithms that reduce the number of required channel measurements compared +with traditional channel estimation [25], [26]. +Sparse multipath channels are characterized by a geometric model involving angles of arrival/departure and +complex gains of the signal paths. The goal of sparse channel recovery is to estimate the parameters of the angular +representation of the channel described in Sec. II-A, Eq. (4), i.e., +G = A2 diag(α) AH +1 +which involves angles of arrival/departure captured in the left and right matrix, and the path strengths captured in +the diagonal matrix. Even though G might be (highly) rank deficient, it is not (yet) expressed in a suitable format +for compressive sampling algorithms. In Eq. (4), the basis vectors (columns of A1 and A2) can take values over an +uncountably infinite set. To recast the problem in a friendly format for compressive sampling, the candidate angles +are restricted to a finite set of size G, often corresponding to a uniform grid in a prescribed coordinate system. +The basis vectors6 corresponding to the discretized angles are collected into dictionary matrices �A1 ∈ CMt×G and +�A2 ∈ CN×G. With sufficiently good quantization of angles, matrices A1 and A2 are approximately7 submatrices +of �A1 and �A2. A sparse matrix Λg can select the appropriate columns from �A1 and �A2 so that: +G = A2 diag(α)AH +1 ≈ �A2 Λg �AH +1 +(23) +The problem is now in a standard form for compressive sampling. With pre-determined dictionaries �A1 and �A2, +the objective is to estimate the sparse matrix Λg from a noisy linear observation of G (via pilots). +If the transmitter-to-RIS channel G has L paths, the discretized grid of angles must have G ≫ L elements. +Ignoring for now any grid mismatch issues +G = �A2Λg �A +H +1 . +6also known as steering vectors +7because the true AoA/AoD may not fall exactly on the quantized grid of angles +January 13, 2023 +DRAFT + +12 +Similarly, let P denote the sparsity of RIS-to-receiver channel H. Consider a discretized set of candidate angles with +size H ≫ P, and collect the steering vectors corresponding to these angles into dictionary matrices �B1 ∈ CN×H, +�B2 ∈ CMr×H. Then, +H = �B2Λh �B +H +1 , +where Λh is a H × H sparse matrix with P non-zero elements. For simplicity of exposition, we assume no direct +path exists between transmitter and receiver. Thus, the received signal in Eq. (1) takes the form +y = √ρ �B2Λh �B +H +1 Φ �A2Λg �A +H +1 x + w += √ρ ¯Hx + w += √ρ(xT ⊗ IMr)vec( ¯H) + w, +(24) +where ¯H ≜ �B2Λh �B +H +1 Φ �A2Λg �A +H +1 . Using a series of vectorization operations, it can be shown that +vec( ¯H) = ( �A +∗ +1 ⊗ �B2)(ΛT +g ⊗ Λh)( �A +T +2 ⋄ �B +H +1 )ψ. +Substituting in Eq. (24), +y=√ρ(xT ⊗ IMr)( �A +∗ +1 ⊗ �B2)(ΛT +g ⊗ Λh)( �A +T +2 ⋄ �B +H +1 )ψ+w += √ρK(x, ψ)λ + w, +(25) +where λ ≜ vec(ΛT +g ⊗ Λh) is the (GH)2 × 1 sparse vector with LP non-zero elements, and +K(x, ψ) ≜ (( �A +T +2 ⋄ �B +H +1 )ψ)T ⊗ ((xT ⊗ IMr)( �A +∗ +1 ⊗ �B2)) +is the effective measurement matrix which is a function of pilots and RIS training states. During the training phase, +the input sequence (xj, ψj) takes J distinct values, and the output sequence yj is observed: +� +���� +y1 +... +yJ +� +���� = √ρ +� +���� +K(x1, ψ1) +... +K(xJ, ψJ) +� +���� λ + +� +���� +w1 +... +wJ +� +���� . +(26) +The sparse channel vector λ can be reconstructed from the measurements in (26) using standard sparse recovery +algorithms such as orthogonal matching pursuit (OMP) [25] and subspace pursuit [27]. Alternating direction method +of multipliers (ADMM) [28] and approximate message passing [29] have also been explored for sparse channel +estimation in RIS. Noh et al. [30] show that, for an RIS-aided single antenna system employing J pilots (J < N) +for sparse channel estimation, using the J equi-spaced columns of the N ×N DFT matrix as training states produce +lower mean squared error compared with canonical training states and the first J columns of the DFT matrix. +Training Overhead and Complexity: To reconstruct an LP-sparse vector of length (GH)2, orthogonal matching +pursuit requires O(LP log(GH)) measurements [31]. Since each pilot provides Mr measurements, the required +number of pilots for sparse RIS channel estimation is O +� +LP +Mr log(GH) +� +. Subspace pursuit requires even fewer +measurements; specifically, it requires O(LP log( GH +√ +LP )) measurements [31], and hence O( LP +Mr log( GH +√ +LP )) pilots. +In general, L and P are small at high frequencies, and hence the contribution of LP to the training overhead is +January 13, 2023 +DRAFT + +13 +small. Also, due to the logarithmic dependence on GH, the induced overhead is low, especially when Mr is large. +A thorough characterization of optimal training with sparse recovery, and the associated training-based capacity (as +in Sec. IV) is still open. The complexity of estimation using orthogonal matching pursuit (and subspace pursuit) is +O((LPGH)2 log(GH)), while linear estimation has a complexity of O((MtMrN)3). With suitable choice of G +and H, sparse recovery can reduce the complexity when compared with linear estimation. +Beyond sparsity, other structural properties can be exploited for further reducing the training overhead. For +broadband channel estimation in RIS-aided mmWave massive MIMO systems, Wan et al. [32] exploit the common +sparsity shared by different sub-carriers and propose a distributed orthogonal matching pursuit to reduce the +overhead. In the context of an RIS-aided multiuser downlink setting, Wei et al. [26] show that the angular cascaded +channels associated with different users have exactly the same non-zero rows and some common non-zero columns +(termed as double structured sparsity). The adaptation of orthogonal matching pursuit to this double-sparse structure +is shown to further reduce overhead. Zhou et al. [33] consider uplink channel estimation in RIS-aided mmWave +massive MIMO and exploit the fact that in many scenarios the angles of arrival/departure between RIS and base +station remain unchanged over multiple coherence blocks. Therefore, the base-station to RIS channel parameters, +which are more numerous in massive MIMO, need fewer updates, which can reduce pilot overhead. Lin et al. [34] +decompose the sparse channel recovery into three components: recovery of angle of arrival, angle of departure, and +complex gains. A semi-passive RIS with a few receiver chains at the RIS was proposed by [35], [36]. A semi-passive +RIS allows for receiving pilots and channel estimation at the RIS. The transmitter-RIS channel is estimated with +the aid of the few RIS on-site measurements, and utilizing compressive sensing. +Matrix factorization and matrix completion [37] can also be used for estimating rank-deficient, sparse RIS +channels. The key idea of this approach can be explained as follows. With pilots Xτ = [x1 . . . xJ], RIS training +states Ψτ = [ψ1 . . . ψJ], and received pilots Yτ = [y1 . . . yJ], the system model in Eq. (1) becomes [37] +Yτ = √ρH(Ψτ ⊙ (GXτ)) + N, +(27) +where N ∈ CMr×J is the additive noise matrix. Equation (27) can be equivalently written in the factored form as +Yτ = √ρHA + N, +(28) +where A ≜ Ψτ ⊙ (GXτ). With this representation, the estimation is achieved using a two stage process. In the +first stage (matrix factorization), the matrices H and A are estimated based on Yτ. In the second stage (matrix +completion), G is estimated based on the estimate of A. The success of this method requires A to be sparse +and H to be a low-rank matrix. The sparsity of A can be satisfied by selecting the RIS training matrix Ψτ to +be sparse, i.e., most of its coefficients set to zero. High frequency (mmWave/THz) channels H have low rank +due to dominance of reflections over scattering. He and Yuan [37] achieve the matrix factorization step using the +bilinear generalized approximate message passing algorithm [38], and the matrix completion step using Riemannian +manifold gradient-based algorithm [39]. Other methods based on similar ideas can be found in [40]–[44]. +The majority of the literature on sparse channel estimation assume that the true angles of arrival/departure lie +on a discretized grid (i.e., on the discrete steering angles of the dictionary matrices). In practice, when the angles +January 13, 2023 +DRAFT + +14 +of arrival/departure do not coincide with the discrete angles in the dictionary, the sampling process leads to many +non-zero sample measurements, degrading the sparse recovery algorithm. The sensitivity of sparse recovery to grid +mismatch was systematically analyzed in [45], but this analysis has not been widely adopted in the sparse channel +estimation literature. In an alternative approach, He et al. [46] propose atomic norm minimization for RIS channel +estimation. Atomic norm is a convex function that generalizes the ℓ1 norm for sparse recovery and nuclear norm +(i.e., sum of singular values) for low-rank matrix completion. Atomic norm minimization works in the continuous +domain and avoids discretization, therefore eliminating the grid mismatch problem [47]. Its solution is often via +semidefinite programming. +VI. WIDEBAND CHANNEL ESTIMATION +The RIS-aided OFDM model was discussed in Sec. II-B, where the frequency domain input-output relation was +provided by Eq. (6). Based on this model, the present section provides the main ideas involved in the channel +estimation for RIS-aided OFDM. The system model in Eq. (6) can be equivalently written as +qy = √ρX +� +Bψ + qhd +� ++ w, +(29) +where B is an Mc × N matrix whose columns are qgn ⊙ qhn. The Mc × T frequency-time frame is divided into two +sub-frames: a training sub-frame of size Mc × (N + 1) and data-transmission sub-frame of size Mc × (T − N − 1). +In the training sub-frame, the received pilot sequence is given by +qyj = √ρXj +� +Bψj + qhd +� ++ wj, j = 1, . . . , N + 1. +Defining Ψ = +� +ψ1 · · · ψN+1 +� +, and using the pilot sequence Xj = IN for j = 1, . . . , N + 1, the received training +sequence qY = [qy1, . . . , qyN+1] is given by +qY = √ρ +� +qhd +B +� +� +�1T +Ψ +� +� + W, +where W = [w1, . . . , wN+1]. For convenience of notation we define: +C ≜ +� +qhd +B +� +�Ψ ≜ +� +�1T +Ψ +� +� +resulting in +qY = √ρ C �Ψ + W. +(30) +From this, the matrix C containing the direct and cascaded channels can be estimated as +�C = +1 +√ρ +qY �Ψ +−1. +(31) +It has been shown in [18] that choosing �Ψ = FN+1 results in the least error variance. +January 13, 2023 +DRAFT + +15 +N+1 +T-(N+1) +MC +... +... +... +... +... +... +... +... +... +... +... +... +... + +: Pilot Symbol +: Data Symbol +Transmission Frame +Subframe 1 +Subframe 2 +Frequency +Time +Fig. 5. Transmission frame structure in RIS-aided OFDM [18]. +In the above method, pilots were inserted on all the Mc sub-carriers of the N + 1 training OFDM symbols. In +practice, when the channel is correlated in the frequency domain, fewer pilots may be employed and then channel +estimates may be interpolation among sub-carriers.8 The system model in Eq. (29) is equivalent to: +qy = √ρXq + w, +(32) +where q ≜ Bψ + qhd. Now, as shown in Fig. 5, in the training sub-frame, Np pilots (Np < Mc) are inserted in +each OFDM symbol with a spacing of ∆ = ⌊ Mc +Np ⌋. Let P = {0, ∆, . . . , (Np − 1)∆} denote the indices of the +sub-carries containing pilots. Let qxP and qyP denote the transmitted and received pilots on sub-carriers indexed by +P, respectively. Also, let XP = diag(qxP). Then, the estimate of qP can be obtained as +ˆqP = +1 +√ρX−1 +P qyP. +Using ˆqP, the estimate of q (denoted by ˆq) is obtained via interpolation along the subcarriers. The work in [18] +applies the DFT/IDFT-based interpolation on the pilot sequence. Now, in order to resolve B and qhd from ˆq, the +RIS training states ψj are adjusted during each training OFDM symbol and the corresponding ˆqj is given by +ˆqj = Bψj + qhd + vj, j = 1, . . . , N + 1 +where vj is the error in estimating q during the pilot slot j. Let �Q = [ˆq1 . . . ˆqN+1] and V = [v1, . . . , vN+1]. +Then, +�Q = C �Ψ + V. +Using the above equation, the estimate of matrix C containing direct and RIS-aided channels can be obtained as +�C = �Q �Ψ +−1. +(33) +8This is a long-standing practice that has been adopted in 3GPP. +January 13, 2023 +DRAFT + +16 +To further reduce the estimation overhead, one may group the neighboring RIS elements and use the same +reflection coefficient for all the elements in each group. This solution has been suggested for both flat and frequency +selective channels [16]–[23]. If the N RIS elements are divided into Ng groups (Ng < N) with each group +containing N/Ng elements, then the channel estimation requires estimating only Ng aggregated RIS-aided channels +corresponding to each group, instead of estimating all the N cascaded channels corresponding to each RIS element. +Therefore, the training overhead is reduced from N +1 OFDM symbol durations to Ng+1 OFDM symbol durations, +where each OFDM symbol duration is composed of (Mc + Lcp) time-slots with Lcp being the length of the cyclic +prefix. The grouping strategy trades-off accuracy for overhead in order to improve the overall spectral efficiency, +which is analytically characterized in [16]. Zheng et al. [48] extend this estimation technique to multiuser orthogonal +frequency-division multiple access (OFDMA) systems. The same authors [49] propose a fast channel estimation +scheme for reducing the training-overhead in RIS-aided OFDM. The key idea is to use short OFDM symbols of +M ′ +c sub-carriers (L ≤ M ′ +c ≪ Mc) during the training phase, which consumes (M ′ +c + Lcp) time-slots per OFDM +training symbol. This reduces the (Mc + Lcp) pilots that were required by [18]. +The above works assume an ideal reflection model in which the RIS elements achieve the same amplitude and +phase response across the entire OFDM band. Wenhao et al. [50] show that the practical response of RIS is tightly +related to the frequency of the signal. Based on this, Yang et al. [51] studies channel estimation for RIS-aided +OFDM under a practical reflection model and finite precision coefficients. While differing in its modeling, the +estimation technique of [51] is similar to [18], as outlined above, and is omitted for brevity. +VII. REDUCING THE RIS ESTIMATION OVERHEAD +We begin by exploring savings in pilots and estimation overhead that arise from the multi-user nature of the RIS +channel. An RIS-aided uplink system with K single-antenna users and M-antenna base station (BS) must estimate +KMN + KM links for RIS beamforming and equalization, which can be prohibitive either under massive MIMO, +or in large cells. Linear estimation techniques (Sec. III) require K(N +1) pilot transmission slots, growing linearly +with the size of RIS. We discuss avenues for reducing the pilot overhead. +A. Common RIS-BS Channel +Consider a scenario where a base station is aided by a single RIS for communication with multiple users. To +describe channel estimation in this multi-user scenario, we adapt the system model from Eq. (1) for the multi-user +uplink channel: +y = +K +� +k=1 +√ρ(hdk + HΦgk)xk + wk, +(34) +where hdk is the direct channel from a single-antenna user k to a multi-antenna base station, and gk ∈ CN +is the channel from the user k to RIS. Recall that the combined (direct and cascaded) RIS-aided channel gains +to be estimated were collected into a single matrix in Eq. (8); a specialization of that matrix for the case of a +single-antenna user is given by: +Hck = [hdk H diag(gk)] +(35) +January 13, 2023 +DRAFT + +17 +Among the quantities participating in this expression, the two vectors hdk and gk are distinct for different users, +but the BS-RIS matrix H is common between users. The user-by-user uplink channel estimation requires one pilot +for estimating the direct channel and N pilots for estimating the cascaded channel, for a total of K(N + 1) total +pilot transmission slots. But the commonality of H among users hints at possible savings in the total number of +needed pilot slots, which we now explore. +To begin with, the direct channels for all users is estimated, by deactivating the RIS. This requires one pilot +slot per user, but this step may not be crucial, because it is often the absence of a direct path that makes the +RIS an attractive choice. In the next step, the cascaded channel (H diag(g1)) is measured by emitting N pilots +from User 1 to the base station. This is accomplished via N successive training states at the RIS, whose details +are omitted for brevity. For User 2, we now need to measure (H diag(g2)). This new matrix has columns that +are co-directional with columns of (H diag(g1)), thus only the magnitude of each column needs to be measured. +For N columns, this requires N new observation samples, however, reception at the multi-antenna base station +provides M independent observations per pilot transmission. Therefore, after obtaining (H diag(g1)), only N +M pilot +transmissions are needed per additional user, as long as training states are designed properly. The design of training +states that ensure the requisite linearly independent observations has been explained in [52], [53]. When M > N, +following the above argument, it is easy to see that one pilot per user is sufficient for estimating the cascaded +channels of Users 2, . . . , K. Therefore, the total training overhead of this scheme is given by +J = K + N + max +� +K − 1, +�(K − 1)N +M +�� +, +(36) +where ⌈·⌉ denotes the smallest integer bigger than or equal to the argument. For massive MIMO systems with +M > N, the overhead J = 2K + N − 1, meaning that each user beyond the first one requires two pilot slots. This +provides significant savings over the N+1 slots needed conventionally. Guan et al. [54] propose a slight modification +of this technique, in which a few stationary nodes called the anchor nodes are assumed to exist in the network. The +anchor nodes transmit pilot signals and the base station estimates anchor-RIS-BS channels. Due to the common +RIS-BS channel, the User-RIS-BS channels are subsequently estimated with fewer pilot transmissions. Since the +anchor nodes are stationary, the estimation of anchor-RIS-BS channels are done less frequently compared with +the earlier, single-reference user in [52], [53] which was not assumed to be stationary, thus resulting in additional +savings. Guo and Lao [55] also explore the possibility of exploiting the common RIS-BS channel without requiring +a reference user. +The methods in [52], [53] estimate the direct channel, and subsequently subtract it from the measurement intended +for the cascaded channel, which is not MMSE optimal. A joint estimate of [hd1 H diag(g1)], i.e., User 1’s direct +and cascaded channels, has a lower mean squared error (MSE). Wei et al. [56] propose this joint estimation, and +then the remaining user channels are estimated via the same technique as [52], [53]. +This modification acknowledges and addresses the propagation of the error in the estimation of hd1 when +estimating the cascaded channel of User 1. The estimate of H is used for constructing the cascaded channels +of other users too, therefore in a sense, the errors committed in estimating the channel of User 1 can propagate +January 13, 2023 +DRAFT + +18 +into the estimation of other users’ channels. However, since User 2 and subsequent users employ fewer pilots than +User 1, it is not obvious that their channel measurements can be used to improve the estimate of H. +B. Slowly Varying BS-RIS Channel +Since the BS and RIS are static, the channel between them varies slowly. In comparison, the BS-user and RIS-user +channels are more dynamic because of the mobility of the user. The high dimensional, but slowly varying BS-RIS +channel can therefore be estimated less frequently, while the low-dimensional BS-user and RIS-user channels are +estimated more frequently. To isolate the estimation of BS-RIS link, [57] assumes a full-duplex base station. The +base station will emit pilots and listen for the reflection from the RIS. The self-interference of the full-duplex +reception must be dealt with, and the BS-RIS channel recovered. Given the BS-RIS channel estimate, the direct +BS-user channel and the RIS-user channel are estimated conventionally. The latter estimates are more frequent, +but also require smaller overhead. If TL denotes the coherence time of the BS-RIS channel and TS denotes the +coherence time of the BS-user and RIS-user channels such that TL = αTS, then the overhead of the two-timescale +method is +J = 2(N + 1) +α ++ K +� N +M +� ++ K. +In practice, α ≫ 1 and hence the first term is small. For massive MIMO systems with many base station antennas +M > N, the overhead becomes 2(N+1) +α ++2K, i.e., after estimating the BS-RIS channel, each user needs two pilots. +Under α > 2, this method has smaller overhead compared with the method of Section VII-A, although one must +be careful that the two methods address different channels and different base station capabilities, so they are not +directly comparable. +C. Infrequent RIS Coefficient Updates +Another source of potential savings in RIS induced channels is to deliberately reduce the frequency with which +RIS reflection coefficients are updated. As long as RIS coefficients are not updated, the RIS blends into the channel +and effectively the system is reduced to a (multi-user) MIMO system, with conventional channel training and pilots. +Of course, this involves a tradeoff: fewer pilot slots are needed, but also, the match of RIS coefficients to the channel +will go stale, therefore part of the beamforming gains of RIS will be lost. +Ideally, an analysis of this situation requires a temporally varying channel model with a corresponding temporal +correlation. However, the work in this area has taken a different direction, via considering a channel model, with +line-of-sight and rich scattering components. For the Tx-RIS channel G, this means the Rician model +G = +� +Kg +1 + Kg +¯G + +� +1 +1 + Kg +�G, +which we also saw in Eq. (2). A similar model is utilized for the RIS-Rx channels. +It is assumed that the infrequent update of RIS is able to fully capture the line-of-sight component, while not +capturing anything about the rich scattering part of the model, even immediately following the pilot transmission. +This approximation is different from the common modeling of temporal variance in most wireless channels, in +January 13, 2023 +DRAFT + +19 +which channel knowledge is accurate at times that are proximate to the pilots, but it has the advantage of removing +the complexities involved in the temporal dynamics of the channel. Thus, it reduces the problem to an equivalent +problem involving a channel state that is partially known. The literature [58]–[62] refers to this new formulation +of channel temporal dynamics as statistical channel state information.9 +The central idea of [58]–[62] is that the line-of-sight component has a longer coherence interval than the rich +scattering component. Single-antenna mobiles estimate the end-to-end uplink channel with a single pilot at the +smaller coherence interval, and the base-station beamforming is also updated at the smaller coherence interval. +However, the RIS coefficient is updated only at the longer line-of-sight coherence intervals. This creates significant +savings, since most of the pilot slots in the RIS-induced channel, especially for single-antenna mobiles, is needed +for estimation and updating of the RIS coefficients. +Several works have attempted to maximize the ergodic downlink rates to single-antenna mobiles with beamforming +f, either in the single-user or multiple user scenarios. For the single-user case, the received signal is given by +y = √ρ(hHΦG + hH +d )fs + w, +The best beamformer f is found for a given set of RIS coefficients Φ, but then utilize as Φ the (fixed) RIS +coefficients that are statistically the best over the variations of the channel. +C∗=max +Φ +Eh,G,hd +� +max +f +� +log2 +� +1 + ρ|(hHΦG + hH +d )f|2��� +(37) +Han et al. [58] achieve the inner maximization in Eq. (37) via maximal ratio transmission, and adjust the reflection +coefficients based on an outer bound on the ergodic rate. Hu et al. [59] maximize Eq. (37) via alternating optimization +method, Zhao et al. [60] uses a penalty dual decomposition method, Zhi et al. [61] achieve minimum user rate +maximization via genetic algorithm, and Gan et al. [62] propose methods based on ADMM, fractional programming, +and alternating optimization. +Several important points and open problems remain for consideration in this area. To begin with, these methods +are based on the assumption that the RIS will be changed infrequently, but also calculate and optimize ergodic +capacity. Therefore, the practical implementation of these techniques requires an outer code that goes across many +coherence intervals of the slower channel. In many such cases, outage capacity or throughput may be a more suitable +metric for optimization, and there is room for future work in this area. +Another useful direction is to find simplifications and approximations of the expression in Eq. (37) in order to +recognize trends and/or suggest different approaches. In this area, there is a need for achievable rate (inner bound) +expressions rather than outer bounds. Inner bounds for this expression have not been developed at the time of the +writing of this paper. +D. Opportunistic RIS +Another strategy for reducing the estimation overhead is inspired by an idea that harks back to the concept of +opportunistic transmission [65]–[67]. Q randomly selected vectors {ψ1, . . . , ψQ} are assigned one-by-one as RIS +9This IRS channel model has weak connections with earlier, well-known work in MIMO channels with statistical CSI [63], [64] that were +driven by CSI impairments due to limited feedback or feedback delay. +January 13, 2023 +DRAFT + +20 +Phase I +Phase II +Pilot +Signal +Pilot +Signal +Direct Channel + (RIS off) + Direct + + Cascaded + Channel +Received + Pilots +Received + Pilots +ChannelNet + Estimate of +Direct Channel + Estimate of +Cascaded Channel +ChannelNet +Fig. 6. RIS channel estimation using ChannelNet [70]. +phase vectors. In each instance, the RIS changes the scattering environment randomly, so there is no beamforming in +the usual sense of the word. For each of these Q scattering conditions, the end-to-end multiple-input single-output +(MISO) channel is measured using a few pilots, and the best one is chosen for one block of transmission. An +and Gan [68] propose the above approach in narrowband channels, and study bounds on its ergodic performance. +The set {ψ1, . . . , ψQ} is called a codebook in [68], however, this is a slight misnomer since this set need not be +determined or agreed upon ahead of time, is statistically independent of signals emitted from transmit antennas, and +is not needed at the receiver for decoding. The connection of this class of techniques with opportunistic transmission +is evidenced by the appearance of the order statistics of (induced) channels in [68, Proposition 1]. An et al. [69] +extend this idea to OFDM transmission. +VIII. MACHINE LEARNING BASED CHANNEL ESTIMATION +Machine learning is being actively investigated for channel estimation; this section explores machine learning +channel estimation in the context of RIS. +Among the early attempts at using machine learning for RIS channel estimation was a non-parametric con- +volutional neural network estimator by Elbir et al. [70], applied to RIS-aided downlink mmWave channels. The +estimation is achieved in two phases (see Fig. 6), somewhat similar to other methods seen earlier. In the first phase, +the base station transmits pilots while the RIS elements are inactive (turned off) so that the direct channel(s) are +estimated at the receivers, each of them operating an instance of the convolutional neural network. In the second +phase, the RIS assumes (several) training states while the base station transmits pilots. Each of the users employs +the received pilots in this phase, and in combination with the estimated direct channels (obtained in the previous +phase), produces an estimate of the cascaded channel using the same convolutional neural network architecture. +Under low SNR conditions, this technique claims better performance than a (corresponding) two-phase least squares +technique. Under high SNR conditions, the neural network technique has a performance ceiling, while the least +squares techniques do not. The experiments involved a 64-antenna base station, 100-element RIS, 8 single-antenna +users, and a geometric channel with 10 paths. The neural network has an input layer, an output regression layer +providing complex valued channel estimates, three convolutional layers each with 256 3 × 3 filters, two fully +connected layers with 1024 and 2048 nodes. The input layer has size +√ +M × +√ +M × 3 for direct channel estimation +January 13, 2023 +DRAFT + +21 ++ +- +Pilot +signal +Direct + +Cascaded +Channel +Received + Pilots +Least Squares + Estimator + Denoising + Neural + Network +Residual + Noise + Refined +Channel Estimate + LS +Estimate +Fig. 7. RIS channel estimation via denoising of least squares estimate using neural networks. +and N × M × 3 for cascaded channel estimation. The output layer has size 2M × 1 for direct and 2NM × 1 for +cascaded channel estimation. +A. Post Processing Least Squares Estimates +Motivated by image denoising using neural networks, Kundu and McKay [71] model the problem of RIS channel +estimation as that of denoising the least squares solution (see Fig. 7). Specifically, in the first step, least squares +estimate of direct and cascaded channels is obtained using pilots and DFT training states. The obtained least squares +estimate is viewed as a noisy version of the original channel. This is followed by a post-processing step in which +the Denoising Convolutional Neural Network (DnCNN) [72] or Fast & Flexible Denoising Network (FFDNet) [73] +are used.10 The least squares channel estimate is the input to the neural network, whose output is an estimate of +the least squares estimation error. The post-processed estimate is obtained by subtracting the estimate of the least +squares estimation error from the least squares estimate. +The optimal minimum mean squared error estimator of the channel gains is the conditional mean, but since the +cascaded channel is non-Gaussian, this estimator is non-linear and difficult to characterize. This has been the main +motivation mentioned in [71] for a neural network approach. One can infer that the LMMSE estimate, being linear, +is akin to a first order term of a Taylor series expansion for the conditional mean estimator, and the neural network +attempts to approximate the higher order terms. +Chang et al. [74] propose convolutional deep residual networks for denoising the least squares solution. Nipuni +et al. [75] employ neural network denoising for wideband channel estimation in RIS-aided OFDM. Shicong et +al. [36] propose an RIS architecture with a few active elements for initially estimating the low-dimensional channel, +compressive sensing reconstruction of the complete high-dimensional channel, and a further refinement with a +convolutional neural network. Mao et al. [76] refine the channel estimates produced by orthogonal matching pursuit +using deep residual networks in RIS-aided mmWave channels. Ye et al. [77] and Jin et al. [78] explore generative +adversarial networks for estimation in RIS-aided mmWave massive MIMO. +January 13, 2023 +DRAFT + +22 ++ +- +Phase I +Phase II +Phase III +Pilots +Direct Channel + (RIS off) +Received + Pilots +Least Squares + Estimator + Direct +Channel +Estimate +Neural Network +Refined + Direct +Channel +Estimate +Pilots +Direct + Cascaded + Channel + +Received + Pilots +Least Squares + Estimator +Partial +Cascaded +Channel +Estimate +Denoising Deep +Neural Network +Residual + Noise +Improved Cascaded +Channel Estimate +Inactive RIS Prediction +Deep Neural Network +Predicted +Channel +of Inactive +Elements +Fig. 8. RIS channel estimation using predictive neural networks. +B. Partial CSI +In order to reduce the training overhead in deep learning RIS channel estimation, Gao et al. [79] propose a +predictive neural network in the context of RIS-aided uplink massive MIMO. As shown in Fig. 8, the proposed +method works in three stages: In the first stage, the RIS is turned off and the direct channel is estimated using least +squares method, which is further refined using a fully connected neural network. In the second stage, only a part +of RIS elements are activated (N1 < N), and the cascaded channels corresponding to the activated RIS elements +are estimated using least squares method with N1 × N1 DFT training states and further refined using a denoising +convolutional neural network. In the third stage, the cascaded channels corresponding to the inactive RIS elements +are predicted using a fully connected inactive RIS channel prediction neural network. A geometric channel model +is employed for the base-station to RIS channel where the channel matrix is generated from a geometric model +that assigns the same gains and angles of arrival/departure to different RIS elements, with an implicit underlying +assumption that the scatterers are far from the RIS and the base station. The proposed estimation needs N1 + 1 +pilots instead of N + 1 pilots, reducing the overhead. +Xu et al. [80] propose an ordinary differential equation (ODE) based convolutional neural network for predicting +the channel corresponding to inactive RIS elements. Shtaiwi et al. [81] assume a multi-user uplink channel in which +the channel of different users is highly correlated, so that only a few users need to transmit pilots, and the channel +of the remaining users may be predicted from the first few. A neural network is employed for the prediction. In +RIS-aided uplink communication, Xu et al. [82] uses spatial correlation between the channels of different RIS +elements, as well as temporal correlation of time-varying channels, to reduce the communication overhead. For +exploiting spatial correlation, some RIS elements are turned off, hence corresponding channels are not directly +10These neural networks are imported from the image denoising literature. +January 13, 2023 +DRAFT + +23 +estimated by pilots, but are interpolated from other RIS channel estimates using a convolutional neural network. +For temporal correlation, a recurrent neural network is used to interpolate the channel values between two pilot +transmissions. +IX. FRONTIERS OF RIS CHANNEL MODELING +Powerful channel estimation techniques depend on accurate, yet convenient, channel models. RISs are relatively +new devices whose channel modeling brings together aspects of electromagnetic engineering, hardware constraints, +and communication system concepts. Certain frontiers of RIS channel modeling are still being explored; this section +outlines several issues of contemporary interest and investigation in RIS channel modeling. A summary of the +contents of this section appears in Table II. +Issue +Cause +Application or Limitation +Channel reciprocity +Angle dependence of RIS phase shifts +Massive MIMO channel estimation +Mutual coupling +Reduced element spacing +Modeling & estimation in large RIS +Perfect absorption +Dissipation and resonance control +Required in some channel estimation methods +Dependence of RIS gain/phase +Metallic/Dielectric/Ohmic losses vs. phase +Passive beamforming, DFT-training +RIS frequency dependence +Load/control impedance vs. frequency +Wideband communications using RIS +Near field issues +Spherical vs. planar wave approximation +Large-RIS, indoor communication +TABLE II: Frontiers of RIS channel modeling +A. Channel Reciprocity +RIS channel estimation in multiuser settings rely on the principle of channel reciprocity for reducing the estimation +and feedback overhead, since reciprocity enables the downlink precoding using uplink pilots/estimation in the time +division duplex operation [57], [83]–[85]. Channel reciprocity holds for many boundary conditions occurring in +wireless communication, including reflections from large objects as well as scattering. However, in the case of +RIS-assisted systems, the literature is inconclusive (see Fig. 9). Chen et al. [86] based on an equivalent circuit +model claims that angle reciprocity only holds for small angles with respect to normal. In the opposite direction, +Tang et al. [87] invokes the Rayleigh-Carson theorem to conclude that RIS enjoy reciprocity, but does not elaborate. +Liang et al. [88] states that the angle reciprocity depends on the design of the RIS surface, and proposes a structure +that achieves reciprocity for wide range of angles. +A resolution of the differences between these results, and a conclusive determination of the conditions under which +RIS-induced channels are reciprocal or non-reciprocal, will be welcomed. The system model and applications for a +non-reciprocal RIS may be of interest in future applications, but is in need of verifiable theory and/or experimental +evidence. +January 13, 2023 +DRAFT + +24 +Tx +Rx +Ө1 +Ө2 +Ө2 +Tx +Rx +Ө3 +Fig. 9. Schematic representation of reciprocity in RIS. +B. Mutual Coupling +A common assumption in RIS modeling is that the passive elements are spaced half-wavelength apart and the +mutual coupling among the elements is negligible, allowing them to be controlled individually and independently. +However, in practical planar RIS structures with fixed aperture, it is desirable to increase the number of elements +by reducing the inter element spacing in order to increase the directivity of the reflected waves and thereby improve +the received power. Reducing the inter-element spacing results in dependency/connection of the impedances of the +neighboring elements that is non-negligible, having its effect on the channel model, estimation, and the design of +reflection coefficients [89], [90]. Gradoni and Di Renzo [91] have proposed an electromagnetic compliant end-to- +end channel model for RIS-aided communication that accounts for mutual impedance among RIS elements, while +also including the effects of antenna elements at the transmitter and receiver. This impedance-based communication +model is utilized in [92] to maximize the end-to-end received power by optimizing the RIS tunable load impedances +in SISO system, which is further generalized in [93] for MIMO systems. In the end-to-end channel modeling of +the aforementioned references, the statistical components of the Tx-RIS and RIS-Rx channels are intertwined with +the circuit model parameters, which is not desirable from the signal processing/system design perspective. A more +interesting and useful model is one that retains the factored form GΦH, separating the statistical part of the Tx- +RIS and RIS-Rx channels, but incorporating the effects of mutual impedances and coupling into the RIS matrix Φ, +making it potentially non-diagonal and function of circuit model parameters. Obtaining such a factored model and +the associated estimation technique is a potential direction for future research. +C. Perfect Absorption/Reflection at RIS +Several channel estimation methods, such as those based on matrix completion and channel decomposition, +depend on the ability to completely deactivate some RIS elements. Some methods depend on separately estimating +the direct channel between the transmitter and the receiver, by eliminating the effect of RIS reflections, which +requires deactivating all the elements. This requires the incident energy to either be completely absorbed by the +RIS, or the incident waves to pass through the RIS. The feasibility of perfect absorption is debatable, with partial +results whose applicability to communication systems remains unverified. Some works that study the electromag- +January 13, 2023 +DRAFT + +25 +netics of metasurfaces suggest that perfect absorption is possible at resonant frequencies with proper tuning of +impedance [94]–[97], but it is unclear if/how the proposed structures and the associated methods can be utilized +in the context of RIS-aided communication. The issue of perfectly deactivating the passive elements is raised in +[10], [37], [70], but the question of its physical feasibility remains unresolved. Mishra and Johansson [98] assume +that perfect reflection and absorption are unrealizable and incorporates two constants as implementation errors in +the system model. To the best of the authors’ knowledge, no currently-available study in the open literature offers +an in-depth and conclusive treatment of the feasibility of perfect deactivation of RIS elements and related design +issues. Also, analyzing the effect of imperfect deactivation on the accuracy of estimation methods is an important +future direction for research. +In a related direction, several works explore whether and how the phase and amplitude response of an RIS +elements are related [99]–[101]. A few studies in reflectarrays and meta surfaces [99], [102]–[104] aim at designing +structures that allow near-independent control of reflection amplitude and phase, however, their applicability to +RIS-aided communication has not been established. +D. Frequency Dependence of RIS Elements +RIS-assisted wideband communications [32], [50], [105] requires RIS elements to efficiently operate throughout +the frequencies of the band. In typical RIS constructions, however, the reflection coefficients are tuned by switching +on or off various reactive elements or patterns that are connected to the RIS element. The effect of these tuning +devices can be modeled by an equivalent circuit whose load impedance varies with the carrier frequency. The phase +shift applied to the incident wave via the tunable elements is calculated at a specific frequency, often the resonant +frequency of the RIS element. Within a small deviation of this center frequency, the phase shift remains linear, +but across wider frequencies of operation, the phase shift might vary nonlinearly. In that case, the array factor +will vary across frequency, and the beam may not retain sharpness across the band of frequencies in which RIS +must operate. Several remedies have been proposed in the neighboring literature in reflectarrays, e.g., coupling the +elements to true time delay lines [106] or by coupling multiple resonance elements [107]–[109]. The applicability +of these methods for RIS-aided communications is yet to be explored, and is a direction for future study. +E. Near Field Issues +The transmitter/receiver is said to be in the far-field of RIS if it is at a distance greater than the Fraunhofer +distance 2D2 +RIS +λc +, where DRIS is the largest aperture of RIS and λc is the wavelength corresponding to the carrier +frequency fc [110]. With the far field assumption, the incident/reflected wave from the RIS can be assumed planar, +which simplifies calculations (see Fig. 10). Larger RIS structures can achieve superior SNR [7], [111], but if the +transmitter and receiver locations are fixed, a sufficiently large RIS array will violate the far field assumption [112]. +The near-field scenario arises, for example, when a large RIS is mounted on a large portion of the facade of a +building for servicing users in the street. Without the far-field assumption, the incident waves at different elements +will have unequal angular directions and polarization. The modeling of RIS in the near field scenario is considered +in [113]–[115]. The work in [116] accounts for the difference in the effective area of the elements from different +January 13, 2023 +DRAFT + +26 +Tx +Rx1 +Rx2 +RIS +Fraunhofer Distance +Fig. 10. Illustration of near/far field issue in RIS-aided communication. +observation angles close to the RIS. Studying suitable channel models and associated estimation techniques for +RIS-aided near field communication is a potential direction for future research. +X. CONCLUSION +This paper provides a comprehensive exposition of the channel estimation techniques for RIS-aided systems, +ranging from classical least squares/MMSE methods to machine learning methods. RIS is often employed with +many reflective elements leading to a channel gain with a huge number of parameters, therefore the estimation of +link gains is a signature challenge of RIS-induced communications. This paper explores the utility of RIS channel +structure for reducing the estimation overhead, including the slow variation of the base-station-to-RIS channel, +sparsity of the mmWave channels, and the spatial correlations among the channels of neighboring RIS elements +and neighboring users. Open problems in the broader area of RIS channel estimation were highlighted. +REFERENCES +[1] M. Shafi, A. F. Molisch, P. J. Smith, T. Haustein, P. Zhu, P. De Silva, F. Tufvesson, A. Benjebbour, and G. Wunder, “5G: A tutorial +overview of standards, trials, challenges, deployment, and practice,” IEEE Journal on Selected Areas in Communications, vol. 35, no. 6, +pp. 1201–1221, 2017. +[2] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. 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Edfors, “Beyond massive MIMO: The potential of data transmission with large intelligent surfaces,” IEEE +Transactions on Signal Processing, vol. 66, no. 10, pp. 2746–2758, 2018. +January 13, 2023 +DRAFT + diff --git a/stE4T4oBgHgl3EQfVwwe/content/tmp_files/load_file.txt b/stE4T4oBgHgl3EQfVwwe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d199e104bbd316216d37ac828b008e9a259ec090 --- /dev/null +++ b/stE4T4oBgHgl3EQfVwwe/content/tmp_files/load_file.txt @@ -0,0 +1,1754 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf,len=1753 +page_content='1 Channel Training & Estimation for Reconfigurable Intelligent Surfaces: Exposition of Principles, Approaches, and Open Problems Bharath Shamasundar, Negar Daryanavardan, and Aria Nosratinia Abstract Reconfigurable intelligent surfaces (RIS) are passive controllable arrays of small reflectors that direct elec- tromagnetic energy towards or away from the target nodes, thereby allowing better management of signals and interference in a wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The RIS has the potential for significantly improving the performance of wireless networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Unfortunately, RIS also multiplies the number of Channel State Information (CSI) coefficients between the transmitter and receiver, which magnifies the challenges in estimating and communicating the channel state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Furthermore, the simplicity and cost-effectiveness of the passive RIS also implies that the incoming links are not locally estimated at the RIS, and fresh pilots are not inserted into outgoing RIS links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This introduces new challenges for training and estimation of channel state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The rapid growth of the literature on CSI acquisition in RIS-aided systems has been accompanied by variations in the underlying assumptions, models, and notation, which can obscure the similarities and differences of various techniques, and their relative merits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This paper presents a comprehensive exposition of principles and approaches in RIS channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The basic ideas underlying each class of techniques are reduced to their simplest form under a unified model and notation, and various approaches within each class are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Several open problems in this area are identified and highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' INTRODUCTION The fifth generation (5G) wireless systems are being successfully deployed across the globe, achieving high data rates using large antenna arrays [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 6G systems will aim for even higher data rates, lower latency, and better reliability, at low cost/complexity and high energy efficiency [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This requires innovations in the physical layer technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The continuing migration to the millimeter wave (mmWave) spectrum, while improving data rates, has challenges including channel blockage and intermittent availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Increasing the network density can solve some of these problems, but it injects more power into the network, increases the interference levels, escalates cost of deployment and operation, and raises scalability concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Reconfigurable intelligent surfaces (RIS) are passive, controllable arrays of small reflectors that direct radio waves toward or away from a target node, enabling better management of signals and interference in wireless networks [5]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' They are often interpreted as a mechanism that achieves software-defined control of the wireless This work was supported in part by the National Science Foundation under Grants 1956213 and 2148211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='05026v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='IT] 12 Jan 2023 2 Controller 0 2π RIS RIS Control Link Transmitter Receiver Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Schematic representation of reconfigurable intelligent surface aided communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' propagation environment (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This approach has the potential to address several of the challenges mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Judiciously altering the channel characteristics in real-time as a part of the system operation, to achieve favorable propagation environment, is an attractive idea to handle challenging channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Unlike a relay, RIS does not inject more transmit power into the network, and its operation is independent of the details of PHY signaling other than operating frequency and bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The combination of lower power consumption and simpler construction makes it cheaper to build and operate, thus helping scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' RIS-induced channel coefficients must be estimated at the receiver for coherent communication, and shared with the transmitter (and RIS) for beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The RIS channel estimation problem is distinct from the MIMO (multiple-input multiple-output) case because: (1) RIS is a two-hop channel observable only end-to-end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' due to passivity of RIS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (2) the number of channel gain coefficients is multiplied by the RIS size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' making for a larger vector to be estimated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' and (3) in RIS channel estimation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' training occurs through pilots plus passive RIS training states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' and the latter is without a direct counterpart in traditional channel training and estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Contributions and Distinctions of the Present Work The present work provides a thorough exposition of the ideas underlying the rapidly expanding literature on RIS channel estimation, in a way that is both comprehensive and accessible to a wider audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' An up-to-date discussion of the available estimation techniques and related issues, ranging from classical least squares to the most recent artificial intelligence and machine learning (AI/ML)-based methods, is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' One of the contributions of the present work is to bring a unified notation and system model to the mathematical expression of various RIS channel estimation problems and algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This goes beyond the enumeration of works in a conventional survey, and is greatly helpful in the interpretation and comparison of results in a rapidly expanding literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Among other features, this work illuminates the commonality and differences between RIS channel estimation results, thus exposing synergies and facilitating the generation of new ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 3 In addition to the exposition of various estimation approaches, this paper also explores critical assumptions, explicit or implicit in RIS channel estimation, that are in need of further investigation and validation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', channel reciprocity or perfect deactivation of RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Other important issues, such as the near field effect in very large RIS, or the dependence of reflection gains on phases, are also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To put this paper in context and highlight its distinctions, we briefly review related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In a concise letter, Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [10] explored sparsity, user correlation, and time scales for RIS channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Swindlehurst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [10] consider the identifiability of the models as a function of the pilots and RIS training states, and further consider special cases such as single-input single-output (SISO) and MIMO, availability or unavailability of a direct link, and narrowband vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' wideband.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Noh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [11] concentrates on characteristics of RIS channels in terahertz (THz) and mmWave channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [12] provides a survey of RIS channel estimation that enumerates the problems and outcomes, but does not dwell on methodology or characterization of the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In comparison, the present work provides a simple yet sufficiently detailed exposition of ideas for a wider audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Compared with the works mentioned above, the present work also addresses new dimensions, including: the impact of RIS channel estimation overhead on spectral efficiency, and the implications of optimizing spectral efficiency (through channel estimation constraints) on the size of RIS array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The present work also includes coverage of machine learning methods for RIS channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Also, as mentioned earlier, underlying assumptions with potential impact on practical implementations are also examined in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This paper also presents several new interpretations and connections that have been unavailable in the literature thus far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Among them, (a) Section VII-A provides an elegant explanation for available savings in multiuser RIS systems, leading to suggestions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (b) Section VII-C discusses the differences and similarities of techniques that infrequently update the RIS coefficients, compared with traditional MIMO statistical CSI methods, opening venues for future work, and (c) Section VII-D clarifies for the first time that the so-called “codebook methods” for RIS actually have deep connections with opportunistic communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section II presents RIS system and channel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section III formulates RIS channel estimation problem and presents discussions on pilots, training states, and linear estimation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section IV presents the effect of training overhead and accuracy on the spectral efficiency of RIS, and dis- cusses its impact on the choice of RIS dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section V presents the formulation of RIS channel estimation as a sparse recovery problem when operating in high frequency (mmWave/THz) channels, and discusses the solutions proposed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section VI presents estimation techniques for RIS-aided orthogonal frequency-division multiplexing (OFDM) in wideband channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section VII presents various approaches for reducing estimation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section VIII presents estimation approaches based on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Section IX outlines practical issues of contemporary interest and investigation in RIS channel modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' SYSTEM AND CHANNEL MODELS For easy reference, the mathematical terms and the key variables used in this paper are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='DRAFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='⊛ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='circular convolution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='⊗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Kronecker product ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='⊙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Hadamard product (element-wise product of two matrices) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='⋄ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Khatri-Rao product (column-wise Kronecker product) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='vector of all-ones ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='matrix transpose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='complex conjugate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='conjugate transpose (Hermitian) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='† ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='psuedo-inverse ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='diag(·) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='diagonal matrix built from the argument ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='vec(·) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='concatenation of columns of an M × N matrix into a vector of size MN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='|| · || ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='vector norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Expected value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='RIS reflection coefficient matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' G incoming & outgoing RIS channel Hc end-to-end RIS-induced channel Hd direct (not through RIS) channel Z Training matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' includes pilots & RIS training states TABLE I: Nomenclature A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Narrowband/Frequency-Flat Channels Consider a narrowband point-to-point communication between a transmitter with Mt antennas and a receiver with Mr antennas, assisted by an RIS with N elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let G ∈ CN×Mt denote the channel between the transmitter and RIS, H ∈ CMr×N the channel between the RIS and the receiver, and Hd ∈ CMr×Mt the direct channel between the transmitter and the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The RIS reflection is represented with a matrix Φ = diag(ψ1, ψ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , ψN) whose passive elements ψi = βiejφi have amplitude βi ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, the received signal is given by y = √ρ(Hd + HΦG)x + w, (1) where the transmit signal x ∈ CMt×1 satisfies E∥x∥2 = 1, the transmit power is ρ, and the receiver Gaussian noise is w ∈ CMr×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the microwave L and S bands, and parts of the C band,1 a suitable model of rich scattering involves many multipath components, resulting in independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=') Gaussian gains for the direct and 1The frequency boundaries of the rich scattering model are subject to a number of variables, which are extensively covered in the channel modeling literature [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 5 RIS-aided channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' When a direct line-of-sight (LoS) path exists between any two nodes, a Rician fading model applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In that case, the Tx-RIS channel can be decomposed as follows: G = � Kg 1 + Kg ¯G + � 1 1 + Kg �G, (2) where Kg is the Rician factor, ¯G is the deterministic LoS component, and �G is the random non-LoS component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Recently, RISs have been studied in higher frequencies (mmWave and THz) where the channels are represented in parametric form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' for example, a transmitter-RIS channel with L resolvable paths is given by G = L−1 � ℓ=0 αℓa2(φ2,ℓ, θ2,ℓ)aH 1 (φ1,ℓ, θ1,ℓ), (3) where αℓ is the complex gain of path ℓ, a1(·) and a2(·) are steering vectors at the transmitter and RIS, respectively, with φ1,ℓ(θ1,ℓ) and φ2,ℓ(θ2,ℓ) being azimuth (elevation) angles of departure and arrival for the path ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In matrix form: G = A2 diag(α) AH 1 (4) This matrix representation will be revisited in Section V for channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Wideband/Frequency-Selective Channels Consider an RIS-aided communication between a single antenna transmitter and receiver (for ease of exposition) in frequency selective channels using OFDM modulation with Mc orthogonal sub-carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let L denote the number of channel taps for both the direct and cascaded channels2, with hd = [hd(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , hd(L−1), 01×(Mc−L)]T ∈ CMc×1 being the zero-padded (time-domain) direct channel vector, gn = [gn(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , gn(L − 1), 01×(Mc−L)]T the channel between transmit antenna and the RIS element n, and hn = [hn(0), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , hn(L − 1), 01×(Mc−L)]T the channel between RIS element n and receive antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let qx be the Mc × 1 transmit frequency-domain OFDM symbol and x be its Mc-point inverse discrete Fourier transform (IDFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' At the OFDM transmitter, qx is first transformed to x and transmitted over the frequency-selective channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The signal reaching the receiver due to direct path is given by x⊛hd, where ⊛ denotes the circular convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The signal reaching the receiver via reflection from RIS element n is ψn(x ⊛ gn) ⊛ hn, where ψn is the induced reflection coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The overall received signal due to the direct path and the RIS reflections is y = √ρ � N � n=1 ψn(x ⊛ gn) ⊛ hn + x ⊛ hd � + w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (5) The OFDM receiver performs DFT operation on y to obtain qy = FMcy, where FMc is the Mc × Mc DFT matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let qhd = FMchd denote the frequency domain Tx-Rx channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Similarly, let qgn and qhn denote the frequency domain RIS-aided channels due to element n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, the received signal in the frequency domain is given by qy = √ρ � N � n=1 ψn(qx ⊙ qgn) ⊙ qhn + qx ⊙ qhd � + w = √ρX � N � n=1 ψn(qgn ⊙ qhn) + qhd � + w, (6) 2Assumed for ease of exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Extensions for the case where direct and cascaded channels have a different number of paths is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 6 where X = diag(qx) and ⊙ denotes the Hadamard (elementwise) product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The channel gains can follow either rich or sparse scattering models depending on the propagation environment and frequency of operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' PILOTS, TRAINING STATES, AND LINEAR ESTIMATION A narrowband RIS-aided system is modeled by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (1), or alternatively, y = √ρ(xT ⊗ IMr)vec(Hd + HΦG) + w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (7) Recall ⊗ and vec(·) denote Kronecker product and vectorization, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Define Hc ≜ [vec(Hd) GT⋄ H] (8) where ⋄ is a columnwise Kronecker product, which is a special case of the Khatri-Rao product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' We further define hc ≜ vec(Hc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' We organize the individual reflection coefficients ψ into a vector ψ, which carries the same information as the diagonal matrix Φ = diag(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Further, define ˜ψ ≜ � �1 ψ � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (7) is expressed as follows y = √ρ(xT ⊗ IMr)Hc ˜ψ + w = √ρ(˜ψ T ⊗ xT ⊗ IMr)hc + w ≜ √ρZhc + w, (9) where Z represents a matrix that includes the transmit vector as well as the RIS training state, hc represents the overall channel (including both cascaded and direct channels), whose estimation is necessary for coherent detection at the receiver and beamforming at the transmitter and RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The transmission frame includes a training phase and a data transmission phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' During the training phase, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , J, the pilot signals xj and RIS training states ψj give rise to the overall training matrix Zj = ˜ψ T j ⊗ xT j ⊗ IMr, at the receiver resulting in yj = √ρZjhc + w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Linear Estimation Since yj is Mr-dimensional and hc is MrMt(N + 1)-dimensional, the least squares and minimum mean-square error (MMSE) estimation requires at least J = Mt(N + 1) pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Define: ˜y ≜ � ���� y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' yJ � ���� �Z ≜ � ���� Z1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' ZJ � ���� Then, the least squares estimate of hc is given by ˆhc = 1 √ρ �Z†˜y, (10) where �Z† is the pseudo-inverse of �Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let Rhc denote the covariance matrix of hc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, the LMMSE estimate of hc is given by ˆhc = √ρRhc �ZH(ρ�ZRhc �ZH + IMrJ)−1˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (11) January 13, 2023 DRAFT 7 Start Training i = 1 Set RIS training state i Transmit orthonormal pilots M t i=N +1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Yes LS/LMMSE Estimation i=i+1 No Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Flowchart representing the channel training process in RIS-aided MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Training Process: As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 2, the channel training occurs by the RIS assuming a training state ψi that remains fixed over Mt consecutive slots, and the transmitter emitting Mt linearly independent (preferably orthonormal) pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This process repeats N + 1 times with different RIS training states3 that generate linearly independent extended training vectors �ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' RIS Training States The accuracy of channel estimates depends on the choice of RIS training states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' We consider the canonical, DFT, and Hadamard training states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For simplicity of exposition, it is assumed that the direct path is absent, and the channel gains follow i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' CN(0, 1) resulting in Rhc = IMtMrN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The following identity is used in the derivation of least squares and MMSE estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' �ZH �Z = Mt(ΨΨH)∗ ⊗ IMrMt, (12) where Ψ ≜ [ψ1 · · · ψN].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 1) Canonical Training: Canonical training activates one RIS element in each training state, deactivating the remaining N −1 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='4 Thus, the RIS training vectors constitute a so-called standard basis or canonical basis, as follows: ψi = ei ≜ [δi,1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' δi,N] i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , N where δi,j is the Kronecker delta function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This results in ΨΨH = IN, and hence, by the identity in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (12), ˜ZH ˜Z = MtIMtMrN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Therefore, from (10), the least squares estimate with canonical training states is ˆhc = 1 Mt√ρ �ZH ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (13) 3because of N RIS elements and one direct path, constituting N + 1 degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 4Nulling the reflection of an RIS element with respect to all angles has not been adequately addressed in the literature and remains an open issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' See Section IX for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 8 From (11), the MMSE estimate with canonical training is ˆhc = 1 Mt√ρ � 1 + 1 ρMt � �ZH ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (14) 2) DFT Training: In DFT training, each RIS training state is a column of the standard N × N DFT matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The orthogonality ΨΨH = NIN combined with the identity (12) gives �ZH �Z = MtNIMtMrN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The least squares estimate with DFT training states is therefore ˆhc = 1 MtN√ρ �ZH ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (15) The MMSE estimate is ˆhc = 1 MtN√ρ � 1 + 1 ρMtN � �ZH ˜y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (16) 3) Hadamard Training: Another choice of RIS training states is to use the columns of N ×N Hadamard matrix, which again results in ΨΨH = NIN and therefore �ZH �Z = MtNIMtMrN via the identity (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The least squares and MMSE estimators with Hadamard training are the same as with DFT training and hence are omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (Ambiguity Problem) For any diagonal matrix D ∈ CN×N, G′ ≜ D−1G, and H′ ≜ HD, HΦG = H′ΦG′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (17) Therefore, G and H cannot be uniquely resolved by observing pilots via the channel HΦG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' However, as noted from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (9), the knowledge of cascaded channel hc is sufficient for RIS beamforming, while also being more efficient to estimate compared with estimating G and H separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (Channel Training under Finite Precision Reflection Coefficients) The precision of RIS coefficients may be limited in practical implementations, with only a finite number of quantized phase shifts being available, affecting both training and beamforming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Quantization of phase shifts has no effect on channel estimation under canonical training states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Hadamard training requires only 1-bit phase precision, without any compromise in estimation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' DFT-training, however, requires N phase shifts, which may not be available in practice for a large RIS array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' When L phase shifts (L < N) are available at the RIS, a quantized-DFT training can be achieved by mapping the phases of the DFT matrix FN to the nearest phases in the quantized phase set P to obtain quantized-DFT matrix Fq,N as follows: ∠Fq,N(k, ℓ) = argminφ∈P|e−jφ − e−j 2π(k−1)(ℓ−1) N |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (18) Unfortunately, the quantized-DFT matrix is non-orthogonal, resulting in degraded channel estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This is espe- cially an issue for low-precision RIS implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (Grouping the RIS Elements) Under some scenarios, the estimation of all channel gain parameters induced by the RIS may be prohibitive in terms of time, power, or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A remedy has been proposed [16]–[23] that constrains groups of RIS elements to have the same reflection coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, it is not difficult to see that the relevant estimation parameter is an January 13, 2023 DRAFT 9 aggregate channel gain corresponding to the total reflection produced by the RIS elements in each group (that have the identical reflection coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This scenario is effectively similar to an RIS with fewer (virtual) reflective elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Each of these virtual elements, representing multiple physical elements, will have a stronger reflection and therefore a stronger channel gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' An example of this kind is discussed in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' ESTIMATION VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' SPECTRAL EFFICIENCY Because the RIS channel is not known ahead of time, channel resources must be spent to train and acquire the channel state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Pilots require transmit power, and transmission time is occupied for generating independent channel observations, commensurate with the number of channel parameters being estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Any channel resource used for training becomes unavailable for data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A larger RIS can improve beamforming gain that is beneficial for capacity, but also requires more training resources, which is detrimental for capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This gives rise to an interesting and important tradeoff in the size of RIS and its effects on spectral efficiency, studied in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This section presents the training-based spectral efficiency results for RIS-aided single-antenna transmitters and receivers without a direct path, whose insights carry over to multiple antenna systems as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The developments in this section follow [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let T be the coherence interval of all channels, also used as block length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Td channel symbols are dedicated to data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the absence of a direct path, a minimum of N temporal degrees of freedom are needed for training, T = N + Td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let ρτ and ρd denote the training and data powers, respectively, and let ρ denote the average power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, by conservation of energy: ρT = ρτN + ρdTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With these conditions, the following rate is achievable under canonical training [24]: Rτ = � 1 − N T � Eˆhc log2 � 1 + ρd � �N i=1 |ˆhc(i)| �2 1 + Nρd 1+ρτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (19) Under DFT training and Hadamard training, the following rate is achievable: Rτ = � 1 − N T � Eˆhc log2 � 1 + ρd � �N i=1 |ˆhc(i)| �2 1 + Nρd 1+Nρτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (20) The spectral efficiency is a function of ρτ and ρd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' the optimizer of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (19) is ρdTd = β∗ 1ρT and ρτN = (1−β∗ 1)ρT, with β∗ 1 = �� 1 + ρT N �� 1 + NρT T −N � − � 1 + ρT N � � NρT T −N − ρT N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (21) Similarly, the optimizer of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (20) is ρdTd = β∗ 2ρT and ρτN = (1 − β∗ 2)ρT, with β∗ 2 = � (1 + ρT) � 1 + NρT T −N � − (1 + ρT) � NρT T −N − ρT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (22) January 13, 2023 DRAFT 10 10 5 0 5 10 15 20 25 30 2 4 6 8 10 12 14 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Training-based bounds on capacity with canonical and DFT training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 10 5 0 5 10 15 20 25 30 15 20 25 30 35 40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Optimum RIS size that maximizes spectral efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Figure 3 shows the training-based bounds on the capacity of RIS-assisted system with 32 RIS elements when the channel coherence interval is T = 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The spectral efficiency with DFT training is higher compared with canonical training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='5 Specifically, with equal power allocation between training and data, DFT training achieves a gain of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='5 bits/s/Hz compared with canonical training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With optimal power allocation for both, DFT training achieves a gain of 2 bits/s/Hz over canonical training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The reason for under-performance of canonical training is that the magnitude of RIS training states multiplies the pilot power, therefore the zero coefficients in canonical training reduce the received signal-to-noise ratio for pilots, and induce a penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' On the other hand, DFT training activates all the RIS elements in each training time slot, thereby efficiently utilizing the available pilot power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 5Hadamard training capacity expressions and numerical results are the same as DFT training, and are omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 11 Figure 4 shows the RIS array size that maximizes the spectral efficiency, at each signal-to-noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' At low-SNR, power is at a premium, while degrees of freedom are less important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Therefore, it is beneficial to estimate the channel induced by the entire (available) RIS array, even though the training requires degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Conversely, at high-SNR, degrees of freedom are more important, therefore from a capacity perspective it may be beneficial to utilize only part of an available RIS, so that fewer time slots are utilized for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' SPARSE CHANNEL ESTIMATION Whenever the channel has a sparse multipath structure, fewer channel parameters need to be estimated, and the overhead incurred in transmission of pilots and feedback of channel coefficients is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This is especially relevant for higher frequencies (mmWave/THz) wherein the RIS has the most impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To capture the efficiencies arising from sparse channel structure, the RIS channel estimation is cast in the form of sparse vector recovery and solved with compressive sensing algorithms that reduce the number of required channel measurements compared with traditional channel estimation [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Sparse multipath channels are characterized by a geometric model involving angles of arrival/departure and complex gains of the signal paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The goal of sparse channel recovery is to estimate the parameters of the angular representation of the channel described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' II-A, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', G = A2 diag(α) AH 1 which involves angles of arrival/departure captured in the left and right matrix, and the path strengths captured in the diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Even though G might be (highly) rank deficient, it is not (yet) expressed in a suitable format for compressive sampling algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (4), the basis vectors (columns of A1 and A2) can take values over an uncountably infinite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To recast the problem in a friendly format for compressive sampling, the candidate angles are restricted to a finite set of size G, often corresponding to a uniform grid in a prescribed coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The basis vectors6 corresponding to the discretized angles are collected into dictionary matrices �A1 ∈ CMt×G and �A2 ∈ CN×G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With sufficiently good quantization of angles, matrices A1 and A2 are approximately7 submatrices of �A1 and �A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A sparse matrix Λg can select the appropriate columns from �A1 and �A2 so that: G = A2 diag(α)AH 1 ≈ �A2 Λg �AH 1 (23) The problem is now in a standard form for compressive sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With pre-determined dictionaries �A1 and �A2, the objective is to estimate the sparse matrix Λg from a noisy linear observation of G (via pilots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' If the transmitter-to-RIS channel G has L paths, the discretized grid of angles must have G ≫ L elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Ignoring for now any grid mismatch issues G = �A2Λg �A H 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 6also known as steering vectors 7because the true AoA/AoD may not fall exactly on the quantized grid of angles January 13, 2023 DRAFT 12 Similarly, let P denote the sparsity of RIS-to-receiver channel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Consider a discretized set of candidate angles with size H ≫ P, and collect the steering vectors corresponding to these angles into dictionary matrices �B1 ∈ CN×H, �B2 ∈ CMr×H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, H = �B2Λh �B H 1 , where Λh is a H × H sparse matrix with P non-zero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For simplicity of exposition, we assume no direct path exists between transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Thus, the received signal in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (1) takes the form y = √ρ �B2Λh �B H 1 Φ �A2Λg �A H 1 x + w = √ρ ¯Hx + w = √ρ(xT ⊗ IMr)vec( ¯H) + w, (24) where ¯H ≜ �B2Λh �B H 1 Φ �A2Λg �A H 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Using a series of vectorization operations, it can be shown that vec( ¯H) = ( �A ∗ 1 ⊗ �B2)(ΛT g ⊗ Λh)( �A T 2 ⋄ �B H 1 )ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Substituting in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (24), y=√ρ(xT ⊗ IMr)( �A ∗ 1 ⊗ �B2)(ΛT g ⊗ Λh)( �A T 2 ⋄ �B H 1 )ψ+w = √ρK(x, ψ)λ + w, (25) where λ ≜ vec(ΛT g ⊗ Λh) is the (GH)2 × 1 sparse vector with LP non-zero elements, and K(x, ψ) ≜ (( �A T 2 ⋄ �B H 1 )ψ)T ⊗ ((xT ⊗ IMr)( �A ∗ 1 ⊗ �B2)) is the effective measurement matrix which is a function of pilots and RIS training states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' During the training phase, the input sequence (xj, ψj) takes J distinct values, and the output sequence yj is observed: � ���� y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' yJ � ���� = √ρ � ���� K(x1, ψ1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' K(xJ, ψJ) � ���� λ + � ���� w1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' wJ � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (26) The sparse channel vector λ can be reconstructed from the measurements in (26) using standard sparse recovery algorithms such as orthogonal matching pursuit (OMP) [25] and subspace pursuit [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Alternating direction method of multipliers (ADMM) [28] and approximate message passing [29] have also been explored for sparse channel estimation in RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Noh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [30] show that, for an RIS-aided single antenna system employing J pilots (J < N) for sparse channel estimation, using the J equi-spaced columns of the N ×N DFT matrix as training states produce lower mean squared error compared with canonical training states and the first J columns of the DFT matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Training Overhead and Complexity: To reconstruct an LP-sparse vector of length (GH)2, orthogonal matching pursuit requires O(LP log(GH)) measurements [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Since each pilot provides Mr measurements, the required number of pilots for sparse RIS channel estimation is O � LP Mr log(GH) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Subspace pursuit requires even fewer measurements;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' specifically, it requires O(LP log( GH √ LP )) measurements [31], and hence O( LP Mr log( GH √ LP )) pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In general, L and P are small at high frequencies, and hence the contribution of LP to the training overhead is January 13, 2023 DRAFT 13 small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Also, due to the logarithmic dependence on GH, the induced overhead is low, especially when Mr is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A thorough characterization of optimal training with sparse recovery, and the associated training-based capacity (as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' IV) is still open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The complexity of estimation using orthogonal matching pursuit (and subspace pursuit) is O((LPGH)2 log(GH)), while linear estimation has a complexity of O((MtMrN)3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With suitable choice of G and H, sparse recovery can reduce the complexity when compared with linear estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Beyond sparsity, other structural properties can be exploited for further reducing the training overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For broadband channel estimation in RIS-aided mmWave massive MIMO systems, Wan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [32] exploit the common sparsity shared by different sub-carriers and propose a distributed orthogonal matching pursuit to reduce the overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the context of an RIS-aided multiuser downlink setting, Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [26] show that the angular cascaded channels associated with different users have exactly the same non-zero rows and some common non-zero columns (termed as double structured sparsity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The adaptation of orthogonal matching pursuit to this double-sparse structure is shown to further reduce overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [33] consider uplink channel estimation in RIS-aided mmWave massive MIMO and exploit the fact that in many scenarios the angles of arrival/departure between RIS and base station remain unchanged over multiple coherence blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Therefore, the base-station to RIS channel parameters, which are more numerous in massive MIMO, need fewer updates, which can reduce pilot overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [34] decompose the sparse channel recovery into three components: recovery of angle of arrival, angle of departure, and complex gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A semi-passive RIS with a few receiver chains at the RIS was proposed by [35], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A semi-passive RIS allows for receiving pilots and channel estimation at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The transmitter-RIS channel is estimated with the aid of the few RIS on-site measurements, and utilizing compressive sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Matrix factorization and matrix completion [37] can also be used for estimating rank-deficient, sparse RIS channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The key idea of this approach can be explained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With pilots Xτ = [x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' xJ], RIS training states Ψτ = [ψ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' ψJ], and received pilots Yτ = [y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' yJ], the system model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (1) becomes [37] Yτ = √ρH(Ψτ ⊙ (GXτ)) + N, (27) where N ∈ CMr×J is the additive noise matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Equation (27) can be equivalently written in the factored form as Yτ = √ρHA + N, (28) where A ≜ Ψτ ⊙ (GXτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With this representation, the estimation is achieved using a two stage process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the first stage (matrix factorization), the matrices H and A are estimated based on Yτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the second stage (matrix completion), G is estimated based on the estimate of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The success of this method requires A to be sparse and H to be a low-rank matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The sparsity of A can be satisfied by selecting the RIS training matrix Ψτ to be sparse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', most of its coefficients set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' High frequency (mmWave/THz) channels H have low rank due to dominance of reflections over scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' He and Yuan [37] achieve the matrix factorization step using the bilinear generalized approximate message passing algorithm [38], and the matrix completion step using Riemannian manifold gradient-based algorithm [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Other methods based on similar ideas can be found in [40]–[44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The majority of the literature on sparse channel estimation assume that the true angles of arrival/departure lie on a discretized grid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', on the discrete steering angles of the dictionary matrices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In practice, when the angles January 13, 2023 DRAFT 14 of arrival/departure do not coincide with the discrete angles in the dictionary, the sampling process leads to many non-zero sample measurements, degrading the sparse recovery algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The sensitivity of sparse recovery to grid mismatch was systematically analyzed in [45], but this analysis has not been widely adopted in the sparse channel estimation literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In an alternative approach, He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [46] propose atomic norm minimization for RIS channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Atomic norm is a convex function that generalizes the ℓ1 norm for sparse recovery and nuclear norm (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', sum of singular values) for low-rank matrix completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Atomic norm minimization works in the continuous domain and avoids discretization, therefore eliminating the grid mismatch problem [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Its solution is often via semidefinite programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' WIDEBAND CHANNEL ESTIMATION The RIS-aided OFDM model was discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' II-B, where the frequency domain input-output relation was provided by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Based on this model, the present section provides the main ideas involved in the channel estimation for RIS-aided OFDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The system model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (6) can be equivalently written as qy = √ρX � Bψ + qhd � + w, (29) where B is an Mc × N matrix whose columns are qgn ⊙ qhn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The Mc × T frequency-time frame is divided into two sub-frames: a training sub-frame of size Mc × (N + 1) and data-transmission sub-frame of size Mc × (T − N − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the training sub-frame, the received pilot sequence is given by qyj = √ρXj � Bψj + qhd � + wj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Defining Ψ = � ψ1 · · · ψN+1 � , and using the pilot sequence Xj = IN for j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , N + 1, the received training sequence qY = [qy1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , qyN+1] is given by qY = √ρ � qhd B � � �1T Ψ � � + W, where W = [w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , wN+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For convenience of notation we define: C ≜ � qhd B � �Ψ ≜ � �1T Ψ � � resulting in qY = √ρ C �Ψ + W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (30) From this, the matrix C containing the direct and cascaded channels can be estimated as �C = 1 √ρ qY �Ψ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (31) It has been shown in [18] that choosing �Ψ = FN+1 results in the least error variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 15 N+1 T-(N+1) MC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' : Pilot Symbol : Data Symbol Transmission Frame Subframe 1 Subframe 2 Frequency Time Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Transmission frame structure in RIS-aided OFDM [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the above method, pilots were inserted on all the Mc sub-carriers of the N + 1 training OFDM symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In practice, when the channel is correlated in the frequency domain, fewer pilots may be employed and then channel estimates may be interpolation among sub-carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='8 The system model in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (29) is equivalent to: qy = √ρXq + w, (32) where q ≜ Bψ + qhd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Now, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 5, in the training sub-frame, Np pilots (Np < Mc) are inserted in each OFDM symbol with a spacing of ∆ = ⌊ Mc Np ⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let P = {0, ∆, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , (Np − 1)∆} denote the indices of the sub-carries containing pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let qxP and qyP denote the transmitted and received pilots on sub-carriers indexed by P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Also, let XP = diag(qxP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, the estimate of qP can be obtained as ˆqP = 1 √ρX−1 P qyP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Using ˆqP, the estimate of q (denoted by ˆq) is obtained via interpolation along the subcarriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The work in [18] applies the DFT/IDFT-based interpolation on the pilot sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Now, in order to resolve B and qhd from ˆq, the RIS training states ψj are adjusted during each training OFDM symbol and the corresponding ˆqj is given by ˆqj = Bψj + qhd + vj, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , N + 1 where vj is the error in estimating q during the pilot slot j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Let �Q = [ˆq1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' ˆqN+1] and V = [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , vN+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Then, �Q = C �Ψ + V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Using the above equation, the estimate of matrix C containing direct and RIS-aided channels can be obtained as �C = �Q �Ψ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (33) 8This is a long-standing practice that has been adopted in 3GPP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 16 To further reduce the estimation overhead, one may group the neighboring RIS elements and use the same reflection coefficient for all the elements in each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This solution has been suggested for both flat and frequency selective channels [16]–[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' If the N RIS elements are divided into Ng groups (Ng < N) with each group containing N/Ng elements, then the channel estimation requires estimating only Ng aggregated RIS-aided channels corresponding to each group, instead of estimating all the N cascaded channels corresponding to each RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Therefore, the training overhead is reduced from N +1 OFDM symbol durations to Ng+1 OFDM symbol durations, where each OFDM symbol duration is composed of (Mc + Lcp) time-slots with Lcp being the length of the cyclic prefix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The grouping strategy trades-off accuracy for overhead in order to improve the overall spectral efficiency, which is analytically characterized in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [48] extend this estimation technique to multiuser orthogonal frequency-division multiple access (OFDMA) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The same authors [49] propose a fast channel estimation scheme for reducing the training-overhead in RIS-aided OFDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The key idea is to use short OFDM symbols of M ′ c sub-carriers (L ≤ M ′ c ≪ Mc) during the training phase, which consumes (M ′ c + Lcp) time-slots per OFDM training symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This reduces the (Mc + Lcp) pilots that were required by [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The above works assume an ideal reflection model in which the RIS elements achieve the same amplitude and phase response across the entire OFDM band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Wenhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [50] show that the practical response of RIS is tightly related to the frequency of the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Based on this, Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [51] studies channel estimation for RIS-aided OFDM under a practical reflection model and finite precision coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' While differing in its modeling, the estimation technique of [51] is similar to [18], as outlined above, and is omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' REDUCING THE RIS ESTIMATION OVERHEAD We begin by exploring savings in pilots and estimation overhead that arise from the multi-user nature of the RIS channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' An RIS-aided uplink system with K single-antenna users and M-antenna base station (BS) must estimate KMN + KM links for RIS beamforming and equalization, which can be prohibitive either under massive MIMO, or in large cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Linear estimation techniques (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' III) require K(N +1) pilot transmission slots, growing linearly with the size of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' We discuss avenues for reducing the pilot overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Common RIS-BS Channel Consider a scenario where a base station is aided by a single RIS for communication with multiple users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To describe channel estimation in this multi-user scenario, we adapt the system model from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (1) for the multi-user uplink channel: y = K � k=1 √ρ(hdk + HΦgk)xk + wk, (34) where hdk is the direct channel from a single-antenna user k to a multi-antenna base station, and gk ∈ CN is the channel from the user k to RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Recall that the combined (direct and cascaded) RIS-aided channel gains to be estimated were collected into a single matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' a specialization of that matrix for the case of a single-antenna user is given by: Hck = [hdk H diag(gk)] (35) January 13, 2023 DRAFT 17 Among the quantities participating in this expression, the two vectors hdk and gk are distinct for different users, but the BS-RIS matrix H is common between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The user-by-user uplink channel estimation requires one pilot for estimating the direct channel and N pilots for estimating the cascaded channel, for a total of K(N + 1) total pilot transmission slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' But the commonality of H among users hints at possible savings in the total number of needed pilot slots, which we now explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To begin with, the direct channels for all users is estimated, by deactivating the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This requires one pilot slot per user, but this step may not be crucial, because it is often the absence of a direct path that makes the RIS an attractive choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the next step, the cascaded channel (H diag(g1)) is measured by emitting N pilots from User 1 to the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This is accomplished via N successive training states at the RIS, whose details are omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For User 2, we now need to measure (H diag(g2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This new matrix has columns that are co-directional with columns of (H diag(g1)), thus only the magnitude of each column needs to be measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For N columns, this requires N new observation samples, however, reception at the multi-antenna base station provides M independent observations per pilot transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Therefore, after obtaining (H diag(g1)), only N M pilot transmissions are needed per additional user, as long as training states are designed properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The design of training states that ensure the requisite linearly independent observations has been explained in [52], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' When M > N, following the above argument, it is easy to see that one pilot per user is sufficient for estimating the cascaded channels of Users 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Therefore, the total training overhead of this scheme is given by J = K + N + max � K − 1, �(K − 1)N M �� , (36) where ⌈·⌉ denotes the smallest integer bigger than or equal to the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For massive MIMO systems with M > N, the overhead J = 2K + N − 1, meaning that each user beyond the first one requires two pilot slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This provides significant savings over the N+1 slots needed conventionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Guan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [54] propose a slight modification of this technique, in which a few stationary nodes called the anchor nodes are assumed to exist in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The anchor nodes transmit pilot signals and the base station estimates anchor-RIS-BS channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Due to the common RIS-BS channel, the User-RIS-BS channels are subsequently estimated with fewer pilot transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Since the anchor nodes are stationary, the estimation of anchor-RIS-BS channels are done less frequently compared with the earlier, single-reference user in [52], [53] which was not assumed to be stationary, thus resulting in additional savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Guo and Lao [55] also explore the possibility of exploiting the common RIS-BS channel without requiring a reference user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The methods in [52], [53] estimate the direct channel, and subsequently subtract it from the measurement intended for the cascaded channel, which is not MMSE optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A joint estimate of [hd1 H diag(g1)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', User 1’s direct and cascaded channels, has a lower mean squared error (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [56] propose this joint estimation, and then the remaining user channels are estimated via the same technique as [52], [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This modification acknowledges and addresses the propagation of the error in the estimation of hd1 when estimating the cascaded channel of User 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The estimate of H is used for constructing the cascaded channels of other users too, therefore in a sense, the errors committed in estimating the channel of User 1 can propagate January 13, 2023 DRAFT 18 into the estimation of other users’ channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' However, since User 2 and subsequent users employ fewer pilots than User 1, it is not obvious that their channel measurements can be used to improve the estimate of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Slowly Varying BS-RIS Channel Since the BS and RIS are static, the channel between them varies slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In comparison, the BS-user and RIS-user channels are more dynamic because of the mobility of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The high dimensional, but slowly varying BS-RIS channel can therefore be estimated less frequently, while the low-dimensional BS-user and RIS-user channels are estimated more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To isolate the estimation of BS-RIS link, [57] assumes a full-duplex base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The base station will emit pilots and listen for the reflection from the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The self-interference of the full-duplex reception must be dealt with, and the BS-RIS channel recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Given the BS-RIS channel estimate, the direct BS-user channel and the RIS-user channel are estimated conventionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The latter estimates are more frequent, but also require smaller overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' If TL denotes the coherence time of the BS-RIS channel and TS denotes the coherence time of the BS-user and RIS-user channels such that TL = αTS, then the overhead of the two-timescale method is J = 2(N + 1) α + K � N M � + K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In practice, α ≫ 1 and hence the first term is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For massive MIMO systems with many base station antennas M > N, the overhead becomes 2(N+1) α +2K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', after estimating the BS-RIS channel, each user needs two pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Under α > 2, this method has smaller overhead compared with the method of Section VII-A, although one must be careful that the two methods address different channels and different base station capabilities, so they are not directly comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Infrequent RIS Coefficient Updates Another source of potential savings in RIS induced channels is to deliberately reduce the frequency with which RIS reflection coefficients are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' As long as RIS coefficients are not updated, the RIS blends into the channel and effectively the system is reduced to a (multi-user) MIMO system, with conventional channel training and pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Of course, this involves a tradeoff: fewer pilot slots are needed, but also, the match of RIS coefficients to the channel will go stale, therefore part of the beamforming gains of RIS will be lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Ideally, an analysis of this situation requires a temporally varying channel model with a corresponding temporal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' However, the work in this area has taken a different direction, via considering a channel model, with line-of-sight and rich scattering components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For the Tx-RIS channel G, this means the Rician model G = � Kg 1 + Kg ¯G + � 1 1 + Kg �G, which we also saw in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A similar model is utilized for the RIS-Rx channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' It is assumed that the infrequent update of RIS is able to fully capture the line-of-sight component, while not capturing anything about the rich scattering part of the model, even immediately following the pilot transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This approximation is different from the common modeling of temporal variance in most wireless channels, in January 13, 2023 DRAFT 19 which channel knowledge is accurate at times that are proximate to the pilots, but it has the advantage of removing the complexities involved in the temporal dynamics of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Thus, it reduces the problem to an equivalent problem involving a channel state that is partially known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The literature [58]–[62] refers to this new formulation of channel temporal dynamics as statistical channel state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='9 The central idea of [58]–[62] is that the line-of-sight component has a longer coherence interval than the rich scattering component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Single-antenna mobiles estimate the end-to-end uplink channel with a single pilot at the smaller coherence interval, and the base-station beamforming is also updated at the smaller coherence interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' However, the RIS coefficient is updated only at the longer line-of-sight coherence intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This creates significant savings, since most of the pilot slots in the RIS-induced channel, especially for single-antenna mobiles, is needed for estimation and updating of the RIS coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Several works have attempted to maximize the ergodic downlink rates to single-antenna mobiles with beamforming f, either in the single-user or multiple user scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For the single-user case, the received signal is given by y = √ρ(hHΦG + hH d )fs + w, The best beamformer f is found for a given set of RIS coefficients Φ, but then utilize as Φ the (fixed) RIS coefficients that are statistically the best over the variations of the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' C∗=max Φ Eh,G,hd � max f � log2 � 1 + ρ|(hHΦG + hH d )f|2��� (37) Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [58] achieve the inner maximization in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (37) via maximal ratio transmission, and adjust the reflection coefficients based on an outer bound on the ergodic rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [59] maximize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (37) via alternating optimization method, Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [60] uses a penalty dual decomposition method, Zhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [61] achieve minimum user rate maximization via genetic algorithm, and Gan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [62] propose methods based on ADMM, fractional programming, and alternating optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Several important points and open problems remain for consideration in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To begin with, these methods are based on the assumption that the RIS will be changed infrequently, but also calculate and optimize ergodic capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Therefore, the practical implementation of these techniques requires an outer code that goes across many coherence intervals of the slower channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In many such cases, outage capacity or throughput may be a more suitable metric for optimization, and there is room for future work in this area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Another useful direction is to find simplifications and approximations of the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' (37) in order to recognize trends and/or suggest different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In this area, there is a need for achievable rate (inner bound) expressions rather than outer bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Inner bounds for this expression have not been developed at the time of the writing of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Opportunistic RIS Another strategy for reducing the estimation overhead is inspired by an idea that harks back to the concept of opportunistic transmission [65]–[67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Q randomly selected vectors {ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , ψQ} are assigned one-by-one as RIS 9This IRS channel model has weak connections with earlier, well-known work in MIMO channels with statistical CSI [63], [64] that were driven by CSI impairments due to limited feedback or feedback delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 20 Phase I Phase II Pilot Signal Pilot Signal Direct Channel (RIS off) Direct + Cascaded Channel Received Pilots Received Pilots ChannelNet Estimate of Direct Channel Estimate of Cascaded Channel ChannelNet Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' RIS channel estimation using ChannelNet [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' phase vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In each instance, the RIS changes the scattering environment randomly, so there is no beamforming in the usual sense of the word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For each of these Q scattering conditions, the end-to-end multiple-input single-output (MISO) channel is measured using a few pilots, and the best one is chosen for one block of transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' An and Gan [68] propose the above approach in narrowband channels, and study bounds on its ergodic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The set {ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' , ψQ} is called a codebook in [68], however, this is a slight misnomer since this set need not be determined or agreed upon ahead of time, is statistically independent of signals emitted from transmit antennas, and is not needed at the receiver for decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The connection of this class of techniques with opportunistic transmission is evidenced by the appearance of the order statistics of (induced) channels in [68, Proposition 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' An et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [69] extend this idea to OFDM transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' MACHINE LEARNING BASED CHANNEL ESTIMATION Machine learning is being actively investigated for channel estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' this section explores machine learning channel estimation in the context of RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Among the early attempts at using machine learning for RIS channel estimation was a non-parametric con- volutional neural network estimator by Elbir et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [70], applied to RIS-aided downlink mmWave channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The estimation is achieved in two phases (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 6), somewhat similar to other methods seen earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the first phase, the base station transmits pilots while the RIS elements are inactive (turned off) so that the direct channel(s) are estimated at the receivers, each of them operating an instance of the convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the second phase, the RIS assumes (several) training states while the base station transmits pilots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Each of the users employs the received pilots in this phase, and in combination with the estimated direct channels (obtained in the previous phase), produces an estimate of the cascaded channel using the same convolutional neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Under low SNR conditions, this technique claims better performance than a (corresponding) two-phase least squares technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Under high SNR conditions, the neural network technique has a performance ceiling, while the least squares techniques do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The experiments involved a 64-antenna base station, 100-element RIS, 8 single-antenna users, and a geometric channel with 10 paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The neural network has an input layer, an output regression layer providing complex valued channel estimates, three convolutional layers each with 256 3 × 3 filters, two fully connected layers with 1024 and 2048 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The input layer has size √ M × √ M × 3 for direct channel estimation January 13, 2023 DRAFT 21 + Pilot signal Direct + Cascaded Channel Received Pilots Least Squares Estimator Denoising Neural Network Residual Noise Refined Channel Estimate LS Estimate Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' RIS channel estimation via denoising of least squares estimate using neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' and N × M × 3 for cascaded channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The output layer has size 2M × 1 for direct and 2NM × 1 for cascaded channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Post Processing Least Squares Estimates Motivated by image denoising using neural networks, Kundu and McKay [71] model the problem of RIS channel estimation as that of denoising the least squares solution (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Specifically, in the first step, least squares estimate of direct and cascaded channels is obtained using pilots and DFT training states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The obtained least squares estimate is viewed as a noisy version of the original channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This is followed by a post-processing step in which the Denoising Convolutional Neural Network (DnCNN) [72] or Fast & Flexible Denoising Network (FFDNet) [73] are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='10 The least squares channel estimate is the input to the neural network, whose output is an estimate of the least squares estimation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The post-processed estimate is obtained by subtracting the estimate of the least squares estimation error from the least squares estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The optimal minimum mean squared error estimator of the channel gains is the conditional mean, but since the cascaded channel is non-Gaussian, this estimator is non-linear and difficult to characterize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This has been the main motivation mentioned in [71] for a neural network approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' One can infer that the LMMSE estimate, being linear, is akin to a first order term of a Taylor series expansion for the conditional mean estimator, and the neural network attempts to approximate the higher order terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [74] propose convolutional deep residual networks for denoising the least squares solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Nipuni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [75] employ neural network denoising for wideband channel estimation in RIS-aided OFDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Shicong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [36] propose an RIS architecture with a few active elements for initially estimating the low-dimensional channel, compressive sensing reconstruction of the complete high-dimensional channel, and a further refinement with a convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [76] refine the channel estimates produced by orthogonal matching pursuit using deep residual networks in RIS-aided mmWave channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Ye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [77] and Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [78] explore generative adversarial networks for estimation in RIS-aided mmWave massive MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 2023 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='DRAFT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='+ Phase I ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Phase II ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Phase III ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Pilots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Direct Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='(RIS off) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Received ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Pilots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Least Squares ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Estimator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Estimate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Refined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Estimate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Pilots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Direct ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='+ Cascaded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Received ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Pilots ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Least Squares ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Estimator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Partial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Cascaded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Estimate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Denoising Deep ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Residual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Noise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Improved Cascaded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Channel Estimate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Inactive RIS Prediction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Deep Neural Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Predicted ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Channel ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='of Inactive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Elements ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' RIS channel estimation using predictive neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Partial CSI In order to reduce the training overhead in deep learning RIS channel estimation, Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [79] propose a predictive neural network in the context of RIS-aided uplink massive MIMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 8, the proposed method works in three stages: In the first stage, the RIS is turned off and the direct channel is estimated using least squares method, which is further refined using a fully connected neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the second stage, only a part of RIS elements are activated (N1 < N), and the cascaded channels corresponding to the activated RIS elements are estimated using least squares method with N1 × N1 DFT training states and further refined using a denoising convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the third stage, the cascaded channels corresponding to the inactive RIS elements are predicted using a fully connected inactive RIS channel prediction neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A geometric channel model is employed for the base-station to RIS channel where the channel matrix is generated from a geometric model that assigns the same gains and angles of arrival/departure to different RIS elements, with an implicit underlying assumption that the scatterers are far from the RIS and the base station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The proposed estimation needs N1 + 1 pilots instead of N + 1 pilots, reducing the overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [80] propose an ordinary differential equation (ODE) based convolutional neural network for predicting the channel corresponding to inactive RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Shtaiwi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [81] assume a multi-user uplink channel in which the channel of different users is highly correlated, so that only a few users need to transmit pilots, and the channel of the remaining users may be predicted from the first few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A neural network is employed for the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In RIS-aided uplink communication, Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [82] uses spatial correlation between the channels of different RIS elements, as well as temporal correlation of time-varying channels, to reduce the communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For exploiting spatial correlation, some RIS elements are turned off, hence corresponding channels are not directly 10These neural networks are imported from the image denoising literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 23 estimated by pilots, but are interpolated from other RIS channel estimates using a convolutional neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' For temporal correlation, a recurrent neural network is used to interpolate the channel values between two pilot transmissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' FRONTIERS OF RIS CHANNEL MODELING Powerful channel estimation techniques depend on accurate, yet convenient, channel models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' RISs are relatively new devices whose channel modeling brings together aspects of electromagnetic engineering, hardware constraints, and communication system concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Certain frontiers of RIS channel modeling are still being explored;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' this section outlines several issues of contemporary interest and investigation in RIS channel modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A summary of the contents of this section appears in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Issue Cause Application or Limitation Channel reciprocity Angle dependence of RIS phase shifts Massive MIMO channel estimation Mutual coupling Reduced element spacing Modeling & estimation in large RIS Perfect absorption Dissipation and resonance control Required in some channel estimation methods Dependence of RIS gain/phase Metallic/Dielectric/Ohmic losses vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' phase Passive beamforming, DFT-training RIS frequency dependence Load/control impedance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' frequency Wideband communications using RIS Near field issues Spherical vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' planar wave approximation Large-RIS, indoor communication TABLE II: Frontiers of RIS channel modeling A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Channel Reciprocity RIS channel estimation in multiuser settings rely on the principle of channel reciprocity for reducing the estimation and feedback overhead, since reciprocity enables the downlink precoding using uplink pilots/estimation in the time division duplex operation [57], [83]–[85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Channel reciprocity holds for many boundary conditions occurring in wireless communication, including reflections from large objects as well as scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' However, in the case of RIS-assisted systems, the literature is inconclusive (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [86] based on an equivalent circuit model claims that angle reciprocity only holds for small angles with respect to normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the opposite direction, Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [87] invokes the Rayleigh-Carson theorem to conclude that RIS enjoy reciprocity, but does not elaborate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' [88] states that the angle reciprocity depends on the design of the RIS surface, and proposes a structure that achieves reciprocity for wide range of angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A resolution of the differences between these results, and a conclusive determination of the conditions under which RIS-induced channels are reciprocal or non-reciprocal, will be welcomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The system model and applications for a non-reciprocal RIS may be of interest in future applications, but is in need of verifiable theory and/or experimental evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' January 13, 2023 DRAFT 24 Tx Rx Ө1 Ө2 Ө2 Tx Rx Ө3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Schematic representation of reciprocity in RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Mutual Coupling A common assumption in RIS modeling is that the passive elements are spaced half-wavelength apart and the mutual coupling among the elements is negligible, allowing them to be controlled individually and independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' However, in practical planar RIS structures with fixed aperture, it is desirable to increase the number of elements by reducing the inter element spacing in order to increase the directivity of the reflected waves and thereby improve the received power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Reducing the inter-element spacing results in dependency/connection of the impedances of the neighboring elements that is non-negligible, having its effect on the channel model, estimation, and the design of reflection coefficients [89], [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Gradoni and Di Renzo [91] have proposed an electromagnetic compliant end-to- end channel model for RIS-aided communication that accounts for mutual impedance among RIS elements, while also including the effects of antenna elements at the transmitter and receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This impedance-based communication model is utilized in [92] to maximize the end-to-end received power by optimizing the RIS tunable load impedances in SISO system, which is further generalized in [93] for MIMO systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In the end-to-end channel modeling of the aforementioned references, the statistical components of the Tx-RIS and RIS-Rx channels are intertwined with the circuit model parameters, which is not desirable from the signal processing/system design perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A more interesting and useful model is one that retains the factored form GΦH, separating the statistical part of the Tx- RIS and RIS-Rx channels, but incorporating the effects of mutual impedances and coupling into the RIS matrix Φ, making it potentially non-diagonal and function of circuit model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Obtaining such a factored model and the associated estimation technique is a potential direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Perfect Absorption/Reflection at RIS Several channel estimation methods, such as those based on matrix completion and channel decomposition, depend on the ability to completely deactivate some RIS elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Some methods depend on separately estimating the direct channel between the transmitter and the receiver, by eliminating the effect of RIS reflections, which requires deactivating all the elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This requires the incident energy to either be completely absorbed by the RIS, or the incident waves to pass through the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The feasibility of perfect absorption is debatable, with partial results whose applicability to communication systems remains unverified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Some works that study the electromag- January 13, 2023 DRAFT 25 netics of metasurfaces suggest that perfect absorption is possible at resonant frequencies with proper tuning of impedance [94]–[97], but it is unclear if/how the proposed structures and the associated methods can be utilized in the context of RIS-aided communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The issue of perfectly deactivating the passive elements is raised in [10], [37], [70], but the question of its physical feasibility remains unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Mishra and Johansson [98] assume that perfect reflection and absorption are unrealizable and incorporates two constants as implementation errors in the system model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' To the best of the authors’ knowledge, no currently-available study in the open literature offers an in-depth and conclusive treatment of the feasibility of perfect deactivation of RIS elements and related design issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Also, analyzing the effect of imperfect deactivation on the accuracy of estimation methods is an important future direction for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In a related direction, several works explore whether and how the phase and amplitude response of an RIS elements are related [99]–[101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' A few studies in reflectarrays and meta surfaces [99], [102]–[104] aim at designing structures that allow near-independent control of reflection amplitude and phase, however, their applicability to RIS-aided communication has not been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Frequency Dependence of RIS Elements RIS-assisted wideband communications [32], [50], [105] requires RIS elements to efficiently operate throughout the frequencies of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In typical RIS constructions, however, the reflection coefficients are tuned by switching on or off various reactive elements or patterns that are connected to the RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The effect of these tuning devices can be modeled by an equivalent circuit whose load impedance varies with the carrier frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The phase shift applied to the incident wave via the tunable elements is calculated at a specific frequency, often the resonant frequency of the RIS element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Within a small deviation of this center frequency, the phase shift remains linear, but across wider frequencies of operation, the phase shift might vary nonlinearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' In that case, the array factor will vary across frequency, and the beam may not retain sharpness across the band of frequencies in which RIS must operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Several remedies have been proposed in the neighboring literature in reflectarrays, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=', coupling the elements to true time delay lines [106] or by coupling multiple resonance elements [107]–[109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The applicability of these methods for RIS-aided communications is yet to be explored, and is a direction for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Near Field Issues The transmitter/receiver is said to be in the far-field of RIS if it is at a distance greater than the Fraunhofer distance 2D2 RIS λc , where DRIS is the largest aperture of RIS and λc is the wavelength corresponding to the carrier frequency fc [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' With the far field assumption, the incident/reflected wave from the RIS can be assumed planar, which simplifies calculations (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Larger RIS structures can achieve superior SNR [7], [111], but if the transmitter and receiver locations are fixed, a sufficiently large RIS array will violate the far field assumption [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The near-field scenario arises, for example, when a large RIS is mounted on a large portion of the facade of a building for servicing users in the street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Without the far-field assumption, the incident waves at different elements will have unequal angular directions and polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The modeling of RIS in the near field scenario is considered in [113]–[115].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' The work in [116] accounts for the difference in the effective area of the elements from different January 13, 2023 DRAFT 26 Tx Rx1 Rx2 RIS Fraunhofer Distance Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Illustration of near/far field issue in RIS-aided communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' observation angles close to the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' Studying suitable channel models and associated estimation techniques for RIS-aided near field communication is a potential direction for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' CONCLUSION This paper provides a comprehensive exposition of the channel estimation techniques for RIS-aided systems, ranging from classical least squares/MMSE methods to machine learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' RIS is often employed with many reflective elements leading to a channel gain with a huge number of parameters, therefore the estimation of link gains is a signature challenge of RIS-induced communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} +page_content=' This paper explores the utility of RIS channel structure for reducing the estimation overhead, including the slow variation of the base-station-to-RIS channel, sparsity of the mmWave channels, and the spatial correlations among the channels of neighboring RIS elements and neighboring users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stE4T4oBgHgl3EQfVwwe/content/2301.05026v1.pdf'} 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Oxford, UK +Yarin Gal +OATML +Department of Computer Science +University of Oxford, UK +Alyson Douglas +AOPP +Department of Physics +University of Oxford, UK +Abstract +Aerosol-cloud interactions (ACI) include various effects that result from aerosols +entering a cloud, and affecting cloud properties. In general, an increase in aerosol +concentration results in smaller droplet sizes which leads to larger, brighter, longer- +lasting clouds that reflect more sunlight and cool the Earth. The strength of the +effect is however heterogeneous, meaning it depends on the surrounding environ- +ment, making ACI one of the most uncertain effects in our current climate models. +In our work, we use causal machine learning to estimate ACI from satellite obser- +vations by reframing the problem as a treatment (aerosol) and outcome (change in +droplet radius). We predict the causal effect of aerosol on clouds with uncertainty +bounds depending on the unknown factors that may be influencing the impact of +aerosol. Of the three climate models evaluated, we find that only one plausibly +recreates the trend, lending more credence to its estimate cooling due to ACI. +1 +Introduction +Aerosol, in the form of pollution from human emissions, enters the atmosphere and eventually +interacts with a cloud leading to aerosol-cloud interactions (ACI). As aerosol enters the cloud, a +causal chain of events catalyzes. It begins with aerosol particles activating as cloud droplet nuclei, +which increases the number of droplets within the cloud, reducing the mean radius of cloud droplets +to redistribute the water vapor, and eventually increasing the cloud’s brightness (Figure 1(a)) [1]. +Overall, an increase in atmospheric aerosol leads to larger, brighter, longer-lasting clouds that reflect +more incoming sunlight. ACI are thus a net cooling process and offset some fraction of warming due +to rising levels of CO2. The strength of the effect is however dependent on the local environment +surrounding the cloud. ACI remain one of the most uncertain effects in our current climate models, +as current models are limited in their ability to simulate ACI with such environmental heterogeneity +[2, 3]. Climate models can only approximate ACI given their low spatial resolution and limited +parameterizations, often dependent on only a few environmental parameters, such as the relative +humidity within a grid cell. These factors lead to increased uncertainty in future projections. Currently, +state-of-the-art climate models estimate that the range of cooling due to ACI may offset 0%-50% of +the warming due to greenhouse gas emissions. +This work uses causal machine learning to estimate ACI from satellite observations, by reframing the +problem as a treatment (aerosol) and outcome (change in droplet radius). We predict the causal effect +of aerosol on clouds and provide uncertainty bounds that we compare to the parameterizations of +Tackling Climate Change with Machine Learning: workshop at NeurIPS 2022. +arXiv:2301.11921v1 [physics.data-an] 30 Nov 2022 + +climate model ACI. We consider uncertainty arising from violations of two assumptions: positivity +(or overlap) and unconfoundedness (or no hidden confounding). Positivity violations are due to +insufficient representation within the data for all treatment levels, such as "treating" cloud with aerosol. +Unmeasured confounding are unobserved factors which influence both the treatment and outcomes, +such as humidity causing aerosol swelling and altering cloud properties. To better understand these +individual sources of uncertainty, we use Overcast [4], a prime example of the needs of a community +such as ACI leading to methodological contributions in machine learning. Compared to prior work +such as [5], we consider aerosol optical depth (AOD), our proxy for aerosol concentration, as a +continuous treatment rather than discrete and perform an uncertainty-aware sensitivity analysis to +study the consequences of possible violations of positivity and unconfoundedness. +(a) Simple Causal Graph of ACI +(b) The Southeast Pacific +(c) The South Atlantic +Figure 1: The causal graph underlining our knowledge of ACI and satellite imagery of the two regions +analyzed, chosen due to their unique aerosol-cloud interactions and breadth of past studies to pull +knowledge from. +2 +Methods +Following [4], we use the potential outcomes framework to estimate the effect of a continuous +treatment T ∈ T (aerosol), on outcomes of interest Y ∈ Y (cloud property), for a unit described +by covariates X ∈ X (environmental information) as shown in Figure 1(a) [6, 7, 8, 9]. We call a +potential outcome and denote by Yt what the outcome would have been if the treatment were t. The +covariates considered are relative humidity at 900, 850 and 700 millibar, sea surface temperature, +vertical motion at 500 millibars, lower tropospheric stability, and effective inversion strength. The +treatment is aerosol optical depth (AOD), a proxy for aerosol concentration. The outcome considered +is the cloud droplet size (re). To estimate the treatment-effect, we study the conditional average +potential outcome (CAPO) and the average potential outcome (APO) +CAPO = µ(x, t) := E [Yt | X = x] , +and +APO = µ(t) := E [µ(X, t)] , +which can be identified from the observational distribution P(X, T, Y) using +˜µ(x, t) := E [Y | T = t, X = x] +and +˜µ(t) := E [˜µ(X, t)] , +and further assumptions (unconfoundedness, positivity, no-interference and consistency). Here, we +study the robustness of treatment-effect estimates to positivity and unconfoundedness violations +(see Appendix A for more detail). We compute uncertainty bounds corresponding to user-specified +relaxations of these assumptions. The parameter Λ, for example, is set by the user to explain an +assumed level of unmeasured confounding [4, 10, 11]. Some confounding influences are impossible +to measure directly with satellites, such as humidity causing aerosol swelling and altering cloud +properties, and the parameter Λ can be used to encode an expert’s belief in the influence of such +confounders. +We use daily mean, 1◦ x 1◦ of satellite observations in order to homogenize the data from the +southeast Pacific and south Atlantic (Figures 1(b) and 1(c)). Mean droplet radius (re) from the +MODIS instrument is used as our outcome for all experiments shown within. We employ aerosol +optical depth from MERRA-2 to approximate the concentration of aerosol. Our environmental +confounders are the relative humidity at 900, 850 and 700 millibars, the stability of the atmosphere, +the sea surface temperature, and the vertical motion at 500 mb, all also from MERRA-2. For more +detail about data and implementation, please refer to Appendix B and Appendix D. +2 + +UnobservedContounding +Aerosol +Optical +T +Depth +X +Environmental Information +Cloud Droplet Radius3 +Results +3.1 +Deducing reasonable treatment-effect bounds using domain knowledge +Unlike past studies which only crudely estimate an uncertainty range due to quantifiable effects, we +are able to derive confidence intervals dependent on the influence of confounding by varying Λ. Since +it is impossible to know the strength of the confounding effect from observed data alone, we propose a +method to select a reasonable Λ by contrasting two geographical regions. We contrast the South-East +Pacific and the South Atlantic because these regions have different environmental confounders of +ACI, for example aerosol type, aerosol hygroscopicity, aerosol size. These are important confounders, +but are unfortunately not included in the available data. So we select the parameter Λ for the Pacific +region such that the treatment effect bounds for the Pacific region cover the effect bounds for the +Atlantic region under the assumption of no hidden confounding (Λ → 1) for the Atlantic region, as +shown in Figure 2(a). Setting Λ to 1.07 gives bounds for the pacific region that reasonably account +for the potential bias induced by the unmodeled confounders. While a larger Λ could still be sensible +due to other drivers of confounding, domain knowledge informs us that these are the main missing +physical mechanisms. +3.2 +Evaluating climate models using machine learning +As we now have a possible range of ACI derived using the real, observed outcomes, we can judge +how well climate models recreate this observed trend by seeing if their responses lie within our +derived interval (Figure 2(b)). We find that the Canadian model CanESM5 simulates ACI better than +the UK models HadGEM3-GC3-LL and UKESM1-0-LL. Our trained machine learning model not +only uses the real, observed relationships to derive the magnitude of the effect, but can consider the +environmental context and confounding influences to derive real, quantifiable bounds of uncertainty. +Therefore, by using the curves found by Overcast as the true response, we know those models which +lie outside of our bounds found by contrasting different regions are likely unphysical and highly +unlikely to occur in the real-world. CanESM5 currently estimates the total cooling effect due to ACI +to offset approximately half of the warming due to greenhouse gases; based on our results, we would +say it is likely that this estimate is closest to the true value observed on Earth [12]. +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +AOD +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +re += 0.05 +Atlantic +Pacific +1.0 for Atlantic += 1.07 for Pacific +(a) Choosing Λ using two geographical regions +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +AOD +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +re += 0.05 +Pacific +HadGEM3-GC3-LL +UKESM1-0-LL +CanESM5 += 1.07 for Pacific +(b) Comparison with ESMs in the Pacific +Figure 2: Plausible range of effects of aerosol (AOD) on mean droplet radius (re). +4 +Discussion and conclusions +4.1 +Machine learning’s place in climate model verification +In this work, we show that machine learning methods offer viable ways to objectively judge how +well global climate models reproduce climate processes such as the effect of aerosol on mean droplet +radius. A drawback of historical studies which utilize satellite observations is their inability to +3 + +quantify how the surrounding environment may affect the magnitude of the aerosol-cloud interactions. +Overcast accounts for such contextual confounding and communicates bounds on the treatment effect +due to an expert-informed influence of hidden-confounding. Utilizing this method gives us insight +into whether climate models reproduce the observed relationships between AOD and re. Climate +models currently only reasonably recreate large scale processes that can be explicitly calculated, +leaving processes like aerosol-cloud interactions, which occur on scales smaller than the grid scale, +poorly parameterized and approximated. In order to improve our climate models, we must understand +in more relatable terms how well they are doing, such as by comparing their outcomes to those +from observations. Machine learning provides not only a way to judge these outcomes, but the +relationships learned by Overcast and similar models could in the future be fine-tuned to replace our +current parameterizations [13]. +4.2 +Collaboration across domains vs. purely data driven +While different sources of confounding due to regional differences alter the outcomes, the choice +of which environmental factors are the main sources of confounding can also be investigated using +Overcast. We perform two experiments, with and without relative humidity at 900, 850 and 700 +millibars, to derive varying outcome shapes and fit Λ to both dose-response curves. When Λ is set to +1.04, both curves are captured by the bounds of uncertainty, allowing us to view how the response may +vary within those bounds due to meteorological uncertainty rather than regional uncertainty, where +Λ = 1.07 was required (Figure 3(a)). In the absence of ground truth, purely data-driven techniques +cannot decide between the model with and without relative humidity, but as domain knowledge is +brought in, it is known that the curve with humidity included is the true response curve (Figure 3(b)). +Purely data-driven approaches may not be the most appropriate for studying climatological processes +such as aerosol-cloud interactions as domain knowledge is essential to select the correct inputs and +verify the outcomes. The most robust model arises from combining data and theory, bringing together +experts in machine learning and climate processes. +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +AOD +0.4 +0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +re += 0.05 +Pacific +Pacific without RH +1.0 for Pacific += 1.04 for Pacific without RH +(a) Scaled, with appropriate Λ +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +AOD +13.50 +13.75 +14.00 +14.25 +14.50 +14.75 +15.00 +15.25 +re += 0.05 +Pacific +Pacific without RH +1.0 +1.0 +(b) Unscaled +Figure 3: Plausible range of effects when omitting relative humidity from the covariates. +4.3 +Limitations and future works +This work uses aerosol optical depth as a proxy for aerosol concentration, which could bias treatment- +effect estimates. For example, it is known that bias can arise from measurement-error in the treatment +[14]. Further, we rely on low resolution data that does not perfectly capture the microphysical +processes. Future work could consider different assumptions on the underlying causal model, and +attempt to include other aerosol properties like size, hygroscopicity and type. +4 + +5 +Acknowledgements +This project was supported by the European Union’s Horizon 2020 research and innovation program +under grant agreement No 821205 (FORCeS), the European Research Council (ERC) project con- +stRaining the EffeCts of Aerosols on Precipitation (RECAP) under the European Union’s Horizon +2020 research and innovation programme with grant agreement no. 724602, and the Turing Institute +post-doctoral enrichment award. +References +[1] +Sean A Twomey, M Piepgrass, and TL Wolfe. “An assessment of the impact of pollution on +global cloud albedo”. In: Tellus B 36.5 (1984), pp. 356–366. +[2] +Olivier Boucher, D Randall, P Artaxo, C Bretherton, G Feingold, P Forster, V-M Kerminen, +Y Kondo, H Liao, U Lohmann, P Rasch, S.K Satheesh, B Stevens, and X.Y Zhang. “Clouds +and Aerosols”. 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In: Atmospheric Chemistry and Physics 22.1 (Jan. 2022), pp. 641–674. +6 + +A +Theoretical background: unconfoundedness and positivity assumptions +Confounding variables are factors that influence both the treatment T and the outcomes Y. The +unconfoundedness assumption states that all confounding variables are observed and controlled for +using X, so that the treatment groups are comparable, that is, YT ⊥⊥ T | X. +The positivity assumption states that all subgroups of the data with different covariates have a non- +zero probability of receiving any dose of treatment, that is, p(t | x) > 0 for any t ∈ T and for any +x ∈ X such that p(x) > 0. +In practice, there is a trade-off between positivity and unconfoundedness due to the curse of dimen- +sionality, as with large X and continuous treatment, it is unlikely that we observe all treatment levels +for each x ∈ X. +B +Data and pre-processing +We work with data which is retrieved from re-analyses of satellite observations. The Moderate +Resolution Imaging Spectroradiometer (MODIS) instruments aboard the Terra and Aqua satellites +observe the Earth at approximately 1 km × 1 km resolution [15]. These observations are fed into the +Modern-Era Retrospective Analysis for Research and Applications version 2 (MERRA-2) real-time +model to emulate the atmosphere and its components, such as aerosol [16]. MERRA-2 calculates +global vertical profiles of temperature, relative humidity, and pressure, and assimilates hyperspectral +and passive microwave satellite observations to enhance its ability to model Earth’s atmosphere. The +data studied are MODIS observations from the Aqua and Terra satellites collocated with MERRA +reanalyses of the environments. We work with two different datasets which are 1◦ × 1◦ daily means +of observations over the South Atlantic and the South East Pacific from 2004 to 2019. The sources +are given in Table 1. +Table 1: Sources of satellite observations +Product Name +Description +Mean Droplet Radius (re) +MODIS (1.6, 2.1, 3.7 µm channels) [15] +Precipitation +NOAA CMORPH CDR [17] +Sea Surface Temperature (SST) +NOAA WHOI CDR [18] +Lower Tropospheric Stability (LTS) +MERRA-2 [16] +Vertical Motion at 500 mb (ω500) +MERRA-2 [19] +Estimated Inversion Strength (EIS) +MERRA-2 [16, 20] +Relative Humidity at x mb (RHx) +MERRA-2 [16] +Aerosol Optical Depth (AOD) +MERRA-2 [16] +We restrict our observations to clouds in the “aerosol limited” regime by applying some filtering [21]. +In “aerosol limited” regimes, we assume that cloud development is limited by the availability of +cloud-condensation nuclei, and thus aerosol. Our choice of filtering is informed by domain knowledge. +CWP are filtered to values below 250µm and re to values below 30µm. AOD values are filtered, only +keeping values between 0.03 and 0.3. We also filter out precipitating clouds to avoid a loop in the +causal graph. Finally, all features are normalized before being fed into the model. +C +Model architecture +The models are neural-network architectures with two components: a feature extractor φ(x; θ) and +a density estimator f(φ, t; θ), represented in Appendix C. The covariates x are given as input to +the feature extractor, whose output is concatenated with the treatment t and given as input to the +density estimator which outputs a Gaussian mixture density, p(y | t, x, θ), from which we can sample. +The feature extractor uses attention mechanisms to model the spatio-temporal correlations between +the covariates on a given day using the geographical coordinates of the observations. The model +architecture is represented in Figure 4. +7 + +covariates +treatment + +Density Estimator +Feature Extractor + +outcomes +EncoderBlock +Linear +MAB +depth +InputEmbedding +DenseLinear +Dropout +PositionEmbedding +DenseLinear +Dropout +DenseLinear +Linear +ResNet +(depth-1) +Gaussian +Mixture Model +DenseFeatureExtractor +spatio- +temporal +coordinates +Figure 4: Overcast model architecture. The inputs are represented by circles, in blue the covariates, +in grey the spatio-temporal coordinates, in purple the treatment. In the red circle is the output of the +model, the outcomes distribution. The model has a feature extractor in green and a density estimator +(in orange). +D +Implementation details +We follow the implementation from [4]. The code is written in python. The packages used include +PyTorch [22], scikit-learn [23], Ray [24], NumPy, SciPy and Matplotlib. +We use ray tune [25] with HyperBand Bayesian Optimization [26] search algorithm to optimise our +network hyper-parameters. The hyper-parameters considered during tuning are given in [4]. The final +hyper-parameters for each dataset are given in Table 2. The hyper-parameter optimization objective +is the batch-wise Pearson correlation averaged across all outcomes on the validation data for a single +dataset realization with random seed 1331. +We split the data into training, validation, and testing sets across different days. [4] splits data in the +following way: datapoints from Mondays to Fridays are in the training set, from Saturdays in the +validation set, and from Sundays in the testing set. In our implementation, we keep the same ratio +between datasets but we randomize the splits, using random seed 42 and having 5/7 of the data in +the training set, 1/7 in the validation set, and 1/7 in the testing set. The randomization is motivated +by the fact that there is a clear weekly cycle of aerosol optical depth [27]. Models are optimized by +maximizing the log likelihood of p(y | t, x, θ). +Table 2: Final hyper-parameters for each dataset and model +Hyper-parameter +South-East Pacific +South Atlantic +Hidden Units +128 +128 +Network Depth +3 +4 +GMM T Components +27 +7 +GMM Y Components +22 +24 +Attention Heads +8 +8 +Negative Slope +0.28 +0.19 +Dropout Rate +0.42 +0.16 +Layer Norm +False +True +Batch Size +128 +160 +Learning Rate +0.0001 +0.0001 +Epochs +500 +500 +8 + diff --git a/stFKT4oBgHgl3EQf1y5_/content/tmp_files/load_file.txt b/stFKT4oBgHgl3EQf1y5_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..52195a6efd0f765398331f502ae26f4eeb5005e3 --- /dev/null +++ b/stFKT4oBgHgl3EQf1y5_/content/tmp_files/load_file.txt @@ -0,0 +1,433 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf,len=432 +page_content='Using uncertainty-aware machine learning models to study aerosol-cloud interactions Maëlys Solal Department of Computer Science University of Oxford, UK maelys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='solal@ens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='psl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='eu Andrew Jesson OATML Department of Computer Science University of Oxford, UK Yarin Gal OATML Department of Computer Science University of Oxford, UK Alyson Douglas AOPP Department of Physics University of Oxford, UK Abstract Aerosol-cloud interactions (ACI) include various effects that result from aerosols entering a cloud, and affecting cloud properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In general, an increase in aerosol concentration results in smaller droplet sizes which leads to larger, brighter, longer- lasting clouds that reflect more sunlight and cool the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The strength of the effect is however heterogeneous, meaning it depends on the surrounding environ- ment, making ACI one of the most uncertain effects in our current climate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In our work, we use causal machine learning to estimate ACI from satellite obser- vations by reframing the problem as a treatment (aerosol) and outcome (change in droplet radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We predict the causal effect of aerosol on clouds with uncertainty bounds depending on the unknown factors that may be influencing the impact of aerosol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Of the three climate models evaluated, we find that only one plausibly recreates the trend, lending more credence to its estimate cooling due to ACI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 1 Introduction Aerosol, in the form of pollution from human emissions, enters the atmosphere and eventually interacts with a cloud leading to aerosol-cloud interactions (ACI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' As aerosol enters the cloud, a causal chain of events catalyzes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' It begins with aerosol particles activating as cloud droplet nuclei, which increases the number of droplets within the cloud, reducing the mean radius of cloud droplets to redistribute the water vapor, and eventually increasing the cloud’s brightness (Figure 1(a)) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Overall, an increase in atmospheric aerosol leads to larger, brighter, longer-lasting clouds that reflect more incoming sunlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' ACI are thus a net cooling process and offset some fraction of warming due to rising levels of CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The strength of the effect is however dependent on the local environment surrounding the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' ACI remain one of the most uncertain effects in our current climate models, as current models are limited in their ability to simulate ACI with such environmental heterogeneity [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Climate models can only approximate ACI given their low spatial resolution and limited parameterizations, often dependent on only a few environmental parameters, such as the relative humidity within a grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' These factors lead to increased uncertainty in future projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Currently, state-of-the-art climate models estimate that the range of cooling due to ACI may offset 0%-50% of the warming due to greenhouse gas emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' This work uses causal machine learning to estimate ACI from satellite observations, by reframing the problem as a treatment (aerosol) and outcome (change in droplet radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We predict the causal effect of aerosol on clouds and provide uncertainty bounds that we compare to the parameterizations of Tackling Climate Change with Machine Learning: workshop at NeurIPS 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='11921v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='data-an] 30 Nov 2022 climate model ACI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We consider uncertainty arising from violations of two assumptions: positivity (or overlap) and unconfoundedness (or no hidden confounding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Positivity violations are due to insufficient representation within the data for all treatment levels, such as "treating" cloud with aerosol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Unmeasured confounding are unobserved factors which influence both the treatment and outcomes, such as humidity causing aerosol swelling and altering cloud properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' To better understand these individual sources of uncertainty, we use Overcast [4], a prime example of the needs of a community such as ACI leading to methodological contributions in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Compared to prior work such as [5], we consider aerosol optical depth (AOD), our proxy for aerosol concentration, as a continuous treatment rather than discrete and perform an uncertainty-aware sensitivity analysis to study the consequences of possible violations of positivity and unconfoundedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' (a) Simple Causal Graph of ACI (b) The Southeast Pacific (c) The South Atlantic Figure 1: The causal graph underlining our knowledge of ACI and satellite imagery of the two regions analyzed, chosen due to their unique aerosol-cloud interactions and breadth of past studies to pull knowledge from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 2 Methods Following [4], we use the potential outcomes framework to estimate the effect of a continuous treatment T ∈ T (aerosol), on outcomes of interest Y ∈ Y (cloud property), for a unit described by covariates X ∈ X (environmental information) as shown in Figure 1(a) [6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We call a potential outcome and denote by Yt what the outcome would have been if the treatment were t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The covariates considered are relative humidity at 900, 850 and 700 millibar, sea surface temperature, vertical motion at 500 millibars, lower tropospheric stability, and effective inversion strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The treatment is aerosol optical depth (AOD), a proxy for aerosol concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The outcome considered is the cloud droplet size (re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' To estimate the treatment-effect, we study the conditional average potential outcome (CAPO) and the average potential outcome (APO) CAPO = µ(x, t) := E [Yt | X = x] , and APO = µ(t) := E [µ(X, t)] , which can be identified from the observational distribution P(X, T, Y) using ˜µ(x, t) := E [Y | T = t, X = x] and ˜µ(t) := E [˜µ(X, t)] , and further assumptions (unconfoundedness, positivity, no-interference and consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Here, we study the robustness of treatment-effect estimates to positivity and unconfoundedness violations (see Appendix A for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We compute uncertainty bounds corresponding to user-specified relaxations of these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The parameter Λ, for example, is set by the user to explain an assumed level of unmeasured confounding [4, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Some confounding influences are impossible to measure directly with satellites, such as humidity causing aerosol swelling and altering cloud properties, and the parameter Λ can be used to encode an expert’s belief in the influence of such confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We use daily mean, 1◦ x 1◦ of satellite observations in order to homogenize the data from the southeast Pacific and south Atlantic (Figures 1(b) and 1(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Mean droplet radius (re) from the MODIS instrument is used as our outcome for all experiments shown within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We employ aerosol optical depth from MERRA-2 to approximate the concentration of aerosol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Our environmental confounders are the relative humidity at 900, 850 and 700 millibars, the stability of the atmosphere, the sea surface temperature, and the vertical motion at 500 mb, all also from MERRA-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' For more detail about data and implementation, please refer to Appendix B and Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 2 UnobservedContounding Aerosol Optical T Depth X Environmental Information Cloud Droplet Radius3 Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='1 Deducing reasonable treatment-effect bounds using domain knowledge Unlike past studies which only crudely estimate an uncertainty range due to quantifiable effects, we are able to derive confidence intervals dependent on the influence of confounding by varying Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Since it is impossible to know the strength of the confounding effect from observed data alone, we propose a method to select a reasonable Λ by contrasting two geographical regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We contrast the South-East Pacific and the South Atlantic because these regions have different environmental confounders of ACI, for example aerosol type, aerosol hygroscopicity, aerosol size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' These are important confounders, but are unfortunately not included in the available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' So we select the parameter Λ for the Pacific region such that the treatment effect bounds for the Pacific region cover the effect bounds for the Atlantic region under the assumption of no hidden confounding (Λ → 1) for the Atlantic region, as shown in Figure 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Setting Λ to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='07 gives bounds for the pacific region that reasonably account for the potential bias induced by the unmodeled confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' While a larger Λ could still be sensible due to other drivers of confounding, domain knowledge informs us that these are the main missing physical mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 Evaluating climate models using machine learning As we now have a possible range of ACI derived using the real, observed outcomes, we can judge how well climate models recreate this observed trend by seeing if their responses lie within our derived interval (Figure 2(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We find that the Canadian model CanESM5 simulates ACI better than the UK models HadGEM3-GC3-LL and UKESM1-0-LL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Our trained machine learning model not only uses the real, observed relationships to derive the magnitude of the effect, but can consider the environmental context and confounding influences to derive real, quantifiable bounds of uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Therefore, by using the curves found by Overcast as the true response, we know those models which lie outside of our bounds found by contrasting different regions are likely unphysical and highly unlikely to occur in the real-world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' CanESM5 currently estimates the total cooling effect due to ACI to offset approximately half of the warming due to greenhouse gases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' based on our results, we would say it is likely that this estimate is closest to the true value observed on Earth [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='30 AOD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='4 re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 Atlantic Pacific 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 for Atlantic = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='07 for Pacific (a) Choosing Λ using two geographical regions 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='30 AOD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='4 re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 Pacific HadGEM3-GC3-LL UKESM1-0-LL CanESM5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='07 for Pacific (b) Comparison with ESMs in the Pacific Figure 2: Plausible range of effects of aerosol (AOD) on mean droplet radius (re).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 4 Discussion and conclusions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='1 Machine learning’s place in climate model verification In this work, we show that machine learning methods offer viable ways to objectively judge how well global climate models reproduce climate processes such as the effect of aerosol on mean droplet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' A drawback of historical studies which utilize satellite observations is their inability to 3 quantify how the surrounding environment may affect the magnitude of the aerosol-cloud interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Overcast accounts for such contextual confounding and communicates bounds on the treatment effect due to an expert-informed influence of hidden-confounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Utilizing this method gives us insight into whether climate models reproduce the observed relationships between AOD and re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Climate models currently only reasonably recreate large scale processes that can be explicitly calculated, leaving processes like aerosol-cloud interactions, which occur on scales smaller than the grid scale, poorly parameterized and approximated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In order to improve our climate models, we must understand in more relatable terms how well they are doing, such as by comparing their outcomes to those from observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Machine learning provides not only a way to judge these outcomes, but the relationships learned by Overcast and similar models could in the future be fine-tuned to replace our current parameterizations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 Collaboration across domains vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' purely data driven While different sources of confounding due to regional differences alter the outcomes, the choice of which environmental factors are the main sources of confounding can also be investigated using Overcast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We perform two experiments, with and without relative humidity at 900, 850 and 700 millibars, to derive varying outcome shapes and fit Λ to both dose-response curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' When Λ is set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='04, both curves are captured by the bounds of uncertainty, allowing us to view how the response may vary within those bounds due to meteorological uncertainty rather than regional uncertainty, where Λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='07 was required (Figure 3(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In the absence of ground truth, purely data-driven techniques cannot decide between the model with and without relative humidity, but as domain knowledge is brought in, it is known that the curve with humidity included is the true response curve (Figure 3(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Purely data-driven approaches may not be the most appropriate for studying climatological processes such as aerosol-cloud interactions as domain knowledge is essential to select the correct inputs and verify the outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The most robust model arises from combining data and theory, bringing together experts in machine learning and climate processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='30 AOD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 Pacific Pacific without RH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 for Pacific = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='04 for Pacific without RH (a) Scaled, with appropriate Λ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='30 AOD 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='50 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='75 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='00 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='25 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='50 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='75 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='00 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='25 re = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='05 Pacific Pacific without RH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0 (b) Unscaled Figure 3: Plausible range of effects when omitting relative humidity from the covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='3 Limitations and future works This work uses aerosol optical depth as a proxy for aerosol concentration, which could bias treatment- effect estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' For example, it is known that bias can arise from measurement-error in the treatment [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Further, we rely on low resolution data that does not perfectly capture the microphysical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Future work could consider different assumptions on the underlying causal model, and attempt to include other aerosol properties like size, hygroscopicity and type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 4 5 Acknowledgements This project was supported by the European Union’s Horizon 2020 research and innovation program under grant agreement No 821205 (FORCeS), the European Research Council (ERC) project con- stRaining the EffeCts of Aerosols on Precipitation (RECAP) under the European Union’s Horizon 2020 research and innovation programme with grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 724602, and the Turing Institute post-doctoral enrichment award.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' References [1] Sean A Twomey, M Piepgrass, and TL Wolfe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' “An assessment of the impact of pollution on global cloud albedo”.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In: Atmospheric Chemistry and Physics 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='16 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 9591–9618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' [13] Andrew Gettelman, David John Gagne, C-C Chen, MW Christensen, ZJ Lebo, Hugh Morrison, and Gabrielle Gantos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' “Machine learning the warm rain process”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In: Journal of Advances in Modeling Earth Systems 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='2 (2021), e2020MS002268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 5 [14] Yuchen Zhu, Limor Gultchin, Arthur Gretton, Matt Kusner, and Ricardo Silva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Causal Infer- ence with Treatment Measurement Error: A Nonparametric Instrumental Variable Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' [15] Bryan A.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Vanderplas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Passos, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Cournapeau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Brucher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Perrot, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Duchesnay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' “Scikit-learn: Machine Learning in Python”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In: Journal of Machine Learning Research 12 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 2825–2830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' [24] Philipp Moritz, Robert Nishihara, Stephanie Wang, Alexey Tumanov, Richard Liaw, Eric Liang, Melih Elibol, Zongheng Yang, William Paul, Michael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Jordan, and Ion Stoica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Ray: A Distributed Framework for Emerging AI Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' [25] Richard Liaw, Eric Liang, Robert Nishihara, Philipp Moritz, Joseph E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Gonzalez, and Ion Stoica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Tune: A Research Platform for Distributed Model Selection and Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' July 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' [26] Stefan Falkner, Aaron Klein, and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' “Combining Hyperband and Bayesian Opti- mization”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In: (), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' [27] Matthew W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Christensen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' “Opportunistic Experiments to Constrain Aerosol Effective Radiative Forcing”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In: Atmospheric Chemistry and Physics 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='1 (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 2022), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 641–674.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 6 A Theoretical background: unconfoundedness and positivity assumptions Confounding variables are factors that influence both the treatment T and the outcomes Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The unconfoundedness assumption states that all confounding variables are observed and controlled for using X, so that the treatment groups are comparable, that is, YT ⊥⊥ T | X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The positivity assumption states that all subgroups of the data with different covariates have a non- zero probability of receiving any dose of treatment, that is, p(t | x) > 0 for any t ∈ T and for any x ∈ X such that p(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In practice, there is a trade-off between positivity and unconfoundedness due to the curse of dimen- sionality, as with large X and continuous treatment, it is unlikely that we observe all treatment levels for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' B Data and pre-processing We work with data which is retrieved from re-analyses of satellite observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The Moderate Resolution Imaging Spectroradiometer (MODIS) instruments aboard the Terra and Aqua satellites observe the Earth at approximately 1 km × 1 km resolution [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' These observations are fed into the Modern-Era Retrospective Analysis for Research and Applications version 2 (MERRA-2) real-time model to emulate the atmosphere and its components, such as aerosol [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' MERRA-2 calculates global vertical profiles of temperature, relative humidity, and pressure, and assimilates hyperspectral and passive microwave satellite observations to enhance its ability to model Earth’s atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The data studied are MODIS observations from the Aqua and Terra satellites collocated with MERRA reanalyses of the environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We work with two different datasets which are 1◦ × 1◦ daily means of observations over the South Atlantic and the South East Pacific from 2004 to 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The sources are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Table 1: Sources of satellite observations Product Name Description Mean Droplet Radius (re) MODIS (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='7 µm channels) [15] Precipitation NOAA CMORPH CDR [17] Sea Surface Temperature (SST) NOAA WHOI CDR [18] Lower Tropospheric Stability (LTS) MERRA-2 [16] Vertical Motion at 500 mb (ω500) MERRA-2 [19] Estimated Inversion Strength (EIS) MERRA-2 [16, 20] Relative Humidity at x mb (RHx) MERRA-2 [16] Aerosol Optical Depth (AOD) MERRA-2 [16] We restrict our observations to clouds in the “aerosol limited” regime by applying some filtering [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In “aerosol limited” regimes, we assume that cloud development is limited by the availability of cloud-condensation nuclei, and thus aerosol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Our choice of filtering is informed by domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' CWP are filtered to values below 250µm and re to values below 30µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' AOD values are filtered, only keeping values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='03 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We also filter out precipitating clouds to avoid a loop in the causal graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Finally, all features are normalized before being fed into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' C Model architecture The models are neural-network architectures with two components: a feature extractor φ(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' θ) and a density estimator f(φ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' θ), represented in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The covariates x are given as input to the feature extractor, whose output is concatenated with the treatment t and given as input to the density estimator which outputs a Gaussian mixture density, p(y | t, x, θ), from which we can sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The feature extractor uses attention mechanisms to model the spatio-temporal correlations between the covariates on a given day using the geographical coordinates of the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The model architecture is represented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' 7 covariates treatment Density Estimator Feature Extractor outcomes EncoderBlock Linear MAB depth InputEmbedding DenseLinear Dropout PositionEmbedding DenseLinear Dropout DenseLinear Linear ResNet (depth-1) Gaussian Mixture Model DenseFeatureExtractor spatio- temporal coordinates Figure 4: Overcast model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The inputs are represented by circles, in blue the covariates, in grey the spatio-temporal coordinates, in purple the treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In the red circle is the output of the model, the outcomes distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The model has a feature extractor in green and a density estimator (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' D Implementation details We follow the implementation from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The code is written in python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The packages used include PyTorch [22], scikit-learn [23], Ray [24], NumPy, SciPy and Matplotlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We use ray tune [25] with HyperBand Bayesian Optimization [26] search algorithm to optimise our network hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The hyper-parameters considered during tuning are given in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The final hyper-parameters for each dataset are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The hyper-parameter optimization objective is the batch-wise Pearson correlation averaged across all outcomes on the validation data for a single dataset realization with random seed 1331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' We split the data into training, validation, and testing sets across different days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' [4] splits data in the following way: datapoints from Mondays to Fridays are in the training set, from Saturdays in the validation set, and from Sundays in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' In our implementation, we keep the same ratio between datasets but we randomize the splits, using random seed 42 and having 5/7 of the data in the training set, 1/7 in the validation set, and 1/7 in the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' The randomization is motivated by the fact that there is a clear weekly cycle of aerosol optical depth [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Models are optimized by maximizing the log likelihood of p(y | t, x, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content=' Table 2: Final hyper-parameters for each dataset and model Hyper-parameter South-East Pacific South Atlantic Hidden Units 128 128 Network Depth 3 4 GMM T Components 27 7 GMM Y Components 22 24 Attention Heads 8 8 Negative Slope 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='19 Dropout Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='16 Layer Norm False True Batch Size 128 160 Learning Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} +page_content='0001 Epochs 500 500 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf'} diff --git a/utFAT4oBgHgl3EQfhx3p/content/tmp_files/2301.08596v1.pdf.txt b/utFAT4oBgHgl3EQfhx3p/content/tmp_files/2301.08596v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bf2ddca086695088c92bb1d51ff0028c30a9b48 --- /dev/null +++ b/utFAT4oBgHgl3EQfhx3p/content/tmp_files/2301.08596v1.pdf.txt @@ -0,0 +1,2169 @@ +Draft version January 23, 2023 +Typeset using LATEX twocolumn style in AASTeX631 +The chromosphere underneath a Coronal Bright Point +Souvik Bose +,1, 2, 3, 4 Daniel N´obrega-Siverio +,5, 6, 3, 4 Bart De Pontieu +,1, 3, 4 and +Luc Rouppe van der Voort +3, 4 +1Lockheed Martin Solar & Astrophysics Laboratory, Palo Alto, CA 94304, USA +2Bay Area Environmental Research Institute, NASA Research Park, Moffett Field, CA 94035, USA +3Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway +4Rosseland Centre for Solar Physics, University of Oslo, PO Box 1029, Blindern 0315, Oslo, Norway +5Instituto de Astrof´ısica de Canarias, E-38205 La Laguna, Tenerife, Spain +6Universidad de La Laguna, Dept. Astrof´ısica, E-38206 La Laguna, Tenerife, Spain +ABSTRACT +Coronal Bright Points (CBPs) are sets of small-scale coronal loops, connecting opposite magnetic +polarities, primarily characterized by their enhanced extreme-ultraviolet (EUV) and X-ray emission. +Being ubiquitous, they are thought to play an important role in heating the solar corona. We aim at +characterizing the barely-explored chromosphere underneath CBPs, focusing on the related spicular +activity and on the effects of small-scale magnetic flux emergence on CBPs. We used high-resolution +observations of a CBP in Hβ and Fe I 617.3 nm from the Swedish 1-m Solar Telescope (SST) in +coordination with the Solar Dynamics Observatory (SDO). This work presents the first high-resolution +observation of spicules imaged in Hβ. The spicules were automatically detected using advanced image +processing techniques, which were applied to the Dopplergrams derived from Hβ. Here we report their +abundant occurrence close to the CBP “footpoints”, and find that the orientation of such spicules is +aligned along the EUV loops, indicating that they constitute a fundamental part of the whole CBP +magnetic structure. Spatio-temporal analysis across multiple channels indicates that there are coronal +propagating disturbances associated with the studied spicules, producing transient EUV intensity +variations of the individual CBP loops. +Two small-scale flux emergence episodes appearing below +the CBP were analyzed; one of them leading to quiet-sun Ellerman bombs and enhancing the nearby +spicular activity. This paper presents unique evidence of the tight coupling between the lower and +upper atmosphere of a CBP, thus helping to unravel the dynamic phenomena underneath CBPs and +their impact on the latter. +Keywords: Solar coronal heating (1989) — Solar spicules (1525) — Solar chromosphere (1479) — Solar +corona (1483) — Solar magnetic flux emergence (2000) — Methods: observational +1. INTRODUCTION +Coronal Bright Points (CBPs) appear as bright, +enhanced, blob-like structures when observed in the +extreme-ultraviolet (EUV) light or X-rays. +First ob- +served in X-rays with the grazing incidence X-ray tele- +scope sounding rocket mission (Vaiana et al. 1973), +CBPs comprise small-scale magnetic loops connecting +opposite polarities where the confined plasma is heated +up to a million degrees presumably by magnetic recon- +nection (see Priest et al. 1994). CBPs are ubiquitously +observed in the coronal holes, quiet-Sun, and in the close +vicinity of active regions alike, which makes them in- +bose@baeri.org +teresting from the perspective of their role in coronal +heating. Their lifetimes range from a few hours to even +a few days (Golub et al. 1974; McIntosh & Gurman +2005) and, depending upon the wavelength of obser- +vation, they appear as roundish blobs with diameters +ranging between 5–30′′on average (Vaiana et al. 1973; +Habbal et al. 1990; Mou et al. 2018). Different stud- +ies based on emission spectroscopy and imaging (as dis- +cussed in the recent review by Madjarska 2019) suggest +that the heights over which CBPs extend in the corona +ranges between 5–10 Mm above the photosphere with +an average of 6.5 Mm during their lifetime. +Though CBPs have been the subject of intensive re- +search ever since their discovery back in the early 1970s +(Madjarska 2019), there are still fundamental open ques- +tions regarding these ubiquitous phenomena. +For in- +arXiv:2301.08596v1 [astro-ph.SR] 20 Jan 2023 + +ID2 +Bose et al. +stance, the CBP chromospheric counterpart remains +largely unexplored to date, which may be attributed to +the lack of adequate observations that target the corona +and chromosphere simultaneously. To the best of our +knowledge, only two observational studies – Habbal & +Withbroe (1981) and Madjarska et al. (2021) have fo- +cused on this particular atmospheric layer, both finding +that strong intensity enhancements in the corona pre- +ceded lower temperature (chromospheric and transition +region (TR)) enhancements, thereby indicating a sce- +nario where the heating takes place first in the corona +and is later conducted toward the TR via thermal con- +duction. Another open question is related to the role of +magnetic flux emergence on CBPs. For example, mag- +netic flux emergence is not only known to be respon- +sible for the origin of nearly half of the CBPs (Mou +et al. 2018), but also to enhance the chromospheric ac- +tivity and associated coronal emission (Madjarska et al. +2021). So far in the CBP literature, the focus has pri- +marily been on large-scale emergence episodes that last +for several tens of minutes to hours, therefore studies +about the impact of small-scale magnetic flux emergence +episodes are scarce: the lack of high-resolution, coordi- +nated magnetograms seems to be a major impediment +in this regard. +The aim of this paper is to better understand the +chromospheric scenery underneath a CBP with a fo- +cus on spicules and the atmospheric responses to small- +scale flux emergence episodes. +Spicules are one of +the most abundant and ubiquitous features observed +in the solar chromosphere. They are highly dynamic, +thin, (multi)threaded, and elongated structures that +permeate both the active and non-active regions alike +(Pereira et al. 2012). +They are broadly divided into +two categories–type I and II, with the latter being more +dynamic, with higher apparent velocities, shorter life- +times, and undergoing vigorous swaying and torsional +motion (de Pontieu et al. 2007; Pereira et al. 2012; Bose +et al. 2021a). The signatures of type-II spicules are of- +ten found in the TR and coronal passbands which makes +their studies exciting from the perspective of heating +and mass-loading of the solar corona (De Pontieu et al. +2009, 2011; Pereira et al. 2014; Rouppe van der Voort +et al. 2015; Henriques et al. 2016; Samanta et al. 2019). +The on-disk counterparts of type-II spicules, termed as +rapid blue-shifted and red-shifted excursions (RBEs and +RREs, see Rouppe van der Voort et al. 2009; Sekse +et al. 2012; Bose et al. 2019), abundantly occur in the +close vicinity of strong magnetic field regions (such as +bi-polar/unipolar field patches, see Sekse et al. 2012; +Bose et al. 2021a, for example). +This makes their +study also interesting in the context of CBPs since their +loops appear to be rooted to strong bi-polar magnetic +field configurations present in the photosphere. Multi- +dimensional numerical models, e.g. +by Wyper et al. +(2018) and more recently by N´obrega-Siverio & Moreno- +Insertis (2022) suggest that the loops associated with +CBPs may have some relationship with jets or spicules +observed deeper in solar the atmosphere, which may +contribute toward transient intensity variations in the +CBPs. Regarding small-scale magnetic flux emergence, +our attempt to explore its effects on already existing +CBPs is motivated by two very recent papers: Tiwari +et al. (2022), which find tiny EUV bright dot-like sub- +structures inside a CBP that seem to be associated with +small flux emergence episodes; and N´obrega-Siverio & +Moreno-Insertis (2022), which argue that flux emergence +occurring in a few granules may be enough to destabilize +a CBP and lead to eruptions. +To achieve our objectives, we use a high-quality, +ground-based dataset from the Swedish 1-m Solar Tele- +scope (SST, Scharmer et al. 2003) in coordination with +the Atmospheric Imaging Assembly (AIA, Lemen et al. +2012) instrument on-board NASA’s Solar Dynamics Ob- +servatory (SDO, Pesnell et al. 2012). For the first time, +we employ high-resolution images of the chromospheric +Hβ spectral line to study the spicule-CBP relationship. +Moreover, the impact of multiple small-scale photo- +spheric flux emergence episodes on the chromospheric +and coronal activity are also investigated from coordi- +nated, high-resolution magnetic field measurements. +The rest of the paper is divided as follows. Section 2 +describes the observations and standard data reduction +processes. Section 3 details the methodology employed +to detect on-disk spicules from SST observations and +enhancing the AIA images. +We show the results and +discuss their significance in Sect. 4, before finally sum- +marizing and concluding the paper in Sect. 5. +2. OBSERVATIONS AND DATA REDUCTION +2.1. Swedish 1-m Solar Telescope +For the purpose of this study, we recorded the chro- +mospheric counterparts of the CBP using observa- +tions from the CHROMospheric Imaging Spectrome- +ter (CHROMIS, Scharmer 2017) and CRisp Imaging +Spectropolarimeter (CRISP, Scharmer et al. 2008) in- +struments at the SST on 4 August 2021, under excel- +lent seeing conditions. +The coordinates of the target +were centered around solar (X,Y ) = (250′′,358′′) with +µ = cos θ = 0.88 (θ being the heliocentric angle), and +the observation sequence lasted for about 11 min start- +ing at 09:56 UTC. Figure 1 shows an overview of the +observed target. +CHROMIS sampled the Hβ spectral line centered at +486.1 nm under imaging spectroscopic mode across 27 +wavelength points between ± 0.21 nm with respect to +the line center. +The sampling was uniform between +± 0.1 nm with 0.01 nm steps. Beyond this a non-uniform +sampling was intentionally chosen so as to avoid the ef- +fect of blends. +Panels (e)–(g) of Fig. 1 show the Hβ +blue (at a Doppler offset of −25 km s−1), red wing (at +a Doppler offset of +25 km s−1), and line core images, +respectively. +The cadence of the data was 6.8 s with +a spatial sampling of 0.′′038. CHROMIS also recorded + +Chromosphere underneath a CBP +3 +Figure 1. Overview of the targeted CBP observed on 4 August 2021 at 10:03:31UT. Panel (a) shows an RGB composite image +of the CBP and its neighboring area at the original SDO/AIA pixel scale. Red, blue, and green colors correspond to 30.4, 19.3 +and 17.1 nm channels, respectively. The SST/CHROMIS pointing and FOV is overlaid as a reference. Panels (b)–(d) illustrate +SDO/AIA 30.4, 17.1 and 19.3 nm channels that are rotated and co-aligned to CHROMIS. Panels (e)–(g) show CHROMIS Hβ +images at blue wing (− 25 km s−1), red wing (+ 25 km s−1) and line center, respectively. These images depict the chromospheric +scene underneath the CBP. Panels (h) and (i) contain the photospheric Hβ and Fe i 617.3 nm WB images, and panel (j) shows +the photospheric LOS magnetic field map (BLOS) saturated between ± 60 G (black indicated positive polarity). The dashed +FOV shown in panels (a)–(j) denotes the region-of-interest associated with the CBP which forms the basis for all investigations +carried out in this paper. +wideband (WB) images with the help of an auxillary +WB channel centered at 484.5 nm (referred to as Hβ +WB in panel h). Besides providing context photospheric +images, the WB serves as an anchor channel that aids +in image restoration. The WB images have the same +cadence as the narrowband Hβ sequence. +CRISP sampled the Fe i 617.3 nm line across 14 wave- +length points under imaging spectropolarimetric mode +between −0.032 nm and +0.068 nm with respect to the +line center. The full Stokes Fe i 617.3 nm data were in- +verted using by using a parallel C++/Python implemen- +tation1 of the Milne-Eddington (ME) inversion scheme +developed by de la Cruz Rodr´ıguez (2019) to infer the +photospheric vector magnetic field information. In ad- +dition, the Ca ii 854.2 nm line was sampled across 4 +wavelength points between −0.1 nm and +0.05 nm with +respect to the line core in steps of 0.05 nm under imag- +ing spectroscopic mode. The overall cadence of the com- +bined observation sequences was measured to be 18.5 s +with a spatial sampling of 0.′′058. In this paper, we only +focus on the line-of-sight (LOS) magnetic fields inferred +1 https://github.com/jaimedelacruz/pyMilne +from the Fe i 617.3 spectral line as shown in panel (j) of +Fig. 1. +The combination of excellent seeing conditions, the +SST adaptive optics system, the high-quality CRISP +and CHROMIS re-imaging systems (Scharmer et al. +2019), and Multi-Object Multi-Frame Blind Deconvolu- +tion (MOMFBD, van Noort et al. 2005) image restora- +tion resulted in high-spatial resolution data down to the +diffraction limit of the telescope (for Hβ 1.22λ/D = 0.′′13 +with D = 0.97 m the effective aperture of SST). The +SSTRED reduction pipeline (de la Cruz Rodr´ıguez et al. +2015; L¨ofdahl et al. 2021) was used to facilitate reduction +of the data, including the spectral consistency technique +described in Henriques (2012). Furthermore, both the +CRISP and CHROMIS time series were destretched to +compensate for the residual warping across the field-of- +view (FOV) which was not accounted for by the image +restoration techniques described earlier. +For this study, the CRISP data (with a lower spatial +and temporal resolution) were co-aligned to CHROMIS +by expanding the former to CHROMIS pixel scale fol- +lowed by a cross-correlation between the respective pho- +tospheric WB channels shown in panels (h) and (i) of +Fig. 1. In other words, the CHROMIS data with a FOV + +40 +(b) +40 +40 +(c) +(d) +30 +30 +30 +20 +20 +20 +Y +10 +10 +10 +17.1 +30.4 +nm +19.3 +(a) +Y1 +wu +0 +nm +0 +0 +390 - +10 +20 +30 +40 +50 +60 +10 +20 +30 +0 +0 +40 +50 +60 +0 +10 +20 +30 +50 +CHROMIS FOV +60 +380 - +40 - +40 +40- +(e +(f) +(g) +[arcsec] +30 +30 +30 +20 +20 +20 +1 +10 +10 +10 +340 - +1B +-25 +km +0. +0 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +30 +40 +50 +60 +330 +19.3 +nm +17.1 +40 +40 - +(i) +(j) +320 +30.4 +240 +220 +260 +280 +30 +30 - +30 - +Solar X [arcsec] +20 +20 +10 +10 +10 +. +0 +0 +0 +10 +20 +30 +40 +50 +60 +0 +10 +20 +40 +50 +60 +10 +20 +30 +40 +0 +50 +60 +0 +Xi [arcsec] +Xi [arcsec] +Xi [arcsec]4 +Bose et al. +of 66′′ × 42′′ and a cadence of 6.8 s served as a reference +for the CRISP data to which the latter was aligned. We +used nearest neighbour interpolation for the temporal +alignment. +2.2. Solar Dynamics Observatory +The coronal part associated with the CBP was ob- +served with the AIA instrument on-board SDO. The +SDO datasets were co-aligned to SST (CHROMIS) +datasets in the following manner. +The SDO image +cutout sequences were first downloaded from the Joint +Science Operations Center’s (JSOC) website2. Next, the +images from all the AIA channels were co-aligned to +HMI continuum images (here the AIA 30.4 nm channel +was aligned to HMI continuum), followed by the lat- +ter’s co-alignment to CHROMIS WB channels via an +iterative cross-correlation algorithm. Finally, the SDO +images were cropped to have the same FOV as SST. The +end result of this pipeline is a co-aligned SDO dataset +that consists of eleven (nine AIA and two HMI) image +sequences that are expanded from their original pixel +scale to CHROMIS pixel scale of 0.′′038 and matched in +time by nearest-neighbour sampling to CHROMIS tem- +poral cadence. We used the publicly available3 Inter- +active Data Language (IDL) based automated pipeline +developed by Rob Rutten for this purpose (Rutten 2020) +and refer to Bose et al. (2021b) for an example of this +pipeline’s application. +An RGB composite image, consisting of AIA 30.4, 17.1 +and 19.3 nm channels, of the CBP target at the origi- +nal AIA resolution is shown in Fig. 1 panel (a), while +panels (b)–(d) show the same three channels but ro- +tated and co-aligned to the CHROMIS data using the +procedure described in this section. +This co-aligned +SST and SDO dataset was then visualized extensively +with CRISPEX (Vissers & Rouppe van der Voort 2012), +an IDL widget-based tool that allows an efficient si- +multaneous exploration of multi-dimensional and multi- +wavelength datasets. +3. METHODS EMPLOYED +3.1. Detecting on-disk spicules from Hβ +We employed an automated detection method based +on the difference between images observed in the blue +and red wings of the Hβ spectral line. This is similar +to constructing Dopplergrams (see Sekse et al. 2012; De +Pontieu et al. 2014; Pereira et al. 2016), but instead of +subtracting fixed wavelengths on opposite sides of the +line center, an average over a range of wavelengths (be- +tween ±20–30 km s−1 on opposite sides of the line cen- +ter) is computed which are then subtracted from one +another as shown in Fig. 2 (a). The difference images +2 http://jsoc.stanford.edu/ +3 https://robrutten.nl/rridl/00-README/sdo-manual.html +are then subjected to unsharp masking which causes an +enhancement in the high spatial frequency components +of the image. In this case, it amplifies the threaded spic- +ular features as seen in the panel (b). RBEs appear as +darker threads with negative intensity values whereas +RREs appear brighter with positive intensity values in +these difference images. +It is important to note that +the difference maps so obtained (as in panel b) do not +correspond to absolute measure of the Doppler velocity +associated with RBEs and RREs. The chief goal is to +obtain a representation of the spatiotemporal evolution +of the velocity patterns associated with these features. +Next, an adaptive intensity thresholding technique +was applied on each of the difference images where pix- +els which had intensities above a certain value on either +side of zero were masked and chosen for further pro- +cessing. As a result, two different binary masks were +generated: one which comprised of pixels that satisfied +IUSM > 2.5σ for RREs and the other one which satisfied +IUSM < −1.5σ were considered for RBEs, where IUSM is +the intensity of the difference image post unsharp mask- +ing (Fig. 2 b). The difference in the threshold is due +to the skewness in the distribution of RREs and RBEs +in the difference maps. In both the masks, pixels which +satisfied the thresholding criterion were assigned a value +of 1 while the remaining was assigned as 0. Once the +binary masks were generated, a morphological opening +followed by a closing operation was applied to each of the +masks (independently for the RBEs and RREs), on a per +time step basis, with a 3×3 diamond-shaped structuring +element. We refer the reader to Bose et al. (2021a) and +appendix A.2 of Bose (2021) for more details on these +morphological operations and the associated reasoning +behind them. +Finally, connected component labeling in 3D (i.e. +combining both spatial and temporal dimensions, see +Rosenfeld & Pfaltz 1966) was performed on the morph +processed images so that the RBEs and RREs can be +uniquely identified based on a given heuristic. +Basi- +cally, this technique allows connected neighboring pix- +els in spatio-temporal domain to be uniquely identified +(labeled). To not bias for a particular direction, we em- +ployed a 26-neighborhood connectivity criterion in 3D +space for this purpose. In other words, two pixels were +“connected” if they shared either an edge, a face or a +corner. Furthermore, to avoid erroneous detections and +focus primarily on the elongated spicular structures a +lower cutoff length of ∼200 km (or 8 CHROMIS pixels) +was also imposed on the labeled events. +The above recipe led to a detection of 6457 uniquely +labeled events (3623 as RREs/downflowing RREs and +2834 as RBEs) in the complete dataset lasting 11 min +over the whole FOV. The occurrence of these (combined) +events is shown in the form of 2D probability density +map in Fig. 2 (c) against a background of temporally +averaged Hβ WB image. + +Chromosphere underneath a CBP +5 +Figure 2. Overview of the automated on-disk spicule detection method described in the text. Panel (a) shows the spatio- +temporal average of the Hβ spectral line computed over the entire CHROMIS FOV, and further illustrates how Dopplergrams +are generated by subtracting signals in the blue wing from the red wing (indicated by the shaded areas on either side of the line +center). Panel (b) shows an example of a generated Dopplergram where RBEs and RREs show up as dark and bright threaded +structures. Panel (c) shows the location and the density distribution of the detected spicules against a background of temporally +averaged Hβ WB image. +3.2. Enhancing the AIA images +To facilitate a better understanding of the dynamic re- +lationship between the chromospheric and coronal coun- +terparts of a CBP, it is crucial to enhance the visibility +of the coronal images and the loops (strands) associ- +ated with the CBP. In this regard, the re-sampled (to +CHROMIS pixel scale) AIA images, like the ones shown +in the leftmost column of Fig. 3, are subjected to a mod- +ified version of the common difference technique where +the temporal average, over the entire 11 min duration, of +each AIA channel is subtracted from an unsharp masked +image of the same channel for each time step. This pro- +cedure results in images where small changes in the in- +tensity are visibly more enhanced– due to unsharp mask- +ing which adjusts the contrast of the edges (see the right- +most column of Fig.3). In addition, the AIA images are +also subjected to a multi-scale Gaussian normalization +(MGN) procedure (Morgan & Druckm¨uller 2014) that +enables a better visualization of the overall topology and +the orientation of the overlying coronal structure which +is not very prominent in original (resampled) AIA im- +ages. They are shown in the middle column of Fig. 3. +The various AIA channels used in this study are MGN +enhanced by using the default (same) values of weights +and coefficients as in Morgan & Druckm¨uller (2014). +The animation associated with Fig. 3 provides a bet- +ter idea of the advantage of employing the two methods +described above and further adds to their comparison +with the original co-aligned AIA images. We immedi- +ately notice an improvement over the coronal images +shown in the left column, where the loops associated +with the CBP are barely noticeable. Consequently, the +variation in the intensity of the CBP associated with +rapid spicular dynamics is shown with the common dif- +ference images while the MGN processed images are +used as a proxy of the intensity variation in the CBP +for all subsequent analysis and results described in this +paper. However, it is important to note that MGN does +not preserve the photometric accuracy of the images and +creates a standardized emission, which is enhanced (sub- +dued) in the regions with lower (higher) intensity. This, +however, does not impact the analysis presented in this +paper. +4. RESULTS AND DISCUSSION +This section presents a detailed description and dis- +cussion of the results obtained from the analysis. We +begin by investigating the chromospheric foootpoints of +the CBP in Sect. 4.1, followed by a description of rep- +resentative examples highlighting the spicule-CBP rela- +tionship in Sects. 4.2 and 4.3. Finally, in Sect. 4.4, we +discuss the impact of two small-scale photospheric flux +emergence episodes in the chromosphere and the hotter +AIA channels. +4.1. The chromospheric ”footpoints” of the CBP +The Hβ wing and the line core images, shown within +the dashed FOV in panels (e)–(g) of Fig. 1, depict the +chromospheric scene underlying the CBP. The images +clearly show multiple dark, elongated, and threaded +structures that resemble spicules (or mottles). A zoom- +in to the dashed FOV is shown in Fig. 4 which focuses +solely on the region in and around the CBP. To aid bet- +ter visualization of the intensity disturbances propagat- +ing in the CBP, we show the common difference images +for the different AIA channels in panels (a) through (c). +Panel (d) shows the Hβ line core (LC) width map which +is basically the wavelength separation at half the inten- +sity range between the minimum of the Hβ line profile +and the average intensities at a displaced wing position +from the line center (following Cauzzi et al. 2009) for +each pixel on the FOV. In this case the displacement pa- +rameter was set at ± 66 pm from the line center which +was determined by converting the displacement param- +eter of 90 pm for the Hα spectral line, chosen by Cauzzi +et al. (2009), into equivalent Doppler units (km s−1). +RBEs and RREs (including downflowing RREs) appear +to be in “emission” (compared to the background fea- +tures as seen in panel d) in these maps since they gener- + +40 +Hβ +2250 +2000 +30 +1750 +３ +1500 +1250 +1000 - +# +10 +750 - +0 +(a) +500 +-100 +-50 +50 +100 +10 +20 +0 +40 +50 +60 +10 +20 +30 +40 +50 +60 +0 +0 +M [km s-1] +Xi [arcsec] +Xi [arcsec]6 +Bose et al. +Figure 3. Methods of enhancing the AIA images. The top row (from left to right) shows a zoom in to dashed FOV of AIA +17.1 nm intensity map indicated in Fig. 1 panel (c), the MGN processed version of the same, and the result of applying the +modified common difference technique (see text for details) to the original AIA map, respectively. Bottom row (from left to +right) illustrates the result of applying the two enhancement techniques to AIA 19.3 nm channel in the same format as the top +row. An animation of this figure is available online, which shows a comparison between the different enhancement techniques +along with the temporal evolution of the disturbances propagating along the loops. The animation shows solar evolution over +11 min. +ally have enhanced opacity owing to their broad LOS ve- +locity distribution (Pereira et al. 2016; Bose et al. 2021a) +and enhanced temperature (Leenaarts et al. 2012). Pan- +els (e) and (f) show the co-temporal Hβ line core in- +tensity and the LOS photospheric magnetic field maps +underneath the CBP. Spicules and/or mottles dominate +the whole FOV and they are seen to be predominantly +rooted in the close vicinity of the strong (negative) po- +larity magnetic field patch which also happens to be the +photospheric magnetic roots of the CBP. +A glance at panels (b)–(e) of Fig. 4 immediately sug- +gests that the CBP loops and their chromopsheric coun- +terparts bear a close morphological resemblance. This +is further highlighted in the animation associated with +the figure where the 17.1 and 19.3 nm loops appear to +have propagating disturbances nearly in tandem with +the rapid changes in the chromosphere, especially to- +ward the later half of the data sequence. The 30.4 nm +common difference image appears to be noisier and it +does not show the loops associated with the CBP as +prominently as in the other AIA channels. +However, +the animation shows clear disturbances associated in the +same region as underlying spicules but the overall mor- +phology is less pronounced (compared to panels b and +c) making them difficult to relate visually. +The lack +of loop-like appearances in the 30.4 nm channel could +be attributed to its relatively lower temperature sensi- +tivity (log T(K) ∼ 4.7) compared to 19.3 and 17.1 nm +channels which have a peak temperature sensitivity of +around log T(K) ∼ 6 (Boerner et al. 2012). Moreover, it +is rather common to observe the relatively cooler foot- +points of the CBPs in the 30.4 nm channel underneath +the hotter loops (Kwon et al. 2012; Madjarska 2019; +Madjarska et al. 2021) which may further justify the +less pronounced morphological resemblance. +Madjarska et al. (2021) report that the chromospheric +counterpart of a CBP largely comprises of elongated, +dark features when observed in the Hα line core im- +ages. They name these features “Hα loops” which also +appear to constitute a fundamental part of the overall +magnetic structure of the CBPs. While we do not find +the existence of such loops likely due to our observations +being limited to a part of the entire CBP (thereby miss- +ing the opposite polarity), spicules dominate our FOV +and plays a central role in driving the dynamics of the +chromosphere underneath the CBP. +Figure 5 shows the occurrence of the detected on-disk +spicules, using the method described in Sect. 3, in the +form of a 2D density map against a background of tem- +porally averaged images for four MGN processed AIA + +Aligned AlA +MGN processed +Unsharp mask/Common difference +17 +19.3 nm +15 +Y[ar +5 - +10:06:09 UTC +10 +20 +5 +25 +0 +15 +X [arcsec]Chromosphere underneath a CBP +7 +Figure 4. The chromosphere and the photosphere underneath the CBP. Panels (a)–(c) show common difference images in the +AIA 30.4, 17.1 and 19.3 nm channels at 10:06:43UT. Panel (d) shows the co-temporal Hβ LC width map saturated between 0.45– +0.82 ˚A. Panel (e) shows the co-temporal Hβ LC image and panel (f) depicts the corresponding photospheric BLOS map saturated +between ± 60 G. An animation of this figure is available online, which shows the temporal evolution of the chromospheric and +photospheric scenery underneath the CBP for the entire duration of 11 min. +Figure 5. Morphological similarities between spicules and the loops associated with the CBP. Panels (a) and (c)–(f) show the +2D density map of the detected spicules overlaid against a background of temporally averaged Hβ WB, MGN enhanced AIA +30.4, 17.1, 19.3, and 21.1 nm channels, respectively, whereas panel (b) shows a temporal average of the underlying photospheric +magnetic field saturated between ± 60 G. The FOV in each of the panels correspond to the dashed FOV indicated in Fig. 1. +channels (panels c–f) and an SST WB channel (Hβ, +panel a). As described in Sect. 3.2, the MGN processed + +06:4 +(b +(C +Difference 30.4 nm +Difference 17.1 nm +Difference 19.3 nm +(d) +(e) +(f) +60 +15 +-0.8 +¥4 +0.7 +20 +10 +0.6 +-20 +-40 +5 +0.5 +-60 +Hβ LC width [A] +Hβ line core +0 +5 +10 +15 +20 +25 +0 +X[arcsec](b) +4 +30.4 nm +15 +2 +10 +5 +19.3.nm +7 +nm +nm +10 +15 +20 +25 +0 +X [arcsec]8 +Bose et al. +images show the intensities in absolute units (though it +fails to preserve the photometric accuracy) unlike the +common difference images. From this figure it is clear +that the distribution of spicules is very well correlated +with the orientation and overall morphology of the CBP +loops, as is evident from the 17.1, 19.3 and 21.1 nm +channels. This provides a compelling observational con- +firmation (in a statistical sense) of spicules tracing the +coronal magnetic field lines which, to the best of our +knowledge, has not been reported before. +Moreover, +we also notice that the number density of the detected +spicules is predominantly located close to the footpoint +of the CBP loops. This scenario seems to suggest that +the studied spicules are the cromospheric components of +the CBP loops which, post heating, appear in the hot- +ter TR and coronal channels, and further contribute to- +ward the transient intensity disturbances in the already +hot CBP loops.(see for example De Pontieu et al. 2011; +Madjarska et al. 2011; Pereira et al. 2014; Rouppe van +der Voort et al. 2015; De Pontieu et al. 2017; Samanta +et al. 2019, and the references therein for studies about +the coronal counterpart of spicules). +We will explore +this aspect further with a few representative examples +in Sect. 4.2. +The +morphological +similarities +between +the +Hβ +spicules and the coronal loops associated with CBP in- +dicates the possibility that the loop structures are asso- +ciated with spicular mass ejections and transient heat- +ing of the plasma from chromospheric to coronal tem- +peratures. A direct investigation of such a connection +would however require a more detailed analysis by comb- +ing high-resolution numerical simulations with spectro- +scopic observations of the CBP. Nonetheless, some stud- +ies such as De Pontieu et al. (2017) already showed an +intriguing connection between spicules in the TR and +the formation of coronal strands in a decayed plage re- +gion with the help of numerical simulations and coordi- +nated IRIS and SDO observations. Moreover, spicules +were also found to be responsible in triggering propa- +gating coronal disturbances (PCDs) along many of the +pre-existing (and newly formed) coronal strands rooted +to the plage. PCDs are rapid recurring intensity fluc- +tuations (∼ 100 km s−1) whose exact nature remains +a mystery, especially outside of the sunspots (see De +Pontieu & McIntosh 2010; de Moortel 2009; De Moortel +et al. 2015; Bryans et al. 2016, for example, on the dis- +cussion whether PCDs are flows or waves). Therefore, +it is likely that the intensity disturbances observed in +the common difference coronal images are linked to the +rapid spicular dynamics in the chromosphere. +From Fig. 5 we also notice a significant overlap be- +tween the widths of the detected chromospheric spic- +ular features and the observed loops associated with +the CBP. Using coordinated observations from Hinode’s +Extreme ultraviolet Imaging Spectrometer and Transi- +tion Region and Coronal Explorer instruments, Dere +(2009) derived the volumetric plasma filling factor in +CBPs, and came to the conclusion that the widths of its +loops can be between 0.′′2–1.′′2 with possible substruc- +tures that are below the resolution limit of the instru- +ments. +Comprehensive statistical analysis carried out +by Pereira et al. (2012) and Bose et al. (2021a), indicate +that spicule widths, for both off-limb and on-disk cases, +are consistent with the range reported by Dere (2009) +which further suggests that the Hβ spicules detected in +this study are likely the chromospheric counterparts of +the CBP. +Numerical modeling efforts led by Mart´ınez-Sykora +et al. (2018) offer key insights into the role of spicules in +determining the widths of the coronal loops. They re- +port that the widths of the simulated spicules (and sub- +sequently the coronal loops) are primarily determined by +the driving mechanism that generates these flows, along +with the overall magnetic topology and heating within +the magnetic field lines. Moreover, they find that the +magnetic field rapidly expands primarily between the +photosphere and middle to upper chromosphere where +spicules are seen to be generated (in the model). The +expansion of the field line is rather insignificant between +the transition region and the corona which may explain +why the CBP loops and spicules appear to have similar +widths. +4.2. Representative examples of spicular-CBP +connection +In this section we further illustrate the spicule-CBP +connection discussed in Sect. 4.1 through two repre- +sentative examples shown in the left and right panels +of Fig. 6 including their signatures in the TR (AIA +30.4 nm) and coronal passbands. +We show the com- +mon difference images for the different AIA channels +(in the left columns of each of the two panels) in or- +der to enhance the visibility of the changes in intensity. +The dashed vertical yellow lines in the left columns of +both panels show the region of interest that is chosen +to construct the x-t maps. Moreover, in addition to the +common difference, we also show the x-t maps derived +from the MGN processed AIA images and Hβ LC width +maps to highlight the temporal evolution of the plasma +emission from each channel. +The left panel shows an example of an RBE in the +blue wing of Hβ (at −25 km s−1). +From the anima- +tion and the x-t maps (top row), it is clear that the +RBE has an outward (away from the bright network re- +gions) apparent motion and propagates from ∼ 2′′ to +6′′ in the vertical direction during its evolution. This +is a commonly observed property of spicules where they +originate from strong magnetic flux concentrations and +tend to shoot outwards. Since spicules often have a wide +range of Doppler shifts associated with them (Pereira +et al. 2016; Bose et al. 2021a), analysis based on images +at fixed wavelength positions can sometimes provide an +incomplete picture of their evolution. In such cases LC +width maps offer a better understanding since they are + +Chromosphere underneath a CBP +9 +Figure 6. Two representative examples highlighting the spicule-CBP connection from the chromosphere to the corona. Left +panel: an example of an RBE observed in the blue wing (− 25 km s−1) of Hβ spectral line and its associated propagation in +the different AIA passbands as indicated. The dashed vertical lines in the leftmost column indicate the region along which the +x-t maps have been extracted for the different channels as shown in the middle and the rightmost columns. The solid vertical +red lines in the x-t maps correspond to the instant at which this figure is shown. The dashed cyan line serves as a reference to +illustrate the direction of propagation. Right panel: another example of a spicule observed in the blue wing (− 30 km s−1) of +the Hβ spectral line is shown along with its impact on the AIA channels in the same format as the left panel. Note that the +apparent direction of propagation of this spicule is opposite to the example presented in the left panel. Animations of the two +panels are available online. They show the spatio-temporal evolution of the two spicules in the chromospheric Hβ, transition +region 30.4 nm and coronal 17.1, 19.3 and 21.1 nm channels during their respective lifetimes. +determined by considering a range of wavelengths on ei- +ther side of the line center (see Sect. 4.1). In the present +example however, the x-t maps derived from the LC +widths and Hβ blue wing images are seen to be well +correlated with each other. +A comparison of the spatio-temporal evolution seen +in the corresponding AIA channels show a noticeable +correlation with the Hβ counterpart. An inspection of +the animation of the AIA difference images shows clear +intensity disturbances propagating in the CBP which +appear to be in tandem with the Hβ spicule. The 19.3 +and 17.1 nm difference images, in particular, show a +clear propagation from the bottom to the top of the +FOV. This is also well highlighted in the difference x-t +maps (middle column). The 30.4 nm difference images, +on the other hand, does not show such a clear prop- +agating disturbance however, the x-t maps reveal clear +signatures which are also in tandem with the other wave- +length channels. +A close look at the different x-t maps associated with +the left panel of Fig. 6 reveals a small but distinct spa- +tial (and/or) temporal offsets among the different chan- +nels (with respect to the dashed cyan line) – with the +TR and coronal emission lying above the cooler chro- +mospheric plasma. Such a scenario is consistent with +the analysis presented by De Pontieu et al. (2011); +Pereira et al. (2014), and it suggests that the RBE +has a multi-thermal nature with temperatures that can +range from chromospheric to coronal temperatures (of at +least 1 MK). In fact, an early study focusing on multi- +wavelength diagnostics of a CBP by Habbal & With- +broe (1981), found that coronal emission in CBPs lie a + +x-t +7-X +Hβ -25 kms +Hβ -25kms- +HBLCwidth +81 +ec] +[arcse +6 +istance +4 +2 +0 +19.3nm +8 +ec] +Distance [arcse +6 +4 +D +Z +0 +Difference 17.1 nm +Difference17.1nm +17.1nm +[arcsec] +6 +Distance +4 +Z +0 +Difference : +30.4 nm +30.4nm +8 +Difference30.4nm +[arcsec] +6 +istance +4 +D +2 ++ 0 +0 +2 +4 +6 +0 +100 +200 +100 +200 +Time [s] +Time [s] +Distance [arcsec]x-t +7-X +Hβ -30 kms +Hβ-30 +HB LCwidth +kms-1 +istance [arcsec] +6 +4 +2 +0 +8 +Difference 19.3 nm +Difference19.3nm +19.3nm +ec] +Distance [arcse +6 +4 +Z +0 +Difference17.1nm +17.1nm +8 + [arcsec] +6 +Distance +4 +Z +0 +8 +Difference 21.1 nm +Difference21.1nm +21.1 +nm +secl +[arcse +6 +istance +4 +2 ++ 0 +0 +2 +4 +6 +0 +100 +200 +100 +200 +Time [s] +Time [s] +Distance [arcsec]10 +Bose et al. +few arcseconds over an above the chromospheric emis- +sion suggesting the hypothesis that magnetic loops in a +CBP are rooted in the chromosphere. The spatial offset +between the TR (30.4 nm) and coronal (17.1/19.3 nm) +emission patterns is indicative of the fact that the emis- +sion in the coronal channels are not caused by relatively +cooler ions (such as O v, see Boerner et al. 2012) which +are sensitive to temperatures of about 0.2 MK under +equilibrium conditions. Moreover, the emission from the +O v is expected to be very faint in comparison to the +dominant Fe ix and Fe xv ions and would have occurred +in the same spatial region as the 30.4 nm emission. This +further adds support in favor of the spicular contribution +to coronal emission associated with the CBP. +The right panel of Fig. 6 shows another example of +spicular connection associated with the CBP in the same +format as the previous example. Unlike the left panel, +the spicule appears to be downward propagating as is +evident from the animation and the x-t maps in the +middle and right columns. A quick glance at the Hβ im- +age would suggest that the example here is a blue-wing +counterpart of downflowing RREs (seen in the red wing +images of Hα, see Bose et al. 2021a,b). However, a closer +inspection of the animation reveals a rather complex sce- +nario where the spicule is rapidly seen to change its ori- +entation (with respect to the LOS of the observer) dur- +ing its propagation, before finally disappearing around +t = 180 s. Such a complex propagation seems to con- +vey that the spicule is downward propagating, which in +reality could be the opposite. +Regardless of the interpretation associated with the +orientation of the spicule and its mass flow, interestingly +(and more importantly), the x-t maps of the coronal +channels show a remarkable correlation with Hβ. More- +over, the spatial offsets among the different channels are +consistent with the discussion presented in the previous +example and conforms with the multi-thermal aspect of +the spicule and its relation to the CBP. This supports +our proposition that spicules in the chromosphere have a +direct relationship with the disturbances propagating in +the CBPs. Although many questions remain, this may +also provide support to the idea that the processes asso- +ciated with spicules may play a role in providing mass +and energy flux necessary to sustain the radiative and +conductive energy loses in the solar corona as suggested +in the numerical simulation studies by Mart´ınez-Sykora +et al. (2017, 2018). +4.3. Twists at the footpoints of the CBP +Spicules are known to undergo twisting (torsional) +motions that are often interpreted as a sign of Alfv´enic +waves responsible for driving the fast solar wind and bal- +ancing the energy losses suffered in the solar corona (De +Pontieu et al. 2007; McIntosh et al. 2011; De Pontieu +et al. 2012). +Moreover, small-scaled twists associated +with spicules are ubiquitously found in the solar atmo- +sphere (active regions and quiet Sun alike), and their +Figure 7. Chromospheric twist and its likely propagation +into the solar corona. This figure is in the same format as +Fig. 6 except that the Hβ wing images is replaced by its +Dopplergram at ± 25 km s−1. Blue (red) color is indicative of +plasma motion toward (away from) the observer. The associ- +ated animation (available online) shows the spatio-temporal +evolution of the twist propagation across the chromospheric +and coronal channels for roughly 300 s. +signatures have also been found in the TR (De Pontieu +et al. 2014). +In Fig. 7, we show a case of twist associated with +spicules present at the footpoints of the CBP and their +influence on the coronal loop above. The top row of the +figure shows an Hβ Dopplergram at ± 25 km s−1 with +blue and red colors indicating plasma motions toward +and away from the observer along the LOS. The cor- +responding x-t map and the animation shows a clear +change of direction (or color from blue to red) indi- +cating a definite twisting motion in the chromosphere. +A close look at the animation indicates that the event +starts with a predominantly positive (red) Doppler shift +at t = 0 s which rapidly converts to a negative (blue) +Doppler shift in roughly 30 s. It remains predominantly +negative until around t = 180 s after which it rapidly +twists toward positive (red) once again. +This behav- +ior is very similar to the examples observed in the Hα +and Ca ii 854.2 spectral lines for both off-limb and on- + +7-X +7-X +Hβ +25 +kms +Hβ ±2 +25kms +HB LCwidth +8 +[arcsec] +6 +istance +4 +2 +0 +19.3nm +Difference 19.3nm +Difference19.3nm +ec] +Distance [arcse +6 +4 +D +Z +0 +Difference17.1nm +17.1nm +8 +Distance [arcsec] +6 +4 +2 +0 +Difference 21.1 nm +Difference21.1nm21.1 +8 +nm +[arcsec] +6 +Distance +4 +Z ++0 +0 +2 +4 +6 +0 +100 +200 +300 +100 +200 +300 +Time [s] +Time [s] +Distance [arcsec]Chromosphere underneath a CBP +11 +disk spicules as outlined in De Pontieu et al. (2012) and +De Pontieu et al. (2014). +The propagation speed (of +roughly 35 km s−1) is fairly consistent (given the uncer- +tainty related to the viewing angle) with Alfv´en speeds +at chromospheric heights (De Pontieu et al. 2007). The +LC width x-t map shows a similar trend as the Doppler- +gram x-t map where additionally we see that the plasma +associated with the twisting motion stands out distinctly +with respect to the background. +The x-t maps associated with the difference and MGN +enhanced AIA 19.3, 17.1 and 21.1 nm channels show +strong emission which are in tandem with their chromo- +spheric counterpart (similar to the examples shown in +Fig. 6). We also notice a significant offset in the emis- +sion among the different channels indicating that the +plasma is heated to temperatures of at least 1–2 MK +(refer to the discussion in the previous section), in asso- +ciation with the twisting spicules at chromospheric tem- +peratures. Of course, a complete analysis of such a twist +propagating in the coronal loops necessitates spectro- +scopic studies of the solar corona which is not possible +with the set of instruments used in this study. Moreover, +current observations suggest the prevalence of Alfv´enic +waves in the corona (e.g., Tomczyk et al. 2007; McIn- +tosh et al. 2011) although the wave energy flux and +wave modes are poorly captured with current instru- +mentation. Alfv´enic waves have the potential to heat +coronal loops (Antolin et al. 2018), in particular when +mass flows are present within a structure in addition to +waves (as shown for example in Taroyan 2009; Williams +et al. 2016). Upcoming space missions, such as Solar- +C/EUVST or the MUlti-slit Solar Explorer (MUSE, De +Pontieu et al. 2020, 2022), can potentially address these +aspects. +4.4. Chromospheric and coronal response to emerging +magnetic flux +In this section, we investigate the chromospheric and +coronal responses to two emerging flux episodes as ob- +served from the LOS magnetic field. +4.4.1. Flux emergence episode 1 +Figure 8 and associated animation show an overview +of the first episode. In the LOS magnetogram of panel +(a), a negative parasitic polarity is seen to emerge in a +predominantly positive region. In the figure, the area +where this episode takes place is bounded by a green +square region of 100×100 pixels. The variation of the +total positive and negative magnetic fluxes within this +green square is shown in panel (f) where we find that +the total negative flux starts to increase steadily af- +ter 10:01:31UT, reaching its peak value of ≈4×1021 Mx +around 10:06:30 UT. +We investigated the subsequent chromospheric re- +sponse to the flux emergence episode by analyzing the +Hβ LC width maps (panel (b)). Compared to looking +simply at the Hβ line core, the LC width is optically +thin toward the dense fibrilar canopies visible in the line +core images, thereby facilitating a better understanding +of the “connection” between the spicules and their pho- +tospheric footpoints. In particular, we obtain the light +curve within the green square since changes in the chro- +mosphere and spicules associated with this flux event +would likely be rooted near this region. The result is +shown as a black curve in panel (e). There is a clear en- +hancement in the Hβ LC width intensity starting from +≈ 10:01 UT (around the same time as the total negative +flux in panel (f) starts to show a marked increase). The +LC intensity continues to increase until ≈10:03:30 UT +after which it starts to decay and reaches a minimum +around 10:05:15 UT, incidentally after which the in- +crease in the total negative flux (by ≈300% from the +start of the event) also tends to stabilize. It seems that +as the flux emerges and subsequently interacts with the +dominant positive polarity through magnetic reconnec- +tion (see Section 4.4.2), there is a significant enhance- +ment in the spicular activity compared to the any of the +previous time steps implying a correlation between the +two. This is consistent with the analysis presented in +Madjarska et al. (2021), in the context of chromospheric +response to flux emergence associated with a CBP, and +also Samanta et al. (2019), in general. +However, un- +like Samanta et al. (2019), it is not implied that the +flux emergence (and subsequent cancellation) leads to +the “generation” of spicules seen in close proximity. In- +stead, it is more appropriate to say that the emergence +likely caused an enhancement in the observed spicular +activity. We notice the presence of spicules both well +before and after the emergence event as is evident from +the animation. +We have also investigated the coronal response using +the SDO/AIA 19.3 and 17.1 channels (panels (c) and +(d)) along with the 21.1 nm channel. In this case, the +corresponding light curves, shown in panel (e), are ob- +tained in a region that is spatially displaced from the +emerging flux region (the blue rectangle of 200×300 +pixels in panels (c) and (d)). +This is because it was +shown earlier on in the paper that spicules lie close to +the footpoints of the CBP structure, whereas the coro- +nal loops extend well beyond and are spatially (and/or +temporally) displaced. +Before 10:01:31 UT, the 17.1, +19.3, 21.1 nm light curves have very similar intensity +levels relative to the maximum of each channel. As soon +as the chromospheric activity starts to increase from +10:01:31 UT, we notice a co-temporal increase in the +intensity of the 17.1, whereas the 19.3 and 21.1 channels +show a steady decrease compared to their respective pre- +event values. However, unlike the illustrative spicule ex- +amples shown in the previous section (i.e. in Figs 6 and +7) and also the many studies conducted in the past (such +as, De Pontieu et al. 2011; McIntosh et al. 2011; Hen- +riques et al. 2016) that do establish a coronal connec- +tion quite convincingly in different SDO/AIA channels, +for this particular enhanced spicular episode, the coro- + +12 +Bose et al. +Figure 8. Chromospheric and coronal response to flux emergence episode 1. Panel (a) shows the photospheric LOS magnetic +field map, (b) shows the Hβ LC width map, (c) and (d) shows the overlying corona of the CBP in AIA 19.3 nm and 17.1 nm +channels. The green square boxes drawn in panels (a)–(d) show the region associated with the flux emergence event. Panels (c) +and (d) also shows a blue rectangular box centered around (X,Y )=(7.′′5,15′′) where the coronal response to the emerging flux +is analyzed. +The black contour in panels (c) and (d) indicates the regions with an absolute LOS magnetic field ≥ 50 G. +Panel (e) shows the chromospheric and coronal light curves obtained from the green and blue FOVs indicated in panels (b)–(d), +respectively. Panel (f) shows the temporal variation of the total positive and negative magnetic flux in the region bounded by +the green colored box in panel (a). The gray shaded intervals in panels (e) and (f) shows the time when the chromospheric +spicular activity is enhanced, and the pink shaded regions indicates the interval when the two QSEBs are observed. Animation +of this figure is available online, which shows the evolution of the magnetic flux and its response in Hβ, AIA 19.3 and 17.1 nm +channels in the form of light curves for the entire 11 min of solar evolution. +nal relation is not clear. To not bias our interpretation, +we have also computed other light curves over different +AIA FOVs (of the same area) spatially displaced from +one another, finding similar results. +4.4.2. Reconnection associated with flux emergence 1 +Figures 9 and 10 show evidences of magnetic recon- +nection through two examples of EBs located in the +footpoint of the CBP associated with the emerging flux +episode described above. +We refer to them as quiet- +Sun EBs (QSEBs) after Rouppe van der Voort et al. +(2016) where these were first described. +QSEBs are +smaller, shorter-lived and less intense brightenings and +are found in relatively quieter areas on the Sun com- +pared to their active region counterparts. With the help +of high-quality Hβ observations from the SST, Joshi +et al. (2020); Joshi & Rouppe van der Voort (2022) +recently showed that QSEBs are ubiquitous in the so- +lar atmosphere and can play an important role in the +energy balance of the chromosphere. Recent numerical +modeling efforts led by Hansteen et al. (2017); Danilovic +(2017) and Hansteen et al. (2019) have confirmed that +(QS)EBs are classic markers of small-scale magnetic re- +connection events in the solar photosphere. +In this study, we base our analysis on the small +100×100 pixel FOV shown in panel (a) of Fig. 8 with a +focus on the flux emergence event. Panel (a) of Fig. 9 +shows the QSEB in the blue-wing Hβ intensity map (in- + +-60 +-40 +-20 +0 +20 +40 +60 +0.4 +0.5 +0.6 +0.7 +0.8 +G1 +width［A +OS +15 +15 +15 +15 +[arcsec] +10 +10 +10 - +10 +5 +5 +() +(p +(b) +0 +0 ++0 +0 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +X[arcsec] +X[arcsec] +X[arcsec] +X [arcsec] +intensities +1.00 +0.99 + width +and +A +0.97 +171 +Norm. +(e) +Hβ Icw +09:56:53 +09:59:12 +10:01:31 +10:03:50 +10:06:09 +Time [UTC] +1e23 +1e21 +4.0 +Positive flux [Mx] +1.6 +3.0 +2.5 +1.5 +2.0 +1.4 +Total +QSEB 1 +QSEB 2 +(f) +1.3 +0.5 +09:56:53 +09:59:12 +10:01:31 +10:03:50 +10:06:09 +Time [UTC]Chromosphere underneath a CBP +13 +Figure 9. Details of QSEB 1 observed during the flux emergence episode 1. Panel (a): QSEB observed in the far blue wing +of Hβ. Panel (b): temporal variation of the Hβ line profile for a location in the QSEB indicated by the magenta marker in +panel (a) in the form of a λ − t diagram. Panel (c): Hβ spectral line at a temporal instant indicated by the marker in panel (b) +and the spatio-temporal average Hβ reference profile (dashed black line). Panel (d): corresponding WB image. Panel (e): +corresponding LOS magnetic field map saturated between ± 60 G, and panel (f) temporal evolution of the total positive and +negative magnetic flux within the cyan box shown in panel (e). The dashed vertical line in (e) indicates the instant when this +figure is shown. Animation of this figure is available online and it shows the evolution of the magnetic field, the QSEB and the +corresponding Hβ spectra before, during and after the appearance of the QSEB for about 2.5 min of solar evolution. +dicated by the magenta marker). The animation shows +a tiny, flame-like brightening lasting for about 40 s. We +note that this period (along with the latter QSEB) is +marked in panels (e) and (f) of Fig. 8 as QSEB 1 and +2 respectively. +The Hβ spectral-time (λ − t) slice in +Fig. 9 (b) shows an enhancement in the line-wings com- +pared to the background which is reflected in the spec- +tra shown in panel (c). The observed spectral shape is +characteristic to an EB. Moreover, the co-temporal Hβ +WB image in panel (d) shows no such brightening which +clearly distinguishes this QSEB from a typical photo- +spheric magnetic bright point. Panels (e) and (f) show +the location of the QSEB on the LOS magnetic field +map and the evolution of the total positive and nega- +tive magnetic flux inside the smaller cyan box shown in +panel (e). From the animation and the light curves, we +see that the negative flux decreases up to about 40 s +from the start of the event after which it stabilizes up +to t = 55 s, following which it starts to decrease right +around the onset of the QSEB. +Figure 10 shows another QSEB event (QSEB 2) asso- +ciated with the same emerging flux region under consid- +eration. QSEB 2 is brighter, but shorter-lived (≈25 s) in +comparison to QSEB 1 and it is shown against a back- +ground of red-wing Hβ intensity map. Both the exam- +ples bear close morphological resemblance with distinct +flame-like brightenings. The λ − t slice and the spectra +for QSEB 2 also show characteristic EB-like behavior +but as is also evident from panel (c), the intensity en- +hancement is stronger compared to QSEB 1. The cor- +responding WB image (panel d) shows that QSEB 2 is +located in the intergranular lane, and from the BLOS +map we find that in this case the QSEB exists in the +intersection of opposite polarities. The variation of the +negative polarity flux in panel (f) shows an increase right +around the onset of the QSEB. +The examples presented in this section clearly show +that the emerging flux episode described in the previous +section have a definite impact not just in the form of en- +hanced chromospheric spicular activity, but also deeper +in the solar atmosphere where it reconnects and subse- +quently releases energy in the form of small-scaled EBs. +4.4.3. Flux emergence episode 2 + +(a) +150 - +(b) +(C) +3 - +125 +2.0 +[counts] +ec] +100 +1.5 +75 +t +103 +> +50 +× 1.0- +1 +25 - +HB-50kms- +0.5 +-0 +0 +0 +1 +2 +3 +-100 +-50 +50 +100 +-100 +-50 +0 +50 +100 +0 +X[arcsec] +△^[km s-1] +△^[km s-1] +1e21 +1e19 +(d) +(e) +2.07 +[Mx] + Positive flux [Mx] +1.5 +3 - +fluxI +1.4 +1.5 +[arcsec] +2 +1.3 +1.0 +> +> +1.2 +1 +1 - +Total +BLOS +1.1 +WB +10:04:24 UTC +±60G +- +-0 +2 +3 +0 +1 +3 +0 +25 +50 +75 +100 +125150 +X[arcsec] +X[arcsec] +t (s)14 +Bose et al. +Figure 10. Details of QSEB 2 associated with the flux emergence event 1. The figure and its associated animation (available +online) showing about 2.5 min of solar evolution displays the temporal evolution of the magnetic field, QSEB and the Hβ spectra +in the same format as Fig. 9. +In this section, we analyze the chromospheric and +coronal responses to another flux emergence episode +that occurred close to the footpoints of the CBP. Fig- +ure 11 depicts the overview of the episode in the same +format as Fig. 8, and the emergence is shown with a +green colored box (occupying the same area as before) +drawn in panel (a). A close inspection of the anima- +tion linked with panel (a) suggests a clear but rela- +tively smaller negative flux emergence episode lasting +≈ 5.5 min. This is also evident from panel (f) which +shows an increase in the total negative flux within the +cyan colored box starting around 09:57UT. The to- +tal negative flux increases by ≈180% compared to the +pre-event values and peaks around 09:59:12UT. It then +starts to decrease steadily reaching a minimum value of +0.1×1021 Mx at 10:03:50UT. The emerging negative po- +larity, however, starts to disappear around 10:02:37UT +(indicated by the gray region in panels e and f) from the +BLOS map. +We found a contrasting evolution of the light curves +associated with the chromospheric and coronal channels +for this emergence episode in comparison to the event +described in Sect. 4.4.1. +The Hβ LC width intensity +level in panel (e) does not show a marked increase in +tandem with the flux emergence and subsequent cancel- +lation; it maintains a steady level during the whole flux +emergence episode and only starts to increase well after +the total negative flux reaches its minimum value. The +dynamical evolution of the spicules seen in panel (b) +complements the variation of the Hβ LC light curve in- +dicated in panel (e) where we do not see any signifi- +cant enhancement in spicular activity compared to pre- +emergence scenario. The coronal channels behave sim- +ilarly where very little (or no) changes in their respec- +tive intensity levels are seen during the entire emergence +event, implying that none of the channels are impacted +directly by this emergence episode unlike the scenario +outlined in Sect. 4.4.1. Again, to not be limited to a +single FOV, we repeated the analysis by choosing dif- +ferent (rectangular) cyan FOVs in panels (c) and (d) +like before. +However, we did not find any differences +in the temporal variation of the AIA light curves. In +addition, we were also not able to find any signature of +QSEBs linked with this event. A possible explanation +could be that the strength of the emerging flux in the +second episode is at least a factor of three lesser than the +first episode, which reinforces our conclusion that there +is likely no impact on the chromosphere and the corona +associated with this weaker flux emergence event. +Identifying small-scale flux emergence events, such as +the ones described in this paper, can be a challenging +task. This is primarily because high-resolution observa- +tions of the solar photosphere reveal a myriad of mag- +netic features especially in the regions close to a net- + +(a) +(b) +2.5 +(C) +150 +3 +125 +[counts] +2.0 +ec] +100 +2 +75 +1.5 +103 +t +> +50 +1 +1.0- +25 +Hβ +50km s +-0 +0.5 +0. +0 +2 +3 +-100 +-50 +50 +1 +0 +100 +-100 +-50 +0 +50 +100 +X[arcsec] +△^[km s-1] +△^ [km s-1] +1e20 +1e19 +(d) +(e) +(f) +5.0 +Total Positive flux [Mx] +3 - +3 + fluxI +[arcsec] +4.5 +7 +2 +4.0 +.6 +> +> +1 +1 - +3.5 +WB +10:05:48UTC +BLOS +60G +1 +0 +:0 +1 +2 +3 +0 +0 +1 +3 +0 +25 +50 +75 +100 +125 +150 +X[arcsec] +X [arcsec] +t (s)Chromosphere underneath a CBP +15 +Figure 11. +Chromospheric and coronal response to flux emergence episode 2 in the same format as Fig. 8. +No QSEBs +were observed during this event. An animation of this figure is available online, which shows the flux emergence episode and +corresponding chromospheric and coronal response for the entire 11 min duration in the same format as Fig. 8. +work or an inter-network. This is somewhat clear from +the LOS magnetic field maps used in this paper where +we see sub-arcsecond fields appearing and disappearing +all over the FOV. Therefore, it is imperative that future +studies warrant the need for high-resolution telescopes +(achieving accurate polarimetry at a resolution of 0.′′2 +or better) to discern the impact of such small-scale flux +emergence episodes on the overlying coronal structures. +5. SUMMARY AND CONCLUSIONS +Despite several decades of scientific research dedicated +to the study of CBP, their chromospheric counterparts +remained largely unexplored with the exception of Hab- +bal & Withbroe (1981), and very recently Madjarska +et al. (2021). This paper is an attempt in that direction +where the focus is on the chromosphere underneath a +CBP observed at spatial and temporal scales that have +never been reported before. +In particular, this study +primarily investigates the relationship between ubiqui- +tous spicules seen at the footpoints of a CBP observed +in the Hβ spectral line, and their coronal counterparts. +The chromospheric scenery reveals a conspicuous mor- +phological and topological resemblance with the loops of +the CBP, indicating that spicules form an integral part +of the overall magnetic structure. This interpretation +is further reinforced by computing 2D density distribu- +tion of over 6000 spicules detected using an automated +procedure, and comparing them against coronal images. +Our analysis reveals that these spicules predominantly +lie close to the footpoints of the CBP and have the same +orientation as their coronal counterparts. +We show illustrative examples indicating the “connec- +tion” between spicules and CBP loops that is sugges- +tive of the scenario that spicular flows are often asso- +ciated with heating to TR and coronal temperatures, +and can propagate into the corona likely in the form +of PCDs (N´obrega-Siverio & Moreno-Insertis 2022, see +Fig. 4). Thus, they can potentially contribute toward +transient intensity perturbations of an already existing +CBP. Furthermore, we also show an example of a twist +propagating in the CBP loops that is directly correlated +with twisting spicules seen in the chromospheric foot- +points. All such examples provide strong indication of +a direct link that exists between the chromosphere and +the corona of a CBP. It is however not straightforward +to explain whether spicules observed at the footpoints of + +-60 +-40 +-20 +0 +20 +40 +60 +0.4 +0.5 +0.6 +0.7 +0.8 +BLOS +G? +HB +NidthA +15 +15 +15 +15 +rcsecl +10 - +10 +10 +10 +5 +5 +5 +(d) +17 +(d) +(b) +(a) +am +0 ++0 +0 +0 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +0 +5 +10 +15 +20 +25 +X[arcsec] +X [arcsec] +X[arcsec] +X[arcsec] +nsities +1.00 - +inten +0.99 - +e) +0.98 +width +0.97 +107 +0.96 +211 +193 +171 +Norm. +Hβ Icw +0.93 +09:56:53 +09:59:12 +10:01:31 +10:03:50 +10:06:09 +Time [UTC] +1e22 +1e21 +9.0 +1.4 + Positive flux [Mx] +(f) +8.5 +1.0 +I Negative +0.8 +8.0 +0.6 +7.5 +-0.4 3 +Total +7.0 +-0.0 +09:56:53 +09:59:12 +10:01:31 +10:03:50 +10:06:09 +Time [UTC]16 +Bose et al. +the CBP are unique compared to the spicules observed +elsewhere. +From past studies, it is expected that the +strength of the magnetic field (and its inclination) in +the lower atmosphere plays a major role in driving the +observational properties of spicules. Statistical analysis +by Pereira et al. (2012) reveal clear differences between +properties of spicules in active regions and quiet-Sun +(and coronal holes), and Heggland et al. (2011) report +similar findings with numerical simulations. Underneath +the CBP, the field strength is distinctly stronger com- +pared the rest of the FOV–where the rapid expansion +of the weaker field lines likely leads to different coronal +impact. +Statistical studies with coordinated chromo- +spheric and higher-resolution coronal observations (e.g. +from MUSE), in addition to detailed quantitative anal- +ysis of mass-energy exchanges, are needed to determine +if the coronal contribution of spicules depend on the +strength of the photospheric magnetic fields. +We also investigate the chromospheric and coronal re- +sponses to two different flux emergence episodes and find +very different results. In the first case, we see a clear +enhancement in the chromospheric spicular activity in +tandem with the flux emergence event. The emission in +the 17.1 nm channel shows a strong correlation with the +chromospheric activity whereas the same cannot be said +for the 19.3 and 21.1 nm channels. The emission in the +latter two channels decreases (but only by about 3%) +almost co-temporally with the enhancement seen in the +17.1 nm channel. +The second flux emergence episode +does not seem to contribute toward either a change in +the chromospheric or coronal activity. This is likely due +to a weaker (and smaller-scaled) flux emergence com- +pared to the previous episode which causes little to no +impact in the upper atmospheres of the Sun. Further +coordinated observations (along with numerical simula- +tions) spanning the photosphere through the corona are +needed to statistically establish as to when and why such +small-scaled emergence episodes impact the CBP above. +We also found distinct signatures of magnetic recon- +nection associated with the stronger flux emergence +episode in the form of multiple QSEBs. Although we +found a slight co-temporal intensity increase in one of +the coronal channels, it is not straightforward to corre- +late that directly with the reconnection happening in the +upper photosphere. As explained before, the likely cause +of such a coronal intensity enhancement is attributed to +the enhanced chromospheric spicular activity seen in the +chromosphere. +The results presented in this paper attempts to de- +scribe the (complex) chromospheric scenery underneath +a CBP from the perspective of high-resolution obser- +vations for the very first time. However further studies, +including both the footpoints of a CBP, are needed in co- +ordination with ground- and space-based observations to +answer some of the outstanding questions in more detail. +“Connecting” the photospheric magnetic footpoints to +the corona through the chromosphere remains a chal- +lenge. +Current instrumentation does not allow simul- +taneous photospheric, chromospheric, and coronal mag- +netic field measurements of sufficient spatial resolution +and quality. Until that becomes feasible, a possible way +forward is the use of non-potential magnetic field extrap- +olations in combination with 3D numerical simulations. +Such comparisons may lead to a better understanding of +how flux emergence impacts the chromosphere and the +corona overlying a CBP. +S.B. and B.D.P. gratefully acknowledge support from +NASA grant 80NSSC20K1272 ”Flux emergence and the +structure, dynamics, and energetics of the solar atmo- +sphere”. We thank Edvarda Harnes for the SDO to SST +alignment. The Swedish 1-m Solar Telescope is oper- +ated on the island of La Palma by the Institute for Solar +Physics of Stockholm University in the Spanish Obser- +vatorio del Roque de Los Muchachos of the Instituto de +Astrof´ısica de Canarias. The Institute for Solar Physics +is supported by a grant for research infrastructures of +national importance from the Swedish Research Coun- +cil (registration number 2017-00625). This research is +supported by the Research Council of Norway, project +numbers 250810, 325491, and through its Centres of +Excellence scheme, project number 262622. D.N.S. ac- +knowledges support by the European Research Coun- +cil through the Synergy Grant number 810218 (“The +Whole Sun”, ERC-2018-SyG) and by the Spanish Min- +istry of Science, Innovation and Universities through +project PGC2018-095832-B-I00. +REFERENCES +Antolin, P., Schmit, D., Pereira, T. M. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' USA 2Bay Area Environmental Research Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' NASA Research Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moffett Field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' CA 94035,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' USA 3Institute of Theoretical Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' University of Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' PO Box 1029,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Blindern 0315,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Norway 4Rosseland Centre for Solar Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' University of Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' PO Box 1029,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Blindern 0315,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Oslo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Norway 5Instituto de Astrof´ısica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' E-38205 La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Tenerife,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Spain 6Universidad de La Laguna,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Astrof´ısica, E-38206 La Laguna, Tenerife, Spain ABSTRACT Coronal Bright Points (CBPs) are sets of small-scale coronal loops, connecting opposite magnetic polarities, primarily characterized by their enhanced extreme-ultraviolet (EUV) and X-ray emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Being ubiquitous, they are thought to play an important role in heating the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We aim at characterizing the barely-explored chromosphere underneath CBPs, focusing on the related spicular activity and on the effects of small-scale magnetic flux emergence on CBPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We used high-resolution observations of a CBP in Hβ and Fe I 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm from the Swedish 1-m Solar Telescope (SST) in coordination with the Solar Dynamics Observatory (SDO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This work presents the first high-resolution observation of spicules imaged in Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The spicules were automatically detected using advanced image processing techniques, which were applied to the Dopplergrams derived from Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Here we report their abundant occurrence close to the CBP “footpoints”, and find that the orientation of such spicules is aligned along the EUV loops, indicating that they constitute a fundamental part of the whole CBP magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Spatio-temporal analysis across multiple channels indicates that there are coronal propagating disturbances associated with the studied spicules, producing transient EUV intensity variations of the individual CBP loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Two small-scale flux emergence episodes appearing below the CBP were analyzed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' one of them leading to quiet-sun Ellerman bombs and enhancing the nearby spicular activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This paper presents unique evidence of the tight coupling between the lower and upper atmosphere of a CBP, thus helping to unravel the dynamic phenomena underneath CBPs and their impact on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Keywords: Solar coronal heating (1989) — Solar spicules (1525) — Solar chromosphere (1479) — Solar corona (1483) — Solar magnetic flux emergence (2000) — Methods: observational 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' INTRODUCTION Coronal Bright Points (CBPs) appear as bright, enhanced, blob-like structures when observed in the extreme-ultraviolet (EUV) light or X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' First ob- served in X-rays with the grazing incidence X-ray tele- scope sounding rocket mission (Vaiana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1973), CBPs comprise small-scale magnetic loops connecting opposite polarities where the confined plasma is heated up to a million degrees presumably by magnetic recon- nection (see Priest et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' CBPs are ubiquitously observed in the coronal holes, quiet-Sun, and in the close vicinity of active regions alike, which makes them in- bose@baeri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='org teresting from the perspective of their role in coronal heating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Their lifetimes range from a few hours to even a few days (Golub et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' McIntosh & Gurman 2005) and, depending upon the wavelength of obser- vation, they appear as roundish blobs with diameters ranging between 5–30′′on average (Vaiana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Habbal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Mou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Different stud- ies based on emission spectroscopy and imaging (as dis- cussed in the recent review by Madjarska 2019) suggest that the heights over which CBPs extend in the corona ranges between 5–10 Mm above the photosphere with an average of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 Mm during their lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Though CBPs have been the subject of intensive re- search ever since their discovery back in the early 1970s (Madjarska 2019), there are still fundamental open ques- tions regarding these ubiquitous phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' For in- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='08596v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='SR] 20 Jan 2023 ID2 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' stance, the CBP chromospheric counterpart remains largely unexplored to date, which may be attributed to the lack of adequate observations that target the corona and chromosphere simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' To the best of our knowledge, only two observational studies – Habbal & Withbroe (1981) and Madjarska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2021) have fo- cused on this particular atmospheric layer, both finding that strong intensity enhancements in the corona pre- ceded lower temperature (chromospheric and transition region (TR)) enhancements, thereby indicating a sce- nario where the heating takes place first in the corona and is later conducted toward the TR via thermal con- duction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Another open question is related to the role of magnetic flux emergence on CBPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' For example, mag- netic flux emergence is not only known to be respon- sible for the origin of nearly half of the CBPs (Mou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2018), but also to enhance the chromospheric ac- tivity and associated coronal emission (Madjarska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' So far in the CBP literature, the focus has pri- marily been on large-scale emergence episodes that last for several tens of minutes to hours, therefore studies about the impact of small-scale magnetic flux emergence episodes are scarce: the lack of high-resolution, coordi- nated magnetograms seems to be a major impediment in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The aim of this paper is to better understand the chromospheric scenery underneath a CBP with a fo- cus on spicules and the atmospheric responses to small- scale flux emergence episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Spicules are one of the most abundant and ubiquitous features observed in the solar chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' They are highly dynamic, thin, (multi)threaded, and elongated structures that permeate both the active and non-active regions alike (Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' They are broadly divided into two categories–type I and II, with the latter being more dynamic, with higher apparent velocities, shorter life- times, and undergoing vigorous swaying and torsional motion (de Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The signatures of type-II spicules are of- ten found in the TR and coronal passbands which makes their studies exciting from the perspective of heating and mass-loading of the solar corona (De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2009, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Rouppe van der Voort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Henriques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Samanta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The on-disk counterparts of type-II spicules, termed as rapid blue-shifted and red-shifted excursions (RBEs and RREs, see Rouppe van der Voort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Sekse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2019), abundantly occur in the close vicinity of strong magnetic field regions (such as bi-polar/unipolar field patches, see Sekse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021a, for example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This makes their study also interesting in the context of CBPs since their loops appear to be rooted to strong bi-polar magnetic field configurations present in the photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Multi- dimensional numerical models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' by Wyper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2018) and more recently by N´obrega-Siverio & Moreno- Insertis (2022) suggest that the loops associated with CBPs may have some relationship with jets or spicules observed deeper in solar the atmosphere, which may contribute toward transient intensity variations in the CBPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Regarding small-scale magnetic flux emergence, our attempt to explore its effects on already existing CBPs is motivated by two very recent papers: Tiwari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2022), which find tiny EUV bright dot-like sub- structures inside a CBP that seem to be associated with small flux emergence episodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' and N´obrega-Siverio & Moreno-Insertis (2022), which argue that flux emergence occurring in a few granules may be enough to destabilize a CBP and lead to eruptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' To achieve our objectives, we use a high-quality, ground-based dataset from the Swedish 1-m Solar Tele- scope (SST, Scharmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2003) in coordination with the Atmospheric Imaging Assembly (AIA, Lemen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012) instrument on-board NASA’s Solar Dynamics Ob- servatory (SDO, Pesnell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' For the first time, we employ high-resolution images of the chromospheric Hβ spectral line to study the spicule-CBP relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, the impact of multiple small-scale photo- spheric flux emergence episodes on the chromospheric and coronal activity are also investigated from coordi- nated, high-resolution magnetic field measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The rest of the paper is divided as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Section 2 describes the observations and standard data reduction processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Section 3 details the methodology employed to detect on-disk spicules from SST observations and enhancing the AIA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We show the results and discuss their significance in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4, before finally sum- marizing and concluding the paper in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' OBSERVATIONS AND DATA REDUCTION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Swedish 1-m Solar Telescope For the purpose of this study, we recorded the chro- mospheric counterparts of the CBP using observa- tions from the CHROMospheric Imaging Spectrome- ter (CHROMIS, Scharmer 2017) and CRisp Imaging Spectropolarimeter (CRISP, Scharmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2008) in- struments at the SST on 4 August 2021, under excel- lent seeing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The coordinates of the target were centered around solar (X,Y ) = (250′′,358′′) with µ = cos θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='88 (θ being the heliocentric angle), and the observation sequence lasted for about 11 min start- ing at 09:56 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Figure 1 shows an overview of the observed target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' CHROMIS sampled the Hβ spectral line centered at 486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm under imaging spectroscopic mode across 27 wavelength points between ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='21 nm with respect to the line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The sampling was uniform between ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='01 nm steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Beyond this a non-uniform sampling was intentionally chosen so as to avoid the ef- fect of blends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (e)–(g) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1 show the Hβ blue (at a Doppler offset of −25 km s−1), red wing (at a Doppler offset of +25 km s−1), and line core images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The cadence of the data was 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='8 s with a spatial sampling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' CHROMIS also recorded Chromosphere underneath a CBP 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Overview of the targeted CBP observed on 4 August 2021 at 10:03:31UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (a) shows an RGB composite image of the CBP and its neighboring area at the original SDO/AIA pixel scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Red, blue, and green colors correspond to 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The SST/CHROMIS pointing and FOV is overlaid as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (b)–(d) illustrate SDO/AIA 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm channels that are rotated and co-aligned to CHROMIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (e)–(g) show CHROMIS Hβ images at blue wing (− 25 km s−1), red wing (+ 25 km s−1) and line center, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' These images depict the chromospheric scene underneath the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (h) and (i) contain the photospheric Hβ and Fe i 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm WB images, and panel (j) shows the photospheric LOS magnetic field map (BLOS) saturated between ± 60 G (black indicated positive polarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The dashed FOV shown in panels (a)–(j) denotes the region-of-interest associated with the CBP which forms the basis for all investigations carried out in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' wideband (WB) images with the help of an auxillary WB channel centered at 484.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 nm (referred to as Hβ WB in panel h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Besides providing context photospheric images, the WB serves as an anchor channel that aids in image restoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The WB images have the same cadence as the narrowband Hβ sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' CRISP sampled the Fe i 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm line across 14 wave- length points under imaging spectropolarimetric mode between −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='032 nm and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='068 nm with respect to the line center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The full Stokes Fe i 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm data were in- verted using by using a parallel C++/Python implemen- tation1 of the Milne-Eddington (ME) inversion scheme developed by de la Cruz Rodr´ıguez (2019) to infer the photospheric vector magnetic field information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In ad- dition, the Ca ii 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2 nm line was sampled across 4 wavelength points between −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm and +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='05 nm with respect to the line core in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='05 nm under imag- ing spectroscopic mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The overall cadence of the com- bined observation sequences was measured to be 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 s with a spatial sampling of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′058.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In this paper, we only focus on the line-of-sight (LOS) magnetic fields inferred 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='com/jaimedelacruz/pyMilne from the Fe i 617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 spectral line as shown in panel (j) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The combination of excellent seeing conditions, the SST adaptive optics system, the high-quality CRISP and CHROMIS re-imaging systems (Scharmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2019), and Multi-Object Multi-Frame Blind Deconvolu- tion (MOMFBD, van Noort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2005) image restora- tion resulted in high-spatial resolution data down to the diffraction limit of the telescope (for Hβ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='22λ/D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′13 with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='97 m the effective aperture of SST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The SSTRED reduction pipeline (de la Cruz Rodr´ıguez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' L¨ofdahl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021) was used to facilitate reduction of the data, including the spectral consistency technique described in Henriques (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Furthermore, both the CRISP and CHROMIS time series were destretched to compensate for the residual warping across the field-of- view (FOV) which was not accounted for by the image restoration techniques described earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' For this study, the CRISP data (with a lower spatial and temporal resolution) were co-aligned to CHROMIS by expanding the former to CHROMIS pixel scale fol- lowed by a cross-correlation between the respective pho- tospheric WB channels shown in panels (h) and (i) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In other words, the CHROMIS data with a FOV 40 (b) 40 40 (c) (d) 30 30 30 20 20 20 Y 10 10 10 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 (a) Y1 wu 0 nm 0 0 390 - 10 20 30 40 50 60 10 20 30 0 0 40 50 60 0 10 20 30 50 CHROMIS FOV 60 380 - 40 - 40 40- (e (f) (g) [arcsec] 30 30 30 20 20 20 1 10 10 10 340 - 1B 25 km 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 0 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 330 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 40 40 - (i) (j) 320 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 240 220 260 280 30 30 - 30 - Solar X [arcsec] 20 20 10 10 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 0 0 0 10 20 30 40 50 60 0 10 20 40 50 60 10 20 30 40 0 50 60 0 Xi [arcsec] Xi [arcsec] Xi [arcsec]4 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' of 66′′ × 42′′ and a cadence of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='8 s served as a reference for the CRISP data to which the latter was aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We used nearest neighbour interpolation for the temporal alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Solar Dynamics Observatory The coronal part associated with the CBP was ob- served with the AIA instrument on-board SDO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The SDO datasets were co-aligned to SST (CHROMIS) datasets in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The SDO image cutout sequences were first downloaded from the Joint Science Operations Center’s (JSOC) website2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Next, the images from all the AIA channels were co-aligned to HMI continuum images (here the AIA 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm channel was aligned to HMI continuum), followed by the lat- ter’s co-alignment to CHROMIS WB channels via an iterative cross-correlation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Finally, the SDO images were cropped to have the same FOV as SST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The end result of this pipeline is a co-aligned SDO dataset that consists of eleven (nine AIA and two HMI) image sequences that are expanded from their original pixel scale to CHROMIS pixel scale of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′038 and matched in time by nearest-neighbour sampling to CHROMIS tem- poral cadence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We used the publicly available3 Inter- active Data Language (IDL) based automated pipeline developed by Rob Rutten for this purpose (Rutten 2020) and refer to Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2021b) for an example of this pipeline’s application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' An RGB composite image, consisting of AIA 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm channels, of the CBP target at the origi- nal AIA resolution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1 panel (a), while panels (b)–(d) show the same three channels but ro- tated and co-aligned to the CHROMIS data using the procedure described in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This co-aligned SST and SDO dataset was then visualized extensively with CRISPEX (Vissers & Rouppe van der Voort 2012), an IDL widget-based tool that allows an efficient si- multaneous exploration of multi-dimensional and multi- wavelength datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' METHODS EMPLOYED 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Detecting on-disk spicules from Hβ We employed an automated detection method based on the difference between images observed in the blue and red wings of the Hβ spectral line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is similar to constructing Dopplergrams (see Sekse et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2016), but instead of subtracting fixed wavelengths on opposite sides of the line center, an average over a range of wavelengths (be- tween ±20–30 km s−1 on opposite sides of the line cen- ter) is computed which are then subtracted from one another as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The difference images 2 http://jsoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='stanford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='edu/ 3 https://robrutten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='nl/rridl/00-README/sdo-manual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='html are then subjected to unsharp masking which causes an enhancement in the high spatial frequency components of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In this case, it amplifies the threaded spic- ular features as seen in the panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' RBEs appear as darker threads with negative intensity values whereas RREs appear brighter with positive intensity values in these difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' It is important to note that the difference maps so obtained (as in panel b) do not correspond to absolute measure of the Doppler velocity associated with RBEs and RREs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The chief goal is to obtain a representation of the spatiotemporal evolution of the velocity patterns associated with these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Next, an adaptive intensity thresholding technique was applied on each of the difference images where pix- els which had intensities above a certain value on either side of zero were masked and chosen for further pro- cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' As a result, two different binary masks were generated: one which comprised of pixels that satisfied IUSM > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5σ for RREs and the other one which satisfied IUSM < −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5σ were considered for RBEs, where IUSM is the intensity of the difference image post unsharp mask- ing (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2 b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The difference in the threshold is due to the skewness in the distribution of RREs and RBEs in the difference maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In both the masks, pixels which satisfied the thresholding criterion were assigned a value of 1 while the remaining was assigned as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Once the binary masks were generated, a morphological opening followed by a closing operation was applied to each of the masks (independently for the RBEs and RREs), on a per time step basis, with a 3×3 diamond-shaped structuring element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We refer the reader to Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2021a) and appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2 of Bose (2021) for more details on these morphological operations and the associated reasoning behind them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Finally, connected component labeling in 3D (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' combining both spatial and temporal dimensions, see Rosenfeld & Pfaltz 1966) was performed on the morph processed images so that the RBEs and RREs can be uniquely identified based on a given heuristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Basi- cally, this technique allows connected neighboring pix- els in spatio-temporal domain to be uniquely identified (labeled).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' To not bias for a particular direction, we em- ployed a 26-neighborhood connectivity criterion in 3D space for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In other words, two pixels were “connected” if they shared either an edge, a face or a corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Furthermore, to avoid erroneous detections and focus primarily on the elongated spicular structures a lower cutoff length of ∼200 km (or 8 CHROMIS pixels) was also imposed on the labeled events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The above recipe led to a detection of 6457 uniquely labeled events (3623 as RREs/downflowing RREs and 2834 as RBEs) in the complete dataset lasting 11 min over the whole FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The occurrence of these (combined) events is shown in the form of 2D probability density map in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2 (c) against a background of temporally averaged Hβ WB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Chromosphere underneath a CBP 5 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Overview of the automated on-disk spicule detection method described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (a) shows the spatio- temporal average of the Hβ spectral line computed over the entire CHROMIS FOV, and further illustrates how Dopplergrams are generated by subtracting signals in the blue wing from the red wing (indicated by the shaded areas on either side of the line center).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (b) shows an example of a generated Dopplergram where RBEs and RREs show up as dark and bright threaded structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (c) shows the location and the density distribution of the detected spicules against a background of temporally averaged Hβ WB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Enhancing the AIA images To facilitate a better understanding of the dynamic re- lationship between the chromospheric and coronal coun- terparts of a CBP, it is crucial to enhance the visibility of the coronal images and the loops (strands) associ- ated with the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In this regard, the re-sampled (to CHROMIS pixel scale) AIA images, like the ones shown in the leftmost column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 3, are subjected to a mod- ified version of the common difference technique where the temporal average, over the entire 11 min duration, of each AIA channel is subtracted from an unsharp masked image of the same channel for each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This pro- cedure results in images where small changes in the in- tensity are visibly more enhanced– due to unsharp mask- ing which adjusts the contrast of the edges (see the right- most column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In addition, the AIA images are also subjected to a multi-scale Gaussian normalization (MGN) procedure (Morgan & Druckm¨uller 2014) that enables a better visualization of the overall topology and the orientation of the overlying coronal structure which is not very prominent in original (resampled) AIA im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' They are shown in the middle column of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The various AIA channels used in this study are MGN enhanced by using the default (same) values of weights and coefficients as in Morgan & Druckm¨uller (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The animation associated with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 3 provides a bet- ter idea of the advantage of employing the two methods described above and further adds to their comparison with the original co-aligned AIA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We immedi- ately notice an improvement over the coronal images shown in the left column, where the loops associated with the CBP are barely noticeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Consequently, the variation in the intensity of the CBP associated with rapid spicular dynamics is shown with the common dif- ference images while the MGN processed images are used as a proxy of the intensity variation in the CBP for all subsequent analysis and results described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' However, it is important to note that MGN does not preserve the photometric accuracy of the images and creates a standardized emission, which is enhanced (sub- dued) in the regions with lower (higher) intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This, however, does not impact the analysis presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' RESULTS AND DISCUSSION This section presents a detailed description and dis- cussion of the results obtained from the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We begin by investigating the chromospheric foootpoints of the CBP in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1, followed by a description of rep- resentative examples highlighting the spicule-CBP rela- tionship in Sects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Finally, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4, we discuss the impact of two small-scale photospheric flux emergence episodes in the chromosphere and the hotter AIA channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The chromospheric ”footpoints” of the CBP The Hβ wing and the line core images, shown within the dashed FOV in panels (e)–(g) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1, depict the chromospheric scene underlying the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The images clearly show multiple dark, elongated, and threaded structures that resemble spicules (or mottles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A zoom- in to the dashed FOV is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4 which focuses solely on the region in and around the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' To aid bet- ter visualization of the intensity disturbances propagat- ing in the CBP, we show the common difference images for the different AIA channels in panels (a) through (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (d) shows the Hβ line core (LC) width map which is basically the wavelength separation at half the inten- sity range between the minimum of the Hβ line profile and the average intensities at a displaced wing position from the line center (following Cauzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2009) for each pixel on the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In this case the displacement pa- rameter was set at ± 66 pm from the line center which was determined by converting the displacement param- eter of 90 pm for the Hα spectral line, chosen by Cauzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2009), into equivalent Doppler units (km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' RBEs and RREs (including downflowing RREs) appear to be in “emission” (compared to the background fea- tures as seen in panel d) in these maps since they gener- 40 Hβ 2250 2000 30 1750 ３ 1500 1250 1000 - # 10 750 - 0 (a) 500 100 50 50 100 10 20 0 40 50 60 10 20 30 40 50 60 0 0 M [km s-1] Xi [arcsec] Xi [arcsec]6 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Methods of enhancing the AIA images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The top row (from left to right) shows a zoom in to dashed FOV of AIA 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm intensity map indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1 panel (c), the MGN processed version of the same, and the result of applying the modified common difference technique (see text for details) to the original AIA map, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Bottom row (from left to right) illustrates the result of applying the two enhancement techniques to AIA 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm channel in the same format as the top row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' An animation of this figure is available online, which shows a comparison between the different enhancement techniques along with the temporal evolution of the disturbances propagating along the loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The animation shows solar evolution over 11 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' ally have enhanced opacity owing to their broad LOS ve- locity distribution (Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021a) and enhanced temperature (Leenaarts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Pan- els (e) and (f) show the co-temporal Hβ line core in- tensity and the LOS photospheric magnetic field maps underneath the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Spicules and/or mottles dominate the whole FOV and they are seen to be predominantly rooted in the close vicinity of the strong (negative) po- larity magnetic field patch which also happens to be the photospheric magnetic roots of the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A glance at panels (b)–(e) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4 immediately sug- gests that the CBP loops and their chromopsheric coun- terparts bear a close morphological resemblance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is further highlighted in the animation associated with the figure where the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm loops appear to have propagating disturbances nearly in tandem with the rapid changes in the chromosphere, especially to- ward the later half of the data sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm common difference image appears to be noisier and it does not show the loops associated with the CBP as prominently as in the other AIA channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' However, the animation shows clear disturbances associated in the same region as underlying spicules but the overall mor- phology is less pronounced (compared to panels b and c) making them difficult to relate visually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The lack of loop-like appearances in the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm channel could be attributed to its relatively lower temperature sensi- tivity (log T(K) ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='7) compared to 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels which have a peak temperature sensitivity of around log T(K) ∼ 6 (Boerner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, it is rather common to observe the relatively cooler foot- points of the CBPs in the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm channel underneath the hotter loops (Kwon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Madjarska 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Madjarska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021) which may further justify the less pronounced morphological resemblance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Madjarska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2021) report that the chromospheric counterpart of a CBP largely comprises of elongated, dark features when observed in the Hα line core im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' They name these features “Hα loops” which also appear to constitute a fundamental part of the overall magnetic structure of the CBPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' While we do not find the existence of such loops likely due to our observations being limited to a part of the entire CBP (thereby miss- ing the opposite polarity), spicules dominate our FOV and plays a central role in driving the dynamics of the chromosphere underneath the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Figure 5 shows the occurrence of the detected on-disk spicules, using the method described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 3, in the form of a 2D density map against a background of tem- porally averaged images for four MGN processed AIA Aligned AlA MGN processed Unsharp mask/Common difference 17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm 15 Y[ar 5 - 10:06:09 UTC 10 20 5 25 0 15 X [arcsec]Chromosphere underneath a CBP 7 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The chromosphere and the photosphere underneath the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (a)–(c) show common difference images in the AIA 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm channels at 10:06:43UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (d) shows the co-temporal Hβ LC width map saturated between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='45– 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='82 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (e) shows the co-temporal Hβ LC image and panel (f) depicts the corresponding photospheric BLOS map saturated between ± 60 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' An animation of this figure is available online, which shows the temporal evolution of the chromospheric and photospheric scenery underneath the CBP for the entire duration of 11 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Morphological similarities between spicules and the loops associated with the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (a) and (c)–(f) show the 2D density map of the detected spicules overlaid against a background of temporally averaged Hβ WB, MGN enhanced AIA 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3, and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels, respectively, whereas panel (b) shows a temporal average of the underlying photospheric magnetic field saturated between ± 60 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The FOV in each of the panels correspond to the dashed FOV indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' channels (panels c–f) and an SST WB channel (Hβ, panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' As described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2, the MGN processed 06:4 (b (C Difference 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm Difference 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm Difference 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm (d) (e) (f) 60 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='8 ¥4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='7 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='6 20 40 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 60 Hβ LC width [A] Hβ line core 0 5 10 15 20 25 0 X[arcsec](b) 4 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm 15 2 10 5 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='nm 7 nm nm 10 15 20 25 0 X [arcsec]8 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' images show the intensities in absolute units (though it fails to preserve the photometric accuracy) unlike the common difference images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' From this figure it is clear that the distribution of spicules is very well correlated with the orientation and overall morphology of the CBP loops, as is evident from the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This provides a compelling observational con- firmation (in a statistical sense) of spicules tracing the coronal magnetic field lines which, to the best of our knowledge, has not been reported before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, we also notice that the number density of the detected spicules is predominantly located close to the footpoint of the CBP loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This scenario seems to suggest that the studied spicules are the cromospheric components of the CBP loops which, post heating, appear in the hot- ter TR and coronal channels, and further contribute to- ward the transient intensity disturbances in the already hot CBP loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (see for example De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Madjarska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Rouppe van der Voort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Samanta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2019, and the references therein for studies about the coronal counterpart of spicules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We will explore this aspect further with a few representative examples in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The morphological similarities between the Hβ spicules and the coronal loops associated with CBP in- dicates the possibility that the loop structures are asso- ciated with spicular mass ejections and transient heat- ing of the plasma from chromospheric to coronal tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A direct investigation of such a connection would however require a more detailed analysis by comb- ing high-resolution numerical simulations with spectro- scopic observations of the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Nonetheless, some stud- ies such as De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2017) already showed an intriguing connection between spicules in the TR and the formation of coronal strands in a decayed plage re- gion with the help of numerical simulations and coordi- nated IRIS and SDO observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, spicules were also found to be responsible in triggering propa- gating coronal disturbances (PCDs) along many of the pre-existing (and newly formed) coronal strands rooted to the plage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' PCDs are rapid recurring intensity fluc- tuations (∼ 100 km s−1) whose exact nature remains a mystery, especially outside of the sunspots (see De Pontieu & McIntosh 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' de Moortel 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' De Moortel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Bryans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2016, for example, on the dis- cussion whether PCDs are flows or waves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Therefore, it is likely that the intensity disturbances observed in the common difference coronal images are linked to the rapid spicular dynamics in the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 5 we also notice a significant overlap be- tween the widths of the detected chromospheric spic- ular features and the observed loops associated with the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Using coordinated observations from Hinode’s Extreme ultraviolet Imaging Spectrometer and Transi- tion Region and Coronal Explorer instruments, Dere (2009) derived the volumetric plasma filling factor in CBPs, and came to the conclusion that the widths of its loops can be between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′2–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′2 with possible substruc- tures that are below the resolution limit of the instru- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Comprehensive statistical analysis carried out by Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2012) and Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2021a), indicate that spicule widths, for both off-limb and on-disk cases, are consistent with the range reported by Dere (2009) which further suggests that the Hβ spicules detected in this study are likely the chromospheric counterparts of the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Numerical modeling efforts led by Mart´ınez-Sykora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2018) offer key insights into the role of spicules in determining the widths of the coronal loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' They re- port that the widths of the simulated spicules (and sub- sequently the coronal loops) are primarily determined by the driving mechanism that generates these flows, along with the overall magnetic topology and heating within the magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, they find that the magnetic field rapidly expands primarily between the photosphere and middle to upper chromosphere where spicules are seen to be generated (in the model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The expansion of the field line is rather insignificant between the transition region and the corona which may explain why the CBP loops and spicules appear to have similar widths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Representative examples of spicular-CBP connection In this section we further illustrate the spicule-CBP connection discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 through two repre- sentative examples shown in the left and right panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 6 including their signatures in the TR (AIA 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm) and coronal passbands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We show the com- mon difference images for the different AIA channels (in the left columns of each of the two panels) in or- der to enhance the visibility of the changes in intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The dashed vertical yellow lines in the left columns of both panels show the region of interest that is chosen to construct the x-t maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, in addition to the common difference, we also show the x-t maps derived from the MGN processed AIA images and Hβ LC width maps to highlight the temporal evolution of the plasma emission from each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The left panel shows an example of an RBE in the blue wing of Hβ (at −25 km s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' From the anima- tion and the x-t maps (top row), it is clear that the RBE has an outward (away from the bright network re- gions) apparent motion and propagates from ∼ 2′′ to 6′′ in the vertical direction during its evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is a commonly observed property of spicules where they originate from strong magnetic flux concentrations and tend to shoot outwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Since spicules often have a wide range of Doppler shifts associated with them (Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021a), analysis based on images at fixed wavelength positions can sometimes provide an incomplete picture of their evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In such cases LC width maps offer a better understanding since they are Chromosphere underneath a CBP 9 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Two representative examples highlighting the spicule-CBP connection from the chromosphere to the corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Left panel: an example of an RBE observed in the blue wing (− 25 km s−1) of Hβ spectral line and its associated propagation in the different AIA passbands as indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The dashed vertical lines in the leftmost column indicate the region along which the x-t maps have been extracted for the different channels as shown in the middle and the rightmost columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The solid vertical red lines in the x-t maps correspond to the instant at which this figure is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The dashed cyan line serves as a reference to illustrate the direction of propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Right panel: another example of a spicule observed in the blue wing (− 30 km s−1) of the Hβ spectral line is shown along with its impact on the AIA channels in the same format as the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Note that the apparent direction of propagation of this spicule is opposite to the example presented in the left panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Animations of the two panels are available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' They show the spatio-temporal evolution of the two spicules in the chromospheric Hβ, transition region 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm and coronal 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels during their respective lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' determined by considering a range of wavelengths on ei- ther side of the line center (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In the present example however, the x-t maps derived from the LC widths and Hβ blue wing images are seen to be well correlated with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A comparison of the spatio-temporal evolution seen in the corresponding AIA channels show a noticeable correlation with the Hβ counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' An inspection of the animation of the AIA difference images shows clear intensity disturbances propagating in the CBP which appear to be in tandem with the Hβ spicule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm difference images, in particular, show a clear propagation from the bottom to the top of the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is also well highlighted in the difference x-t maps (middle column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm difference images, on the other hand, does not show such a clear prop- agating disturbance however, the x-t maps reveal clear signatures which are also in tandem with the other wave- length channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A close look at the different x-t maps associated with the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 6 reveals a small but distinct spa- tial (and/or) temporal offsets among the different chan- nels (with respect to the dashed cyan line) – with the TR and coronal emission lying above the cooler chro- mospheric plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Such a scenario is consistent with the analysis presented by De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2014), and it suggests that the RBE has a multi-thermal nature with temperatures that can range from chromospheric to coronal temperatures (of at least 1 MK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In fact, an early study focusing on multi- wavelength diagnostics of a CBP by Habbal & With- broe (1981), found that coronal emission in CBPs lie a x-t 7-X Hβ -25 kms Hβ -25kms- HBLCwidth 81 ec] [arcse 6 istance 4 2 0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3nm 8 ec] Distance [arcse 6 4 D Z 0 Difference 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm Difference17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm [arcsec] 6 Distance 4 Z 0 Difference : 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4nm 8 Difference30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4nm [arcsec] 6 istance 4 D 2 + 0 0 2 4 6 0 100 200 100 200 Time [s] Time [s] Distance [arcsec]x-t 7-X Hβ -30 kms Hβ-30 HB LCwidth kms-1 istance [arcsec] 6 4 2 0 8 Difference 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm Difference19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3nm 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3nm ec] Distance [arcse 6 4 Z 0 Difference17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm 8 [arcsec] 6 Distance 4 Z 0 8 Difference 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm Difference21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm secl [arcse 6 istance 4 2 + 0 0 2 4 6 0 100 200 100 200 Time [s] Time [s] Distance [arcsec]10 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' few arcseconds over an above the chromospheric emis- sion suggesting the hypothesis that magnetic loops in a CBP are rooted in the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The spatial offset between the TR (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm) and coronal (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1/19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm) emission patterns is indicative of the fact that the emis- sion in the coronal channels are not caused by relatively cooler ions (such as O v, see Boerner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012) which are sensitive to temperatures of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2 MK under equilibrium conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, the emission from the O v is expected to be very faint in comparison to the dominant Fe ix and Fe xv ions and would have occurred in the same spatial region as the 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 nm emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This further adds support in favor of the spicular contribution to coronal emission associated with the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 6 shows another example of spicular connection associated with the CBP in the same format as the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Unlike the left panel, the spicule appears to be downward propagating as is evident from the animation and the x-t maps in the middle and right columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A quick glance at the Hβ im- age would suggest that the example here is a blue-wing counterpart of downflowing RREs (seen in the red wing images of Hα, see Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2021a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' However, a closer inspection of the animation reveals a rather complex sce- nario where the spicule is rapidly seen to change its ori- entation (with respect to the LOS of the observer) dur- ing its propagation, before finally disappearing around t = 180 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Such a complex propagation seems to con- vey that the spicule is downward propagating, which in reality could be the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Regardless of the interpretation associated with the orientation of the spicule and its mass flow, interestingly (and more importantly), the x-t maps of the coronal channels show a remarkable correlation with Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' More- over, the spatial offsets among the different channels are consistent with the discussion presented in the previous example and conforms with the multi-thermal aspect of the spicule and its relation to the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This supports our proposition that spicules in the chromosphere have a direct relationship with the disturbances propagating in the CBPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Although many questions remain, this may also provide support to the idea that the processes asso- ciated with spicules may play a role in providing mass and energy flux necessary to sustain the radiative and conductive energy loses in the solar corona as suggested in the numerical simulation studies by Mart´ınez-Sykora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2017, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Twists at the footpoints of the CBP Spicules are known to undergo twisting (torsional) motions that are often interpreted as a sign of Alfv´enic waves responsible for driving the fast solar wind and bal- ancing the energy losses suffered in the solar corona (De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, small-scaled twists associated with spicules are ubiquitously found in the solar atmo- sphere (active regions and quiet Sun alike), and their Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Chromospheric twist and its likely propagation into the solar corona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This figure is in the same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 6 except that the Hβ wing images is replaced by its Dopplergram at ± 25 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Blue (red) color is indicative of plasma motion toward (away from) the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The associ- ated animation (available online) shows the spatio-temporal evolution of the twist propagation across the chromospheric and coronal channels for roughly 300 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' signatures have also been found in the TR (De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 7, we show a case of twist associated with spicules present at the footpoints of the CBP and their influence on the coronal loop above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The top row of the figure shows an Hβ Dopplergram at ± 25 km s−1 with blue and red colors indicating plasma motions toward and away from the observer along the LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The cor- responding x-t map and the animation shows a clear change of direction (or color from blue to red) indi- cating a definite twisting motion in the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A close look at the animation indicates that the event starts with a predominantly positive (red) Doppler shift at t = 0 s which rapidly converts to a negative (blue) Doppler shift in roughly 30 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' It remains predominantly negative until around t = 180 s after which it rapidly twists toward positive (red) once again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This behav- ior is very similar to the examples observed in the Hα and Ca ii 854.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2 spectral lines for both off-limb and on- 7-X 7-X Hβ 25 kms Hβ ±2 25kms HB LCwidth 8 [arcsec] 6 istance 4 2 0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3nm Difference 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3nm Difference19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3nm ec] Distance [arcse 6 4 D Z 0 Difference17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm 8 Distance [arcsec] 6 4 2 0 Difference 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm Difference21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1nm21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 8 nm [arcsec] 6 Distance 4 Z +0 0 2 4 6 0 100 200 300 100 200 300 Time [s] Time [s] Distance [arcsec]Chromosphere underneath a CBP 11 disk spicules as outlined in De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2012) and De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The propagation speed (of roughly 35 km s−1) is fairly consistent (given the uncer- tainty related to the viewing angle) with Alfv´en speeds at chromospheric heights (De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The LC width x-t map shows a similar trend as the Doppler- gram x-t map where additionally we see that the plasma associated with the twisting motion stands out distinctly with respect to the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The x-t maps associated with the difference and MGN enhanced AIA 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3, 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels show strong emission which are in tandem with their chromo- spheric counterpart (similar to the examples shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We also notice a significant offset in the emis- sion among the different channels indicating that the plasma is heated to temperatures of at least 1–2 MK (refer to the discussion in the previous section), in asso- ciation with the twisting spicules at chromospheric tem- peratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Of course, a complete analysis of such a twist propagating in the coronal loops necessitates spectro- scopic studies of the solar corona which is not possible with the set of instruments used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, current observations suggest the prevalence of Alfv´enic waves in the corona (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=', Tomczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' McIn- tosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2011) although the wave energy flux and wave modes are poorly captured with current instru- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Alfv´enic waves have the potential to heat coronal loops (Antolin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2018), in particular when mass flows are present within a structure in addition to waves (as shown for example in Taroyan 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Upcoming space missions, such as Solar- C/EUVST or the MUlti-slit Solar Explorer (MUSE, De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2020, 2022), can potentially address these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Chromospheric and coronal response to emerging magnetic flux In this section, we investigate the chromospheric and coronal responses to two emerging flux episodes as ob- served from the LOS magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Flux emergence episode 1 Figure 8 and associated animation show an overview of the first episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In the LOS magnetogram of panel (a), a negative parasitic polarity is seen to emerge in a predominantly positive region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In the figure, the area where this episode takes place is bounded by a green square region of 100×100 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The variation of the total positive and negative magnetic fluxes within this green square is shown in panel (f) where we find that the total negative flux starts to increase steadily af- ter 10:01:31UT, reaching its peak value of ≈4×1021 Mx around 10:06:30 UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We investigated the subsequent chromospheric re- sponse to the flux emergence episode by analyzing the Hβ LC width maps (panel (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Compared to looking simply at the Hβ line core, the LC width is optically thin toward the dense fibrilar canopies visible in the line core images, thereby facilitating a better understanding of the “connection” between the spicules and their pho- tospheric footpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In particular, we obtain the light curve within the green square since changes in the chro- mosphere and spicules associated with this flux event would likely be rooted near this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The result is shown as a black curve in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' There is a clear en- hancement in the Hβ LC width intensity starting from ≈ 10:01 UT (around the same time as the total negative flux in panel (f) starts to show a marked increase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The LC intensity continues to increase until ≈10:03:30 UT after which it starts to decay and reaches a minimum around 10:05:15 UT, incidentally after which the in- crease in the total negative flux (by ≈300% from the start of the event) also tends to stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' It seems that as the flux emerges and subsequently interacts with the dominant positive polarity through magnetic reconnec- tion (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2), there is a significant enhance- ment in the spicular activity compared to the any of the previous time steps implying a correlation between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is consistent with the analysis presented in Madjarska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2021), in the context of chromospheric response to flux emergence associated with a CBP, and also Samanta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2019), in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' However, un- like Samanta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2019), it is not implied that the flux emergence (and subsequent cancellation) leads to the “generation” of spicules seen in close proximity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In- stead, it is more appropriate to say that the emergence likely caused an enhancement in the observed spicular activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We notice the presence of spicules both well before and after the emergence event as is evident from the animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We have also investigated the coronal response using the SDO/AIA 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 channels (panels (c) and (d)) along with the 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In this case, the corresponding light curves, shown in panel (e), are ob- tained in a region that is spatially displaced from the emerging flux region (the blue rectangle of 200×300 pixels in panels (c) and (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is because it was shown earlier on in the paper that spicules lie close to the footpoints of the CBP structure, whereas the coro- nal loops extend well beyond and are spatially (and/or temporally) displaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Before 10:01:31 UT, the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1, 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3, 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm light curves have very similar intensity levels relative to the maximum of each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' As soon as the chromospheric activity starts to increase from 10:01:31 UT, we notice a co-temporal increase in the intensity of the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1, whereas the 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 channels show a steady decrease compared to their respective pre- event values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' However, unlike the illustrative spicule ex- amples shown in the previous section (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' in Figs 6 and 7) and also the many studies conducted in the past (such as, De Pontieu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' McIntosh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Hen- riques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 2016) that do establish a coronal connec- tion quite convincingly in different SDO/AIA channels, for this particular enhanced spicular episode, the coro- 12 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Chromospheric and coronal response to flux emergence episode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (a) shows the photospheric LOS magnetic field map, (b) shows the Hβ LC width map, (c) and (d) shows the overlying corona of the CBP in AIA 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 nm and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The green square boxes drawn in panels (a)–(d) show the region associated with the flux emergence event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (c) and (d) also shows a blue rectangular box centered around (X,Y )=(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′5,15′′) where the coronal response to the emerging flux is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The black contour in panels (c) and (d) indicates the regions with an absolute LOS magnetic field ≥ 50 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (e) shows the chromospheric and coronal light curves obtained from the green and blue FOVs indicated in panels (b)–(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (f) shows the temporal variation of the total positive and negative magnetic flux in the region bounded by the green colored box in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The gray shaded intervals in panels (e) and (f) shows the time when the chromospheric spicular activity is enhanced, and the pink shaded regions indicates the interval when the two QSEBs are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Animation of this figure is available online, which shows the evolution of the magnetic flux and its response in Hβ, AIA 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels in the form of light curves for the entire 11 min of solar evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' nal relation is not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' To not bias our interpretation, we have also computed other light curves over different AIA FOVs (of the same area) spatially displaced from one another, finding similar results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Reconnection associated with flux emergence 1 Figures 9 and 10 show evidences of magnetic recon- nection through two examples of EBs located in the footpoint of the CBP associated with the emerging flux episode described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We refer to them as quiet- Sun EBs (QSEBs) after Rouppe van der Voort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2016) where these were first described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' QSEBs are smaller, shorter-lived and less intense brightenings and are found in relatively quieter areas on the Sun com- pared to their active region counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' With the help of high-quality Hβ observations from the SST, Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Joshi & Rouppe van der Voort (2022) recently showed that QSEBs are ubiquitous in the so- lar atmosphere and can play an important role in the energy balance of the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Recent numerical modeling efforts led by Hansteen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Danilovic (2017) and Hansteen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2019) have confirmed that (QS)EBs are classic markers of small-scale magnetic re- connection events in the solar photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In this study, we base our analysis on the small 100×100 pixel FOV shown in panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 8 with a focus on the flux emergence event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (a) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 9 shows the QSEB in the blue-wing Hβ intensity map (in- 60 40 20 0 20 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='8 G1 width［A OS 15 15 15 15 [arcsec] 10 10 10 - 10 5 5 () (p (b) 0 0 +0 0 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 X[arcsec] X[arcsec] X[arcsec] X [arcsec] intensities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='99 width and A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='97 171 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (e) Hβ Icw 09:56:53 09:59:12 10:01:31 10:03:50 10:06:09 Time [UTC] 1e23 1e21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 Positive flux [Mx] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 Total QSEB 1 QSEB 2 (f) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 09:56:53 09:59:12 10:01:31 10:03:50 10:06:09 Time [UTC]Chromosphere underneath a CBP 13 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Details of QSEB 1 observed during the flux emergence episode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (a): QSEB observed in the far blue wing of Hβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (b): temporal variation of the Hβ line profile for a location in the QSEB indicated by the magenta marker in panel (a) in the form of a λ − t diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (c): Hβ spectral line at a temporal instant indicated by the marker in panel (b) and the spatio-temporal average Hβ reference profile (dashed black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (d): corresponding WB image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panel (e): corresponding LOS magnetic field map saturated between ± 60 G, and panel (f) temporal evolution of the total positive and negative magnetic flux within the cyan box shown in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The dashed vertical line in (e) indicates the instant when this figure is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Animation of this figure is available online and it shows the evolution of the magnetic field, the QSEB and the corresponding Hβ spectra before, during and after the appearance of the QSEB for about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 min of solar evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' dicated by the magenta marker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The animation shows a tiny, flame-like brightening lasting for about 40 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We note that this period (along with the latter QSEB) is marked in panels (e) and (f) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 8 as QSEB 1 and 2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The Hβ spectral-time (λ − t) slice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 9 (b) shows an enhancement in the line-wings com- pared to the background which is reflected in the spec- tra shown in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The observed spectral shape is characteristic to an EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Moreover, the co-temporal Hβ WB image in panel (d) shows no such brightening which clearly distinguishes this QSEB from a typical photo- spheric magnetic bright point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Panels (e) and (f) show the location of the QSEB on the LOS magnetic field map and the evolution of the total positive and nega- tive magnetic flux inside the smaller cyan box shown in panel (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' From the animation and the light curves, we see that the negative flux decreases up to about 40 s from the start of the event after which it stabilizes up to t = 55 s, following which it starts to decrease right around the onset of the QSEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Figure 10 shows another QSEB event (QSEB 2) asso- ciated with the same emerging flux region under consid- eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' QSEB 2 is brighter, but shorter-lived (≈25 s) in comparison to QSEB 1 and it is shown against a back- ground of red-wing Hβ intensity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Both the exam- ples bear close morphological resemblance with distinct flame-like brightenings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The λ − t slice and the spectra for QSEB 2 also show characteristic EB-like behavior but as is also evident from panel (c), the intensity en- hancement is stronger compared to QSEB 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The cor- responding WB image (panel d) shows that QSEB 2 is located in the intergranular lane, and from the BLOS map we find that in this case the QSEB exists in the intersection of opposite polarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The variation of the negative polarity flux in panel (f) shows an increase right around the onset of the QSEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The examples presented in this section clearly show that the emerging flux episode described in the previous section have a definite impact not just in the form of en- hanced chromospheric spicular activity, but also deeper in the solar atmosphere where it reconnects and subse- quently releases energy in the form of small-scaled EBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Flux emergence episode 2 (a) 150 - (b) (C) 3 - 125 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 [counts] ec] 100 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 75 t 103 > 50 × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0- 1 25 - HB-50kms- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 0 0 0 1 2 3 100 50 50 100 100 50 0 50 100 0 X[arcsec] △^[km s-1] △^[km s-1] 1e21 1e19 (d) (e) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='07 [Mx] Positive flux [Mx] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 3 - fluxI 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 [arcsec] 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 > > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='2 1 1 - Total BLOS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 WB 10:04:24 UTC ±60G 0 2 3 0 1 3 0 25 50 75 100 125150 X[arcsec] X[arcsec] t (s)14 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Details of QSEB 2 associated with the flux emergence event 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The figure and its associated animation (available online) showing about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 min of solar evolution displays the temporal evolution of the magnetic field, QSEB and the Hβ spectra in the same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In this section, we analyze the chromospheric and coronal responses to another flux emergence episode that occurred close to the footpoints of the CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Fig- ure 11 depicts the overview of the episode in the same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 8, and the emergence is shown with a green colored box (occupying the same area as before) drawn in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A close inspection of the anima- tion linked with panel (a) suggests a clear but rela- tively smaller negative flux emergence episode lasting ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is also evident from panel (f) which shows an increase in the total negative flux within the cyan colored box starting around 09:57UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The to- tal negative flux increases by ≈180% compared to the pre-event values and peaks around 09:59:12UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' It then starts to decrease steadily reaching a minimum value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1×1021 Mx at 10:03:50UT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The emerging negative po- larity, however, starts to disappear around 10:02:37UT (indicated by the gray region in panels e and f) from the BLOS map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We found a contrasting evolution of the light curves associated with the chromospheric and coronal channels for this emergence episode in comparison to the event described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The Hβ LC width intensity level in panel (e) does not show a marked increase in tandem with the flux emergence and subsequent cancel- lation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' it maintains a steady level during the whole flux emergence episode and only starts to increase well after the total negative flux reaches its minimum value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The dynamical evolution of the spicules seen in panel (b) complements the variation of the Hβ LC light curve in- dicated in panel (e) where we do not see any signifi- cant enhancement in spicular activity compared to pre- emergence scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The coronal channels behave sim- ilarly where very little (or no) changes in their respec- tive intensity levels are seen during the entire emergence event, implying that none of the channels are impacted directly by this emergence episode unlike the scenario outlined in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Again, to not be limited to a single FOV, we repeated the analysis by choosing dif- ferent (rectangular) cyan FOVs in panels (c) and (d) like before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' However, we did not find any differences in the temporal variation of the AIA light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In addition, we were also not able to find any signature of QSEBs linked with this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' A possible explanation could be that the strength of the emerging flux in the second episode is at least a factor of three lesser than the first episode, which reinforces our conclusion that there is likely no impact on the chromosphere and the corona associated with this weaker flux emergence event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Identifying small-scale flux emergence events, such as the ones described in this paper, can be a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is primarily because high-resolution observa- tions of the solar photosphere reveal a myriad of mag- netic features especially in the regions close to a net- (a) (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 (C) 150 3 125 [counts] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 ec] 100 2 75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 103 t > 50 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0- 25 Hβ +50km s 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 0 2 3 100 50 50 1 0 100 100 50 0 50 100 X[arcsec] △^[km s-1] △^ [km s-1] 1e20 1e19 (d) (e) (f) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 Total Positive flux [Mx] 3 - 3 fluxI [arcsec] 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 7 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='6 > > 1 1 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 WB 10:05:48UTC BLOS 60G 1 0 :0 1 2 3 0 0 1 3 0 25 50 75 100 125 150 X[arcsec] X [arcsec] t (s)Chromosphere underneath a CBP 15 Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Chromospheric and coronal response to flux emergence episode 2 in the same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' No QSEBs were observed during this event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' An animation of this figure is available online, which shows the flux emergence episode and corresponding chromospheric and coronal response for the entire 11 min duration in the same format as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' work or an inter-network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is somewhat clear from the LOS magnetic field maps used in this paper where we see sub-arcsecond fields appearing and disappearing all over the FOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Therefore, it is imperative that future studies warrant the need for high-resolution telescopes (achieving accurate polarimetry at a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='′′2 or better) to discern the impact of such small-scale flux emergence episodes on the overlying coronal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' SUMMARY AND CONCLUSIONS Despite several decades of scientific research dedicated to the study of CBP, their chromospheric counterparts remained largely unexplored with the exception of Hab- bal & Withbroe (1981), and very recently Madjarska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This paper is an attempt in that direction where the focus is on the chromosphere underneath a CBP observed at spatial and temporal scales that have never been reported before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In particular, this study primarily investigates the relationship between ubiqui- tous spicules seen at the footpoints of a CBP observed in the Hβ spectral line, and their coronal counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The chromospheric scenery reveals a conspicuous mor- phological and topological resemblance with the loops of the CBP, indicating that spicules form an integral part of the overall magnetic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This interpretation is further reinforced by computing 2D density distribu- tion of over 6000 spicules detected using an automated procedure, and comparing them against coronal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Our analysis reveals that these spicules predominantly lie close to the footpoints of the CBP and have the same orientation as their coronal counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We show illustrative examples indicating the “connec- tion” between spicules and CBP loops that is sugges- tive of the scenario that spicular flows are often asso- ciated with heating to TR and coronal temperatures, and can propagate into the corona likely in the form of PCDs (N´obrega-Siverio & Moreno-Insertis 2022, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Thus, they can potentially contribute toward transient intensity perturbations of an already existing CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Furthermore, we also show an example of a twist propagating in the CBP loops that is directly correlated with twisting spicules seen in the chromospheric foot- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' All such examples provide strong indication of a direct link that exists between the chromosphere and the corona of a CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' It is however not straightforward to explain whether spicules observed at the footpoints of 60 40 20 0 20 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='8 BLOS G?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' HB NidthA 15 15 15 15 rcsecl 10 - 10 10 10 5 5 5 (d) 17 (d) (b) (a) am 0 +0 0 0 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 0 5 10 15 20 25 X[arcsec] X [arcsec] X[arcsec] X[arcsec] nsities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='00 - inten 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='99 - e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='98 width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='97 107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='96 211 193 171 Norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Hβ Icw 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='93 09:56:53 09:59:12 10:01:31 10:03:50 10:06:09 Time [UTC] 1e22 1e21 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 Positive flux [Mx] (f) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 I Negative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='4 3 Total 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='0 09:56:53 09:59:12 10:01:31 10:03:50 10:06:09 Time [UTC]16 Bose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' the CBP are unique compared to the spicules observed elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' From past studies, it is expected that the strength of the magnetic field (and its inclination) in the lower atmosphere plays a major role in driving the observational properties of spicules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Statistical analysis by Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2012) reveal clear differences between properties of spicules in active regions and quiet-Sun (and coronal holes), and Heggland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' (2011) report similar findings with numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Underneath the CBP, the field strength is distinctly stronger com- pared the rest of the FOV–where the rapid expansion of the weaker field lines likely leads to different coronal impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Statistical studies with coordinated chromo- spheric and higher-resolution coronal observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' from MUSE), in addition to detailed quantitative anal- ysis of mass-energy exchanges, are needed to determine if the coronal contribution of spicules depend on the strength of the photospheric magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We also investigate the chromospheric and coronal re- sponses to two different flux emergence episodes and find very different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' In the first case, we see a clear enhancement in the chromospheric spicular activity in tandem with the flux emergence event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The emission in the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channel shows a strong correlation with the chromospheric activity whereas the same cannot be said for the 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='3 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The emission in the latter two channels decreases (but only by about 3%) almost co-temporally with the enhancement seen in the 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='1 nm channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The second flux emergence episode does not seem to contribute toward either a change in the chromospheric or coronal activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This is likely due to a weaker (and smaller-scaled) flux emergence com- pared to the previous episode which causes little to no impact in the upper atmospheres of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Further coordinated observations (along with numerical simula- tions) spanning the photosphere through the corona are needed to statistically establish as to when and why such small-scaled emergence episodes impact the CBP above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We also found distinct signatures of magnetic recon- nection associated with the stronger flux emergence episode in the form of multiple QSEBs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Although we found a slight co-temporal intensity increase in one of the coronal channels, it is not straightforward to corre- late that directly with the reconnection happening in the upper photosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' As explained before, the likely cause of such a coronal intensity enhancement is attributed to the enhanced chromospheric spicular activity seen in the chromosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The results presented in this paper attempts to de- scribe the (complex) chromospheric scenery underneath a CBP from the perspective of high-resolution obser- vations for the very first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' However further studies, including both the footpoints of a CBP, are needed in co- ordination with ground- and space-based observations to answer some of the outstanding questions in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' “Connecting” the photospheric magnetic footpoints to the corona through the chromosphere remains a chal- lenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Current instrumentation does not allow simul- taneous photospheric, chromospheric, and coronal mag- netic field measurements of sufficient spatial resolution and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Until that becomes feasible, a possible way forward is the use of non-potential magnetic field extrap- olations in combination with 3D numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' Such comparisons may lead to a better understanding of how flux emergence impacts the chromosphere and the corona overlying a CBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' gratefully acknowledge support from NASA grant 80NSSC20K1272 ”Flux emergence and the structure, dynamics, and energetics of the solar atmo- sphere”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' We thank Edvarda Harnes for the SDO to SST alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The Swedish 1-m Solar Telescope is oper- ated on the island of La Palma by the Institute for Solar Physics of Stockholm University in the Spanish Obser- vatorio del Roque de Los Muchachos of the Instituto de Astrof´ısica de Canarias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' The Institute for Solar Physics is supported by a grant for research infrastructures of national importance from the Swedish Research Coun- cil (registration number 2017-00625).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' This research is supported by the Research Council of Norway, project numbers 250810, 325491, and through its Centres of Excellence scheme, project number 262622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' ac- knowledges support by the European Research Coun- cil through the Synergy Grant number 810218 (“The Whole Sun”, ERC-2018-SyG) and by the Spanish Min- istry of Science, Innovation and Universities through project PGC2018-095832-B-I00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/utFAT4oBgHgl3EQfhx3p/content/2301.08596v1.pdf'} +page_content=' REFERENCES Antolin, P.' metadata={'source': 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Computer Science and Technology +University of Science and Technology of China +Anhui, China +Ruomin Huang +hrm@mail.ustc.edu.cn +School of Data Science +University of Science and Technology of China +Anhui, China +Kai Liu +liukai0010@mail.ustc.edu.cn +School of Computer Science and Technology +University of Science and Technology of China +Anhui, China +Haikuo Yu +yhk7786@mail.ustc.edu.cn +School of Computer Science and Technology +University of Science and Technology of China +Anhui, China +Zixiu Wang +wzx2014@mail.ustc.edu.cn +School of Computer Science and Technology +University of Science and Technology of China +Anhui, China +Abstract +In this paper, we study the problem of k-center clustering with outliers. The problem has +many important applications in real world, but the presence of outliers can significantly +increase the computational complexity. Though a number of methods have been developed +in the past decades, it is still quite challenging to design quality guaranteed algorithm with +low complexity for this problem. Our idea is inspired by the greedy method, Gonzalez’s +algorithm, that was developed for solving the ordinary k-center clustering problem. Based +on some novel observations, we show that a simple randomized version of this greedy strat- +egy actually can handle outliers efficiently. We further show that this randomized greedy +approach also yields small coreset for the problem in doubling metrics (even if the doubling +dimension is not given), which can greatly reduce the computational complexity. Moreover, +together with the partial clustering framework proposed by Guha et al. (2019), we prove +that our coreset method can be applied to distributed data with a low communication +complexity. The experimental results suggest that our algorithms can achieve near optimal +solutions and yield lower complexities comparing with the existing methods. +Keywords: +k-center clustering, outliers, coreset, doubling metrics, distributed algorithms +∗. This work +was +supported in +part +by National +Key R&D program +of +China through +grant +2021YFA1000900. A preliminary version of this paper has appeared in 27th Annual European Sym- +posium on Algorithms (ESA2019) (Ding et al., 2019). +1 + +Ding, Huang, Liu, Yu, and Wang +1. Introduction +Clustering is one of the most fundamental problems that has been widely applied in the +fields of machine learning and data mining (Jain, 2010). Given a set of elements, the goal +of clustering is to partition the input set into several groups based on their similarities or +dissimilarities. Several clustering models have been extensively studied, such as the k-center, +k-median, and k-means clusterings (Awasthi and Balcan, 2014). In practice, the data sets +often contain outliers. In particular, the outliers can be arbitrarily located in the space, +e.g., an adversarial attacker can inject a small number of specially crafted samples into the +data (Biggio and Roli, 2018). Even a small number of outliers could seriously destroy the +final clustering result (Chandola et al., 2009). The clustering with outliers problem is also +closely related to the topics like robust statistics (Diakonikolas et al., 2019) and outliers +removal (Schubert et al., 2017). The key difference with these topics is that the focus of +clustering with outliers is to optimize the clustering objective function via excluding a small +number of outliers. +In this paper, we focus on the problem of k-center clustering with outliers. Given a +metric space with n vertices and a pre-specified number of outliers z < n, the problem is to +find k balls to cover at least n − z vertices and minimize the maximum radius of the balls. +The problem can be also defined in Euclidean space so that the cluster centers can be any +points in the space (i.e., not restricted to be selected from the input points). The k-center +clustering with outliers problem can be viewed as a generalization of the ordinary k-center +clustering problem (i.e., the number of outliers z = 0). The ordinary k-center clustering has +many important applications in machine learning, such as deep learning (Coleman et al., +2020), active learning (Sener and Savarese, 2018), and fairness (Kleindessner et al., 2019). +The 2-approximation algorithms for ordinary k-center clustering (without outliers) were +given by Gonzalez (1985) and Hochbaum and Shmoys (1985), where the “approximation +ratio” is the ratio of the obtained radius to the optimal one. It was also proved that any +approximation ratio lower than “2” implies P = NP. +Comparing with the ordinary k-center clustering problem, the challenge for solving +the case with outliers can be greatly increased. +For example, there are +�n +z +� +different +cases that need to consider for optimizing the objective if we do not know who are the +outliers in advance. +The number +�n +z +� +can be quite large even if z is a constant num- +ber. +So existing algorithms often suffer from the issue of high computational complex- +ity. A 3-approximation algorithm for k-center clustering with outliers in arbitrary met- +rics was proposed by Charikar et al. (2001). +The time complexity of their algorithm is +O(kn2 log n) (or O(kn2D log n) in a D-dimensional Euclidean space) which is quadratic in +the input size n. A following streaming (4 + ǫ)-approximation algorithm was proposed by +McCutchen and Khuller (2008). The time complexity is O +�1 +ǫ (kzn + (kz)2 log Φ) +� +, where Φ +is the ratio of the optimal radius to the smallest pairwise distance among the vertices (e.g., +if z = 5%n, the complexity is quadratic in the input size n). de Berg et al. (2021) proposed +the first streaming algorithm in the sliding-window model based on the static approxima- +tion algorithm of Charikar et al. (2001). Recently, Chakrabarty et al. (2016) proposed a +2-approximation algorithm for metric k-center clustering with outliers, but the algorithm +needs to solve a complicated model of linear programming and the exact time complexity +is not provided. +2 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Obviously, when the input data size is large, these existing algorithms cannot be effi- +ciently implemented in practice. Therefore, from both the theoretical and practical perspec- +tives, an interesting question is that whether we can reduce the computational complexity +of k-center clustering with outliers with preserving the clustering quality guarantee. +1.1 Our Contributions +In this paper, our contributions are threefold. +(1) First, we show that a simple randomized greedy data selection strategy can yield a +quality guaranteed solution with linear time complexity (Section 3). Our idea is inspired by +the greedy method from Gonzalez (1985) which was developed for solving the ordinary k- +center clustering. The Gonzalez’s algorithm greedily selects k points iteratively, where each +iteration takes the point that has the largest distance to the set of already selected points. +Based on some novel insights, we show that a randomized version of this greedy method also +works for the problem with outliers. Roughly speaking, we replace each greedy selection +step by a bi-level “greedy selection+random sampling” step: select the farthest (1+ǫ)z +points (rather than the farthest single point) with a small parameter ǫ ∈ (0, 1), and then +take a random sample from this selected set. Our approach can achieve the approximation +ratio “2” with respect to the clustering cost (i.e., the radius), if (1 + ǫ)z (slightly more +than the pre-specified number z) outliers are allowed to be discarded; moreover, the time +complexity is linear in the input size. Another advantage of our method is that it can be +further improved to be sublinear time, that is, the time complexity can be independent of +the input data size n. Thus our result is a significantly improvement upon the previous +approximation algorithms on time complexity. +Being independent of our preliminary work (Ding et al., 2019), Bhaskara et al. (2019) +proposed a similar greedy algorithm for k-center clustering with outliers, but their clustering +approximation ratio is 2 + δ (δ ∈ (0, 1)). Also, it is unclear that whether their runtime can +be improved to be sublinear. +(2) We then study the coreset construction problem for k-center clustering with outliers. +Given a large data set X, the technique of “coreset” is to generate a much smaller set S that +can approximately preserve the structure of X; therefore we can run any existing algorithm +on S so as to reduce the total complexity (Feldman, 2020). +We consider the uniform sampling approach for coreset construction first. Charikar et al. +(2003) showed that the uniform random sampling technique can be applied to reduce the +data size for metric k-center clustering with outliers. Recently, Huang et al. (2018) showed +a similar result for the problem in Euclidean space. In Section 4.1, we revisit the result of +Huang et al. (2018) and provide a more careful analysis. In particular, we show that the +sample size can be reduced by a factor of 1 +γ where γ = +z +n. This improvement could be +important for the case z ≪ n, e.g., z = √n. +Although the uniform sampling approach is very easy to implement, it is not a standard +coreset since it always incurs an inevitable error on the number of discarded outliers. So +we further consider to build a coreset that can remedy this issue, but we need to add some +mild assumption first. Many real-world data sets have low intrinsic dimensions (Belkin, +2003). For example, image sets usually can be represented in low dimensional manifold +though the Euclidean dimension of the image vectors can be very high. The “doubling +3 + +Ding, Huang, Liu, Yu, and Wang +Methods +Size +Construction Time +Uniform sampling +Huang et al. (2018) +˜O +� +n2 +ǫ2z2kD +� +This paper (Theorem 14) +˜O +� n +ǫ2zkD +� +µ-Coreset +Ceccarello et al. (2019) +O +� +(k + z) +� +24 +µ +�ρ� +O +� +(k + z) +� +24 +µ +�ρ +n +� +This paper (Theorem 17) +2z + ˜O +�� +2 +µ +�ρ +k +� +˜O +�� +2 +µ +�ρ +kn +� +Table 1: Existing and our data compressing method for k-center clustering with z outliers. +“D” and “ρ” stand for the dimension of the Euclidean space and doubling dimen- +sion, respectively. +dimension” is widely used for measuring the intrinsic dimensions of data sets (Talwar, +2004) (the formal definition is given in Section 2). With the “low doubling dimension” +assumption, we show that our aforementioned randomized greedy approach can be used to +construct a coreset that incurs no error on the number of outliers (Section 4.2). The size +of our coreset is 2z + ˜O +� +( 2 +µ)ρk +� +, where ρ is the doubling dimension and µ ∈ (0, 1) is the +small parameter measuring the quality of the coreset; the construction time is ˜O(( 2 +µ)ρkn). +Recently, Ceccarello et al. (2019) also provided a coreset for k-center clustering with z +outliers in doubling metrics, where their coreset size is T = O((k + z)(24 +µ )ρ) with O(nT) +construction time. So our result is a significant improvement upon their result in terms of +both coreset size and construction time. Please see Table 1 for details. Comparing with the +results of Ceccarello et al. (2019), another advantage of our approach is that we only assume +that the inliers of the given data have a low doubling dimension ρ > 0. We do +not have any assumption on the outliers; namely, the outliers can scatter arbitrarily in the +space (e.g., the outliers may be added by an adversarial attacker (Biggio and Roli, 2018)). +We believe that this assumption captures a large range of high dimensional instances in +practice. +(3) Due to the rapid increase of real-world data volume, the study on distributed com- +puting has received a great amount of attention. Several distributed algorithms for k-center +clustering with outliers were proposed recently (Malkomes et al., 2015; Guha et al., 2019; +Ceccarello et al., 2019; Li and Guo, 2018), but most of them have large approximation ra- +tios, e.g., the algorithm of Li and Guo (2018) has the approximation ratio > 19. Therefore, +it is necessary to develop a communication-efficient composable coreset (Indyk et al., 2014) +so that one can compute an approximate solution with higher accuracy in the central cen- +tral server. Namely, the input data is partitioned to be stored in s sites, and each site can +compute an individual coreset and send it to the central central server; finally, the central +server computes an approximation result on the union of the collected coresets. Let B be +the information encoding a point. A straightforward implementation of our proposed core- +set of Section 4.2 yields a communication cost s +� +2z + O +� +( 2 +µ)ρk +�� +B, which can be too high +if z is large (e.g., if z = 5%n and s = 10, the cost can be larger than nB). In Section 5, we +prove that the communication cost can be reduced to be (roughly) +� +4z +s·O +� +( 2 +µ)ρk +�� +B by +using the partial clustering framework of Guha et al. (2019); so we reduce the item “2sz” to +be “4z”. To the best of our knowledge, this is the first communication-efficient composable +4 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Approx. +Total Comm. (B) +Rounds +Local Time +Malkomes et al. (2015) +3α + 2 +s(k + z) +1 +O ((k + z) ni) +Guha et al. (2019) +5α + 4 +sk + z +2 +O((k + z)ni) +Li and Guo (2018) +((5α + 4)(1 + µ), 1 + µ) +O +� +sk +µ · log ∆ +µ +� +2 +O +� +n2 +i · log ∆ +µ +� +Ceccarello et al. (2019) +Deterministic +3 + µ +s(k + z)(24 +µ )ρ +1 +O((k + z)ni(24 +µ )ρ) +Ceccarello et al. (2019) +Randomized +(sk + 6z + s log n)(24 +µ )ρ +1 +˜O((k + z/s)ni(24 +µ )ρ) +This paper (Theorem 21) +α × 1+2µ +1−2µ = α × +� +1 + O(µ) +� +4z + ˜O((sk)( 2 +µ)ρ) +2 +˜O +� +k +� +2 +µ +�ρ +ni log2 z +� +Table 2: Existing and our results for distributed k-center clustering with z outliers. The +“local time” column illustrates the running time on each site, where ni is the data +size in site i. “∆” and “ρ” stand for the aspect ratio and the doubling dimension, +respectively. “α” is the approximation ratio of the algorithm run on the union +of the collected coresets in the central server (e.g., if we run the 3-approximation +algorithm of Charikar et al. 2001, α = 3). The result of Li and Guo (2018) is a +bi-criteria approximation that discards (1 + µ)z outliers. +coreset for k-center clustering with outliers that guarantees a (1 + O(µ))-approximation +error. Please see Table 2 for details. +1.2 Other Related Works +Clustering with outliers. Besides the aforementioned prior works for k-center clus- +tering with outliers, a number of results for other clustering with outliers problems were +also proposed in recent years. For example, the k-means/median clustering with outliers +algorithms with provable guarantees have been proposed by Charikar et al. (2001); Chen +(2008); Krishnaswamy et al. (2018); Friggstad et al. (2018), but they are difficult to im- +plement due to their high complexities. +The heuristic but practical algorithms without +provable guarantees have also been studied, such as Chawla and Gionis (2013). +By us- +ing the local search method, Gupta et al. (2017) provided a constant factor approximation +algorithm for k-means clustering with outliers. Furthermore, +Bhaskara et al. (2019) and +Deshpande et al. (2020) respectively showed that the quality can be improved by modifying +the k-means++ seeding. Other recent clustering with outliers algorithms include Chen et al. +(2018); Im et al. (2020); Chakrabarty et al. (2022). +Coresets. +The study on coresets was initiated by Agarwal et al. (2004), and the +technique has been extensively applied for dealing with large-scale data sets in many +different areas. +For example, it can be used to reduce the computational complexities +for clustering and regression problems in machine learning (Cohen-Addad et al., 2021; +Munteanu et al., 2018). +To handle the problems with distributed data, the techniques +like “mergeable summaries ”(Agarwal et al., 2013) and “composable coresets” (Indyk et al., +2014; Mirrokni and Zadimoghaddam, 2015) were introduced recently. Aghamolaei and Ghodsi +(2018) also considered the composable coreset in doubling metrics but their method is only +for the ordinary k-center clustering problem (without outliers). +5 + +Ding, Huang, Liu, Yu, and Wang +2. Preliminaries +We consider the problem of k-center with outliers in arbitrary metrics and Euclidean space +RD. Let (X, d) be an abstract metric, where X contains n vertices and d(·, ·) is the distance +function; with a slight abuse of notation, we also use the function d to denote the shortest +distance between two subsets X1, X2 ⊆ X, i.e., d(X1, X2) = minp∈X1,q∈X2 d(p, q). In RD, we +use ||p−q|| to denote the Euclidean distance between any two points p and q. For simplicity, +we assume that the distance between any pair of vertices in X can be obtained in O(1) time; +for the problem in Euclidean space, it takes O(D) time to compute the distance between any +pair of points. Below, we introduce several important definitions that are used throughout +this paper. +Definition 1 (k-Center Clustering with Outliers) Given a metric (X, d) with two pos- +itive integers k and z < n, the k-center clustering with outliers problem is to find a subset +X′ ⊆ X, where |X′| ≥ n − z, and k centers {c1, · · · , ck} ⊆ X, such that +max +p∈X′ min +1≤j≤k d(p, cj) +is minimized. If given a set P of n points in RD, the problem is to find a subset P ′ ⊆ P, +where |P ′| ≥ n − z, and k centers {c1, · · · , ck} ⊂ RD, such that maxp∈P ′ min1≤j≤k ||p − cj|| +is minimized. +In this paper, we always use Xopt, a subset of X with size n − z, to denote the subset +yielding the optimal solution. Also, let {C1, · · · , Ck} be the k clusters forming Xopt, and +the resulting clustering cost be ropt; that is, each Cj is covered by an individual ball with +radius ropt. +Usually, the optimization problems with outliers are challenging to solve. Thus we often +relax our goal and allow to remove slightly more than the pre-specified number of outliers. +Actually the same relaxation idea has been adopted by a number of works on clustering with +outliers problems before (Charikar et al., 2003; Huang et al., 2018; Li and Guo, 2018). So +we introduce Definition 2. For the sake of convenience, we describe the following Definition 2 +and Definition 3 only for metric space. In fact, the definitions can be easily modified for +the problem in Euclidean space. +Definition 2 ((k, z)ǫ-Center Clustering) Let (X, d) be an instance of k-center clustering +with z outliers, and ǫ ≥ 0. (k, z)ǫ-center clustering is to find a subset X′ of X, where +|X′| ≥ n − (1 + ǫ)z, such that the corresponding clustering cost of Definition 1 on X′ is +minimized. +(i) Given a set A of cluster centers (|A| could be larger than k), we define the clustering +cost +φǫ(X, A) := min +� +max +p∈X′ min +c∈A d(p, c) | X′ ⊆ X, |X′| ≥ n − (1 + ǫ)z +� +. +(ii) If |A| = k and φǫ(X, A) ≤ αropt with α > 01, the set A is called an α-approximation; +if |A| = βk with β > 1, the set A is called an (α, β)-approximation. +1. Since we discard more than z outliers, it is possible to have an approximation ratio α < 1, i.e., φǫ(X, A) < +ropt. +6 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Obviously, the problem in Definition 1 is a special case of (k, z)ǫ-center clustering with +ǫ = 0. Also, Definition 1 and Definition 2 can be naturally extended to the weighted case: +each vertex p has a non-negative weight wp and the total weight of outliers should be equal +to z. Then we have the following definition for coreset. +Definition 3 (Coreset) Given a small parameter µ ∈ (0, 1) and an instance (X, d) of +k-center clustering with z outliers, a set S ⊆ X is called a µ-coreset of X, if each vertex of +S is assigned a non-negative weight and φ0(S, H) ∈ (1 ± µ)φ0(X, H) for any set H ⊆ X of +k vertices. +Given a large-scale instance (X, d), we can run an existing algorithm on its coreset S to +compute an approximate solution for X. If |S| ≪ n, the running time can be significantly +reduced. Formally, we have the following claim (see the proof in Section A). +Claim 4 If the set H yields an α-approximation of the µ-coreset S, it yields an α × 1+µ +1−µ- +approximation of X. +As mentioned before, we also consider the case with low doubling dimension. Roughly +speaking, the doubling dimension describes the expansion rate of the metric. For any p ∈ X +and r ≥ 0, we use Ball(p, r) to denote the ball centered at p with radius r. +Definition 5 (Doubling Dimension) The doubling dimension of a metric (X, d) is the +smallest number ρ > 0, such that for any p ∈ X and r ≥ 0, X ∩ Ball(p, 2r) is always +covered by the union of at most 2ρ balls with radius r. +The rest of this paper is organized as follows. +In Section 3, we present our +constant factor approximations for k-center clustering with outliers, where the main idea +is a randomized greedy approach based on the Gonzalez’s algorithm. +In Section 4, we +study the coreset for k-center clustering with outliers in Euclidean space and doubling +metrics, respectively. In Section 5, we show that our proposed coreset in Section 4 can +be constructed by a communication-efficient way for distributed setting. In Section 6, we +conduct the experiments to evaluate the proposed methods. +3. Randomized Greedy Algorithms for (k, z)ǫ-Center Clustering +For the sake of completeness, we briefly introduce the algorithm of Gonzalez (1985) for +ordinary k-center clustering first. +Initially, it arbitrarily selects a vertex from X, and +iteratively selects the following k − 1 vertices, where each j-th step (2 ≤ j ≤ k) chooses the +vertex having the largest minimum distance to the already selected j − 1 vertices; finally, +each input vertex is assigned to its nearest neighbor of these selected k vertices. This greedy +strategy yields a 2-approximation of k-center clustering; the algorithm also works for the +problem in Euclidean space and yields the same approximation ratio. In this section, we +show that a randomized version of the Gonzalez’s algorithm can solve the (k, z)ǫ-center +clustering problem with quality guarantee. +7 + +Ding, Huang, Liu, Yu, and Wang +Algorithm 1 Bi-criteria Approximation Algorithm +Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n; +parameters ǫ > 0, η ∈ (0, 1/2), and t ∈ Z+. +1. Let γ = z/n and initialize a set E = ∅. +2. Initially, j = 1; randomly select +1 +1−γ log 1 +η vertices from X and add them to E. +3. Run the following steps until j = t: +(a) Update j = j + 1 and let Qj be the subset of X that are the farthest (1 + +ǫ)z vertices to E (for each vertex p ∈ X, its distance to E is defined as +minq∈E d(p, q)). +(b) Randomly select 1+ǫ +ǫ log 1 +η vertices from Qj and add them to E. +Output E. +3.1 (2, O(1 +ǫ ))-Approximation +We consider the bi-criteria approximation that returns more than k cluster centers. Our +high-level idea is as follows. +The main challenge for implementing the Gonzalez’s algorithm is that the outliers and +inliers are mixed in X. For example, the selected vertex, which has the largest minimum +distance to the already selected vertices, is very likely to be an outlier, and then the cluster- +ing quality could be arbitrarily bad. To resolve this issue, we replace each greedy selection +step by a bi-level “greedy selection+random sampling” step: select the farthest (1 + ǫ)z +points (rather than the farthest single point) with a small parameter ǫ ∈ (0, 1), and then +take a random sample from this selected set. Such a combined strategy can guarantee us +to successfully sample a sufficient number of inliers from the k optimal clusters, and mean- +while restrict the number of sampled outliers. We implement our idea in Algorithm 1. For +simplicity, let γ denote z/n in the algorithm. +Theorem 6 Let ǫ > 0 and η ∈ (0, 1/2). If we set t = +ck +1−η with c = 2 + +2 +k(1−η) ln 1 +η in +Algorithm 1, with probability at least 1 − 2η, φǫ(X, E) ≤ 2ropt. +If 1 +η and +1 +1−γ are constant numbers, the size |E| = +1 +1−γ log 1 +η + 1+ǫ +ǫ log 1 +η × t = O(k +ǫ ). So +Theorem 6 implies that E is a +� +2, O(1 +ǫ ) +� +-approximation for (k, z)ǫ-center clustering of X +with probability at least 1 − 2η. To prove Theorem 6, we need Lemma 7 and Lemma 10 +first. +Lemma 7 With probability at least 1 − η, the set E in Step 2 of Algorithm 1 contains at +least one point from Xopt. +Since |Xopt|/|X| = 1 − γ, Lemma 7 can be easily obtained by the following claim. +Claim 8 Let U be a set of elements and V ⊆ U with |V | +|U| = τ > 0. Given η ∈ (0, 1), if +one randomly samples 1 +τ log 1 +η elements from U, with probability at least 1 − η, the sample +contains at least one element from V . +8 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Actually Claim 8 is a folklore result that has been presented in several papers before (such +as Ding and Xu 2014). Since each sampled element falls in V with probability τ, we know +that the sample S contains at least one element from V with probability 1 − (1 − τ)|S|. +Therefore, if we want 1 − (1 − τ)|S| ≥ 1 − η, |S| should be at least +log 1/η +log 1/(1−τ) ≤ 1 +τ log 1 +η. +Recall that {C1, C2, · · · , Ck} are the k clusters forming Xopt. +Denote by λj(E) the +number of the clusters which have non-empty intersection with E at the beginning of j-th +round in Step 3 of Algorithm 1. For example, through Lemma 7 we know that λ1(E) should +be at least 1. Obviously, if λj(E) = k, i.e., Cl ∩ E ̸= ∅ for any 1 ≤ l ≤ k, E will yield a +2-approximate solution by using the triangle inequality. +Claim 9 If λj(E) = k, then φ0(X, E) ≤ 2ropt. +Lemma 10 In each round of Step 3 of Algorithm 1, either the event (1) d(Qj, E) ≤ 2ropt +happens, or with probability at least 1 − η, the event (2) λj(E) ≥ λj−1(E) + 1 happens. +Proof +Suppose that the event (1) does not happen, i.e., d(Qj, E) > 2ropt, and then we +prove that the event (2) should happen with probability at least 1 − η. Let J include all +the indices l ∈ {1, 2, · · · , k} with E ∩ Cl ̸= ∅. We claim that Qj ∩ Cl = ∅ for each l ∈ J . +Otherwise, we arbitrarily select p ∈ Qj ∩Cl and p′ ∈ E ∩Cl; by using the triangle inequality, +we know that d(p, p′) ≤ 2ropt which is in contradiction to the assumption d(Qj, E) > 2ropt. +Thus, Qj ∩ Xopt only contains the vertices from Cl with l /∈ J . Note that the number of +outliers is z. So we have |Qj \ Xopt| ≤ z and |Qj∩Xopt| +|Qj| +≥ +ǫ +1+ǫ. By Claim 8, if randomly +selecting 1+ǫ +ǫ log 1 +η vertices from Qj, with probability at least 1 − η, the sample contains +at least one vertex from Qj ∩ Xopt; also, the vertex must come from ∪l/∈J Cl. That is, the +event (2) λj(E) ≥ λj−1(E) + 1 happens. +If the event (1) of Lemma 10 happens, i.e., d(Qj, E) ≤ 2ropt, then it implies that +max +p∈X\Qj +d(p, E) ≤ 2ropt; +moreover, since |Qj| = (1 + ǫ)z, we have φǫ(X, E) ≤ 2ropt. +Next, we assume that the +event (1) in Lemma 10 never happens, and prove that λj(E) = k with constant probability +when j = Θ(k). The following idea is inspired from Aggarwal et al. (2009) which achieves +a bi-criteria approximation for k-means clustering. We define a random variable xj: xj = 1 +if λj(E) = λj−1(E), or xj = 0 if λj(E) ≥ λj−1(E) + 1, for j = 1, 2, · · · . So E[xj] ≤ η by +Lemma 10 and +� +1≤s≤j +(1 − xs) ≤ λj(E). +(1) +Also, let Jj = � +1≤s≤j(xs −η) and J0 = 0. Then, {J0, J1, J2, · · · } is a super-martingale with +Jj+1 −Jj < 1. Through the Azuma-Hoeffding inequality (Alon and Spencer, 2004), we have +Prob(Jt ≥ J0 + h) ≤ e− h2 +2t for any t ∈ Z+ and h > 0. Let t = +ck +1−η with c = 2 + +2 +k(1−η) ln 1 +η +9 + +Ding, Huang, Liu, Yu, and Wang +Algorithm 2 2-Approximation Algorithm +Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n; a +parameter ǫ > 0. +1. Initialize a set E = ∅. +2. Let j = 1; randomly select one vertex from X and add it to E. +3. Run the following steps until j = k: +(a) Update j = j + 1 and let Qj be the subset of X that are the farthest (1 + ǫ)z +vertices to E. +(b) Randomly select one vertex from Qj and add it to E. +Output E. +and h = (c − 1)k, the inequality implies +Prob( +� +1≤s≤t +(1 − xs) ≥ t(1 − η) − h) ≥ 1 − e− h2 +2t +=⇒ +Prob( +� +1≤s≤t +(1 − xs) ≥ k) ≥ 1 − e− k(c−1)2(1−η) +2c +≥ 1 − e−(c/2−1)k(1−η) +=⇒ +Prob( +� +1≤s≤t +(1 − xs) ≥ k) ≥ 1 − η. +(2) +Combining (1) and (2), we know that λt(E) ≥ k with probability at least 1 − η. Moreover, +when λt(E) = k, it is easy to know that E is a 2-approximate solution by Claim 9. Together +with Lemma 7, we immediately have Theorem 6 where the overall success probability is at +least (1 − η)2 > 1 − 2η. +Time complexity. In each round of Step 3, there are O(1 +ǫ) new vertices added to E, +thus it takes O(1 +ǫn) time to update the distances from the vertices of X to E; to select the +set Qj, we can apply the linear time selection algorithm of Blum et al. (1973). Overall, the +running time of Algorithm 1 is O(k +ǫ n). If the given instance is in RD, the running time will +be O(k +ǫ nD). +3.2 2-Approximation for Constant k +If k is a constant number, we show that a single-criterion 2-approximation can be achieved. +Actually, we use the same strategy as Section 3.1, but only run k rounds with each round +sampling only one vertex. See Algorithm 2 for the details. +Denote by {v1, · · · , vk} the k sampled vertices of E. Actually, the proof of Theorem 11 +is similar to the analysis in Section 3.1. The only difference is that the probability that +the event (2) λj(E) ≥ λj−1(E) + 1 in Lemma 10 happens is changed to be at least +ǫ +1+ǫ. +Also note that v1 ∈ Xopt with probability 1 − γ (because γ = z/n). If all of these events +happen, either we obtain a 2-approximation before k steps (i.e., d(E, X \ Qj) ≤ 2ropt for +some j < k), or {v1, · · · , vk} fall into the k optimal clusters C1, C2, · · · , Ck separately (i.e., +10 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Algorithm 3 Sublinear Time Implementation of Algorithm 1 +Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n; +parameters ǫ > 0, η ∈ (0, 1/2), and t ∈ Z+. +1. Let γ = z/n, σ = +2 +1+ +� +1+ 4(1+ǫ) +3ǫ +, n′ = +3 +σ2(1+ǫ)γ log 4 +η and initialize a set E = ∅. +2. Initially, j = 1; randomly select +1 +1−γ log 1 +η vertices from X and add them to E. +3. Run the following steps until j = t: +(a) Update j = j+1; uniformly sample n′ vertices from X and denote the sampled +set as Aj. +(b) Let ˆrj be the (1 + σ)(1 + ǫ)γn′-th farthest distance from Aj to E. Let ˆAj = +{p ∈ Aj | d(p, E) ≥ ˆrj}. +(c) Add ˆAj to E. +Output E. +λk(E) = k). No matter which case happens, we always obtain a 2-approximation with +respect to the (k, z)ǫ-center clustering problem. So we have the following Theorem 11. +Theorem 11 Algorithm 2 returns a 2-approximation for the problem of (k, z)ǫ-center clus- +tering on X, with probability at least (1− γ)( +ǫ +1+ǫ)k−1. The time complexity is O(kn). If the +given instance is in RD, the time complexity will be O(knD). +To boost the probability of Theorem 11, we just need to repeatedly run the algorithm. +The success probability is easy to calculate by taking the union bound. +Corollary 12 If we run Algorithm 2 O +� +1 +1−γ (1+ǫ +ǫ )k−1� +times, with constant probability, at +least one time the algorithm returns a 2-approximation for the problem of (k, z)ǫ-center +clustering. +3.3 Sublinear Time Implementation of Algorithm 1 +The input data size n can be quite large in practice, so in this section we consider to +implement Algorithm 1 with a lower time complexity. We present a modified version of +Algorithm 1 that only needs an O( k2 +γǫ2) time complexity, where γ = +z +n. When the data +contains heavy noise and the outliers takes a constant factor of n (e.g., z = 5%n and +1 +γ = 20), the algorithm has a sublinear time complexity that is independent of n. Moreover, +the quality of Algorithm 1 presented in Theorem 6 can be guaranteed exactly. +The key observation and analysis. Recall that in each round of Algorithm 1, we +need to scan the whole data set and select the farthest (1 + ǫ)z vertices to E. Thus it +takes linear time in each round. Our key observation is that we can actually avoid this step +by simple random sampling. The new idea is shown in Algorithm 3 (Step 3). We still let +Qj be the set of (1 + ǫ)z farthest vertices to E from X in the j-th round (as Step 3(a) in +11 + +Ding, Huang, Liu, Yu, and Wang +Algorithm 1). We randomly sample n′ vertices from X and use Aj to denote this sampled +set. We can view each sampled vertex of Aj as an independent random variable ∈ {0, 1}: +each sampled vertex is labeled by “1” if it belongs to Qj; otherwise, it is labeled by “0”. +Let σ ∈ (0, 1). Through the Chernoff bound, we have +Prob +� +|Aj ∩ Qj| ∈ (1 ± σ)(1 + ǫ)γn′� +≥ 1 − 2e− 1 +3(σ2(1+ǫ)γn′). +(3) +We select the farthest (1 + σ)(1 + ǫ)γn′ vertices to E from Aj, where the selected set is +denoted by ˆAj. We set the parameter σ = +2 +1+ +� +1+ 4(1+ǫ) +3ǫ +, and then the right hand-side of (3) +becomes 1 − η +2. So with probability at least 1 − η +2, |Aj ∩ Qj| ≤ (1 + σ)(1 + ǫ)γn′. Because +| ˆAj| = (1 + σ)(1 + ǫ)γn′, we have +| ˆAj| ≥ |Aj ∩ Qj|. +(4) +Denote by rj the distance d(Aj ∩ Qj, E). Since Qj is the set of (1 + ǫ)z farthest vertices to +E from X, we have +{p ∈ Aj | d(p, E) ≥ rj} += +Aj ∩ Qj; +(5) +{p ∈ Aj | d(p, E) > rj} +⫋ +Aj ∩ Qj. +(6) +Now we claim that ˆrj ≤ rj where ˆrj is the (1 + σ)(1 + ǫ)γn′-th farthest distance from Aj +to E (as defined in Step 3(b) of Algorithm 3). Otherwise, if ˆrj > rj, we can deduce that +ˆAj ⊆ {p ∈ Aj | d(p, E) > rj} ⫋ Aj ∩ Qj from (6) the fact ˆAj = {p ∈ Aj | d(p, E) ≥ ˆrj}; so +we have | ˆAj| < |Aj ∩ Qj| which is contradictory to (4). Therefore we have ˆrj ≤ rj; together +with (5), it implies +Aj ∩ Qj ⊆ {p ∈ Aj | d(p, E) ≥ ˆrj} = ˆAj. +Hence Aj ∩ Qj ⊆ ˆAj ∩ Qj. On the other hand, since ˆAj ⊆ Aj, it is easy to know ˆAj ∩ Qj ⊆ +Aj ∩ Qj. Therefore, +Aj ∩ Qj = ˆAj ∩ Qj. +(7) +Moreover, from (3) again we know that +|Aj ∩ Qj| ≥ (1 − σ)(1 + ǫ)γn′ = 1 + ǫ +ǫ +log 2 +η +(8) +with probability at least 1− η +2, where we set n′ = +3 +σ2(1+ǫ)γ log 4 +η. From (7) and (8), we know +that +| ˆAj ∩ Qj| = |Aj ∩ Qj| ≥ 1 + ǫ +ǫ +log 2 +η. +Therefore, ˆAj contains at least 1+ǫ +ǫ log 2 +η vertices from Qj. Then we can obtain the similar +result as Lemma 10: in each round, the event “ either (1) d(Qj, E) ≤ 2ropt or (2) λj(E) ≥ +λj−1(E) + 1” happens with probability at least 1 − η +2 − η +2 = 1 − η (recall the probability in +(3) is 1 − η +2, so the overall probability is at least 1 − η +2 − η +2). +Together with the same super-martingale argument of Theorem 6, we have the following +result. +12 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Theorem 13 Let ǫ > 0. If we set t = +ck +1−η with c = 2 + +2 +k(1−η) ln 1 +η for Algorithm 3, with +probability at least 1 − 2η, φǫ(X, E) ≤ 2ropt. +Quality and time complexity. We assume that γ and 1/η are constants. Algorithm 3 +adds O( 1 +σ2 ) vertices to E at each iteration. Note that σ = Θ(√ǫ), which implies the number +of vertices added to E at each iteration is O(1 +ǫ). So |E| = O(k +ǫ ) at the end of the algorithm. +Then Theorem 13 implies that E is a +� +2, O(1 +ǫ ) +� +-approximation for (k, z)ǫ-center clustering +of X with constant probability. Each round we compute the distances from the vertices of +Aj to E and select ˆAj. Since |Aj| = O( 1 +γǫ) and |E| = O(k +ǫ ), we have the time complexity +O( k +γǫ2 ) for computing the distances in each round of Algorithm 3. The selection of ˆAj takes +O(|Aj|) time. Overall, the time complexity of Algorithm 3 is O( k2 +γǫ2), which is independent +of n. If the given distance is in RD, the time complexity will be O( k2 +γǫ2D). If the input data +size n is large, Algorithm 3 can significantly reduce the time complexity and meanwhile +preserve the same clustering quality of Algorithm 1. +4. Coresets for k-Center Clustering with Outliers +In this section, we consider the coreset construction problem for k-center clustering with +outliers. First, we show that the simple uniform sampling approach can yield a slightly +weaker coreset for (k, z)ǫ-center clustering in Euclidean space, where the number of dis- +carded outliers is amplified from (1 + ǫ)z to be (1 + O +� +ǫ) +� +z. Then we consider the coreset +construction in doubling metrics. We show that the idea of Algorithm 1 can be extended +for building the coreset efficiently, even if the doubling dimension is not given. +4.1 Uniform Sampling in Euclidean Space +Given a metric (X, d), Charikar et al. (2003) showed that we can use a random sample S +to replace X. Recall γ = z/n. Let |S| = O( k +ǫ2γ ln n) and E be an α-approximate solution +of (k, z)ǫ-center clustering on (S, d), then E is an α-approximate solution of (k, z)O(ǫ)- +center clustering on (X, d) with constant probability. In a D-dimensional Euclidean space, +Huang et al. (2018) showed a similar result, where the sample size |S| = ˜O( +1 +ǫ2γ2 kD)2. In +this section, we show that the sample size of Huang et al. (2018) can be further improved +by a factor 1 +γ and the new sample size is ˜O( 1 +ǫ2γ kD). This improvement could be important +for the case z ≪ n, e.g., z = √n. Below We revisit their idea first, and then provide a more +careful analysis to achieve the improvement. +Let P be a set of n points in RD. Consider the range space Σ = (P, Π) where each +range π ∈ Π is the complement of union of k balls in RD. We know that the VC dimension +of balls is O(D) (Alon and Spencer, 2004), and therefore the VC dimension of union of k +balls is O(kD log k) (Blumer et al., 1989). That is, the VC dimension of the range space Σ +is O(kD log k). Let ǫ ∈ (0, 1), and an “ǫ-sample” S of P is defined as follows: +∀π ∈ Π, +��|π ∩ P| +|P| +− |π ∩ S| +|S| +�� ≤ ǫ. +2. The asymptotic notation ˜O(f) = O +� +f · polylog( kD +ǫγ ) +� +. +13 + +Ding, Huang, Liu, Yu, and Wang +Roughly speaking, S is an approximation of P with an additive error within each range π. +Given a range space with the VC dimension dvc, an ǫ-sample can be easily obtained via +uniform sampling (Alon and Spencer, 2004), where the success probability is 1 − λ and the +sample size is O +� 1 +ǫ2(dvc log dvc +ǫ + log 1 +λ) +� +for any 0 < λ < 1. For our problem, we need to +replace the “ǫ” of the “ǫ-sample” by ǫγ to guarantee that the number of uncovered points is +bounded by +� +1+O(ǫ) +� +γn (we show the details below). Since dvc = O(kD log k), the sample +size is ˜O( +1 +ǫ2γ2 kD) (Huang et al., 2018). +Actually, the front factor +1 +ǫ2γ2 of the sample size can be further reduced to be +1 +ǫ2γ by +a more careful analysis. We observe that there is no need to guarantee the additive error +for each range π (as the definition of ǫ-sample). Instead, only a multiplicative error for the +ranges covering at least γn points should be sufficient. Note that when a range covers more +points, the multiplicative error is weaker than the additive error and thus the sample size is +reduced. For this purpose, we use the relative approximation (Har-Peled and Sharir, 2011; +Li et al., 2001): let S ⊆ P be a subset of size ˜O( 1 +ǫ2γ kD) chosen uniformly at random, then +with constant probability, +∀π ∈ Π, +���|π ∩ P| +|P| +− |π ∩ S| +|S| +��� ≤ ǫ × max +�|π ∩ P| +|P| +, γ +� +. +(9) +We formally state our result below. Theorem 14 shows that if we have an α-approximation +algorithm, we can run it on the sample S to obtain a solution E, which is also an α- +approximate solution for (k, z)O(ǫ)-center clustering on P. +Because |S| ≪ |P|, we can +reduce a great amount of runtime. +Theorem 14 Let P be an instance for the problem of k-center clustering with outliers in +RD as described in Definition 1, and S ⊆ P be a subset of size ˜O( 1 +ǫ2γ kD) chosen uniformly +at random. Suppose ǫ ≤ 0.5. Let S be a new instance for the problem of k-center clustering +with outliers where the number of outliers is set to be z′ = (1 + ǫ)γ|S|. +If E is an α- +approximate solution of (k, z′)ǫ-center clustering on S, then E is an α-approximate solution +of (k, z)O(ǫ)-center clustering on P, with constant probability. +Proof +We assume that S is a relative approximation of P and (9) is true (this happens +with constant probability). Let Bopt be the set of k balls covering (1 − γ)n points induced +by the optimal solution for P, and BS be the set of k balls induced by an α-approximate +solution of (k, z′)ǫ-center clustering on S. Suppose the radius of each ball in Bopt (resp., BS) +is ropt (resp., rS). We denote the complements of Bopt and BS as πopt and πS, respectively. +First, since Bopt covers (1 − γ)n points of P and S is a relative approximation of P, we +have +��πopt ∩ S +�� +|S| +≤ +��πopt ∩ P +�� +|P| ++ ǫ × max +�|πopt ∩ P| +|P| +, γ +� += (1 + ǫ)γ +by (9). That is, the set balls Bopt cover at least +� +1 − (1 + ǫ)γ +� +|S| points of S, and therefore +it is a feasible solution for the instance S with respect to the problem of k-center clustering +with z′ outliers. Since BS is an α-approximate solution of (k, z′)ǫ-center clustering on S, we +have +rS ≤ αropt; +|πS ∩ S| ≤ (1 + ǫ)z′ = (1 + ǫ)2γ|S|. +(10) +14 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Now, we claim that +��πS ∩ P +�� ≤ (1 + ǫ)2 +1 − ǫ γ|P|. +(11) +Assume that (11) is not true, then (9) implies +���|πS ∩ P| +|P| +− |πS ∩ S| +|S| +��� ≤ ǫ × max +�|πS ∩ P| +|P| +, γ +� += ǫ|πS ∩ P| +|P| +. +So |πS∩S| +|S| +≥ (1−ǫ)|πS∩P | +|P | +> (1+ǫ)2γ, which is in contradiction with the second inequality of +(10), and thus (11) is true. We assume ǫ ≤ 0.5, so +1 +1−ǫ ≤ 1+2ǫ and (1+ǫ)2 +1−ǫ += 1+O(ǫ). Con- +sequently (11) and the first inequality of (10) together imply that BS is an α-approximate +solution of (k, z)O(ǫ)-center clustering on P. +4.2 Coreset Construction in Doubling Metrics +Actually the sample obtained in Theorem 14 is not a standard coreset as Definition 3, since +it always incurs an error on the number of discarded outliers. In this section, we consider +constructing the coreset that strictly satisfies Definition 3. +We introduce the following +assumption first. +Assumption 15 Given an instance (X, d) of k-center clustering with outliers, the metric +(Xopt, d), i.e., the metric formed by the set of inliers, has a constant doubling dimension +ρ > 0. +We do not have any restriction on the outliers X \ Xopt. Thus the above assumption is +more relaxed and practical than assuming the whole (X, d) has a constant doubling dimen- +sion (e.g., the previous coreset construction algorithm of Ceccarello et al. (2019) assumed +that the whole (X, d) has a constant doubling dimension ρ). From Definition 5, we directly +know that each optimal cluster Cj of Xopt can be covered by 2ρ balls with radius ropt/2 +(see the left figure in Figure 1). So we can imagine that the instance (X, d) has 2ρk clusters, +where the optimal radius is at most ropt/2. Therefore, we can just replace k by 2ρk in +Algorithm 1, so as to reduce the approximation ratio (i.e., the ratio of the obtained radius +to ropt) from 2 to 1. +Theorem 16 If we set t = 2ρck +1−η with c = 2+ +2 +k(1−η) ln 1 +η for Algorithm 1, with probability at +least 1 − 2η, φǫ(X, E) ≤ ropt. So the set E is a +� +1, O(2ρ +ǫ ) +� +-approximation for the problem +of (k, z)ǫ-center clustering, and the time complexity is O((k + ln 1 +2η)2ρ +ǫ n ln 1 +2η). +Theorem 16 is a warm-up, and we can further construct the coreset for k-center clustering +with outliers. +Let µ ∈ (0, 1), and for simplicity we assume that log 2 +µ is an integer. +If +applying Definition 5 recursively, we know that each Cj is covered by 2ρ log 2/µ = ( 2 +µ)ρ balls +with radius µ +2 ropt, and Xopt is covered by ( 2 +µ)ρk such balls in total. See the right figure in +Figure 1. Then we have Algorithm 4 based on this observation. +15 + +Ding, Huang, Liu, Yu, and Wang +✦ +�✧ ★ +✦ +�✧ ★ +✁ +✂ +✄☎ +�✧ ★ +✁ +✂ +✄☎ +�✧ ★ +✁ +✂ +✦ +� +✧ +★ +✁ +✂ +✦ +� +✧ +★ +Figure 1: Illustrations for Theorem 16 and Theorem 17. +Algorithm 4 Coreset Construction in Doubling Metrics +Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n; +parameters η ∈ (0, 1/2) and µ ∈ (0, 1). +1. Let l = ( 2 +µ)ρk, c = 2 + +2 +k(1−η) ln 1 +η. +2. Set ǫ = 1 and run Algorithm 1 with t = +cl +1−η rounds. Denote by ˜r = φ1(X, E) the +maximum distance between E and X by excluding the farthest 2z vertices, after +the final round of Algorithm 1. +3. Let X˜r = {p | p ∈ X and d(x, E) ≤ ˜r}. +4. For each vertex p ∈ X˜r, assign it to its nearest neighbor in E; for each vertex q ∈ E, +let its weight be the number of vertices assigning to it. +5. Add X \ X˜r to E; each vertex of X \ X˜r has weight 1. +Output E as the coreset. +Theorem 17 Let η ∈ (0, 1/2). +With probability at least 1 − 2η, Algorithm 4 returns a +µ-coreset E of k-center clustering with z outliers. The size of E is at most 2z +O +� +( 2 +µ)ρ(k + +ln 1 +2η) ln 1 +2η +� +, and the construction time is O(n( 2 +µ)ρ(k + ln 1 +2η) ln 1 +2η). +Proof Similar to Theorem 16, we know that |X˜r| = n − 2z and ˜r ≤ 2× µ +2ropt = µropt with +probability at least 1 − 2η. The size of E is +|X \ X˜r| + O +� +( 2 +µ)ρ(k + ln 1 +2η ) ln 1 +2η +� += 2z + O +� +( 2 +µ)ρ(k + ln 1 +2η ) ln 1 +2η +� +. +Moreover, it is easy to see that the running time of Algorithm 4 is O +� +( 2 +µ)ρ(k+ln 1 +2η)n ln 1 +2η +� +. +Next, we show that E is a qualified µ-coreset of X. +For each vertex q ∈ E, denote by w(q) the weight of q; for the sake of convenience in +our proof, we view each q as a set of w(q) overlapping unit weight vertices. Thus, from the +construction of E, we can see that there is a bijective mapping f between X and E, where +d (p, f(p)) ≤ ˜r ≤ µropt, +∀p ∈ X. +(12) +16 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Let H = {c1, c2, · · · , ck} be any k vertices of X. Suppose that H induces k clusters +{A1, A2, · · · , Ak} (resp., {B1, B2, · · · , Bk}) with respect to the problem of k-center cluster- +ing with z outliers on E (resp., X), where each Aj (resp., Bj) has the cluster center cj +for 1 ≤ j ≤ k. Let rE = φ0(E, H) and rX = φ0(X, H), respectively. Also, let r′ +E (resp., +r′ +X) be the smallest value r, such that for any 1 ≤ j ≤ k, f(Bj) ⊆ Ball(cj, r) (resp., +f −1(Aj) ⊆ Ball(cj, r)). We need the following claim (see the proof in Section B). +Claim 18 |r′ +E − rX| ≤ µropt and |r′ +X − rE| ≤ µropt. +In addition, since {f(B1), · · · , f(Bk)} also form k clusters for the instance E with the fixed +k cluster centers of H, we know that r′ +E ≥ φ0(E, H) = rE. Similarly, we have r′ +X ≥ rX. +Combining Claim 18, we have +rX − µropt ≤ r′ +X − µropt ≤ rE +� +�� +� +by Claim 18 +≤ r′ +E ≤ rX + µropt +� +�� +� +by Claim 18 +. +So |rX − rE| ≤ µropt, i.e., φ0(E, H) ∈ φ0(X, H) ± µropt ⊆ (1 ± µ)φ0(X, H). Therefore E is +a µ-coreset of (X, d). +Remark 19 (1) It is worth emphasizing that the uniform sampling idea in Section 4.1 +cannot avoid the error on the number of excluded outliers; the sample size will become +infinity if not allowing to remove more than z outliers (i.e., 1 +ǫ = ∞). But our proposed +coreset method in Theorem 17 can guarantee the clustering quality for excluding exactly z +outliers. +(2) The coefficient “2” of z in the coreset size actually can be further reduced by modi- +fying the value of ǫ in Step 2 of Algorithm 4 (we set ǫ = 1 just for simplicity). In general, +the size of E is +(1 + ǫ)z + O +�1 +ǫ ( 2 +µ)ρ(k + ln 1 +2η ) ln 1 +2η +� +and the construction time is O(n 1 +ǫ( 2 +µ)ρ(k + ln 1 +2η) ln 1 +2η). +4.3 When the Doubling Dimension ρ Is Not Given +In Algorithm 4, we run Algorithm 1 t = +cl +1−η rounds. But when the doubling dimension ρ +is not given, we cannot determine the values of l and t. We are aware of several techniques +for estimating the doubling dimension of a given data set (Har-Peled and Mendel, 2006). +Ceccarello et al. (2019) also mentioned that their coreset construction method can be ap- +plied to the case that even ρ is not given. These ideas mainly rely on the fact that if one +runs the Gonzalez’s k-center clustering algorithm on the data, the obtained radius can be +significantly reduced due to the property of doubling metrics. However, we need to empha- +size that these doubling dimension estimation techniques cannot be applied to our problem +under Assumption 15, since the outliers and inliers are mixed and only the inliers have the +nice property of doubling metrics. We perform the following modification for Algorithm 4. +Roughly speaking, we decompose Step 2 of Algorithm 4 into two substeps. +17 + +Ding, Huang, Liu, Yu, and Wang +(1) First, we run Algorithm 1 ˜k = +ck +1−η rounds and then obtain the radius ˜r = φ1(E, X) ≤ +2ropt. Now X is partitioned into ˜k clusters H1, H2, · · · , H˜k with excluding 2z outliers. Each +Hj ∩ Xopt has a constant doubling dimension ρ (note that each Hj may also contain some +points from X \ Xopt). Also, the size +��� +� +∪˜k +j=1 Hj +� +\ Xopt +��� ≤ z, and it implies +��� +� +∪ +˜k +j=1 Hj +� +∩ Xopt +��� = +��� ∪ +˜k +j=1 Hj +��� − +��� +� +∪ +˜k +j=1 Hj +� +\ Xopt +��� ≥ n − 2z − z = n − 3z. +Therefore, if we view the instance (X, d) as an instance of ˜k-center clustering with 3z +outliers, the optimal radius (denote by r(−3z) +opt +) should be at most ˜r. Overall, we have the +upper and lower bounds for ˜r: +r(−3z) +opt +≤ ˜r ≤ 2ropt. +(2) Then, if we run Step 3 of Algorithm 1 (replacing “z” by “3z”) with at most t = +cl′ +1−η +rounds where +l′ = +�r(−3z) +opt +1 +4µ˜r +�ρ˜k ≤ +� r(−3z) +opt +1 +4µr(−3z) +opt +�ρ˜k = O +� +( 4 +µ)ρk +� +, +the obtained radius (excluding the farthest 6z vertices) should be at most +2 × 1 +4µ˜r ≤ 2 × 1 +2µropt = µropt. +Then we can use the similar idea of the proof of Theorem 17 to show that the obtained set +E is a qualified µ-coreset. +Overall, we have Algorithm 5 for the case that the doubling dimension ρ is not given. +The time complexity is O( cl′ +1−ηn) = O +� +n( 4 +µ)ρ(k + ln 1 +2η) ln 1 +2η +� +, and the coreset size is 6z + +O +� +( 4 +µ)ρ(k + ln 1 +2η) ln 1 +2η +� +. +Theorem 20 With probability at least 1−2η, Algorithm 5 outputs a µ-coreset E of k-center +clustering with z outliers. The size of E is at most 6z + O +� +( 4 +µ)ρ(k + ln 1 +2η) ln 1 +2η +� +, and the +construction time is O(( 4 +µ)ρ(k + ln 1 +2η)n ln 1 +2η). +We can apply the same idea of Remark 19 (2) to reduce the coreset size to be 3(1 + +ǫ)z + O +�1 +ǫ( 4 +µ)ρ(k + ln 1 +2η) ln 1 +2η +� +, and meanwhile, the time complexity becomes O +� n +ǫ ( 4 +µ)ρ(k + +ln 1 +2η) ln 1 +2η +� +. +5. Coreset for Distributed Data +In this section, we consider the coreset for distributed clustering in the coordinator model +(ˇDuriˇs and Rolim, 1998). Suppose the data X = ⊔s +i=1Xi are distributed disjointly among +s ≥ 2 sites; all the sites can communicate with a central server. Let O = ⊔s +i=1Oi, where +Oi ⊂ Xi, be the set of outliers in the optimal solution; also suppose each |Oi| = z∗ +i . Let +X \ O = ⊔k +j=1Cj be the k optimal clusters. Note that the value of each z∗ +i is unknown. +18 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Algorithm 5 Coreset Construction in Doubling Metrics with Unknown ρ +Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n; +parameters µ ∈ (0, 1) and η ∈ (0, 1/2). +1. Set ǫ = 1 and run Algorithm 1 t = +ck +1−η rounds where c = 2+ +2 +k(1−η) ln 1 +η. Denote by +˜r = φ1(X, E) the maximum distance between E and X by excluding the farthest +2z vertices, after the final round of Algorithm 1. +2. Continue to run Step 3 of Algorithm 1 (but replacing “z” by “3z”) until φ5(X, E) ≤ +1 +2µ˜r (i.e., excluding 6z outliers). +3. Set ˜r′ = φ5(X, E). Let X˜r′ = {p | p ∈ X and d(p, E) ≤ ˜r′}. +4. For each vertex p ∈ X˜r′, assign it to its nearest neighbor in E; for each vertex +q ∈ E, let its weight be the number of vertices assigning to it. +5. Add X \ X˜r′ to E; each vertex of X \ X˜r′ has weight 1. +Output E as the coreset. +Thus a straightforward approach is to compute a coreset for the k-center clustering with z +outliers on each Xi, and directly send the obtained coresets to the central server. Let B be +the information encoding a point. Obviously this approach takes a communication cost +� +2sz + s · O +� +( 2 +µ)ρ(k + log 1 +2η ) ln 1 +2η +�� +B, +(13) +which can be to too high if z is large (e.g., if z = 5%n and s = 10, the cost can be larger +than nB). +In this section, we show that the framework for distributed k-median/means clustering +with outliers developed by Guha et al. (2019) can also be applied to the k-center clustering +with z outliers problem with our proposed coreset method in Section 4.2; in particular, the +term “2sz” of (13) can be reduced to be “4z”. The high level idea of Guha et al. (2019) is +as follows. First, we need to design a set of numbers {z1, z2, . . . , zs}, where each zi is an +upper bound of z∗ +i for 1 ≤ i ≤ s and their sum �s +i=1 zi ≤ 2z. Note that the requirement +“�s +i=1 zi ≤ 2z” is important for bounding the total communication cost. Each site i runs +the coreset algorithm for k-center clustering with zi outliers on Xi to construct a local +coreset. Then each site sends the weighted points of the local coreset to the central server. +Finally the central server aggregates the weighted points to form a global coreset. The key +challenge is to compute the set {z1, z2, . . . , zs} that are suitable for our coreset method. +Below we introduce some notations first. +1. ropt(Xi, k, z∗ +i ): the optimal radius of k-center clustering with z∗ +i outliers on Xi. +2. Given a set of 2-dimensional points A = {(x1, y1), . . . , (xl, yl)} ⊂ R2 where x1 < +x2 < · · · < xl, we define the corresponding piecewise function hA(·) from [x1, ∞) to +R: hA(x) = yi if xi ≤ x < xi+1. Here we define xl+1 = ∞. See Figure 2 for an +illustration. +19 + +Ding, Huang, Liu, Yu, and Wang +Figure 2: An illustration for a non-increasing piecewise function hA(·) with A += +{(x1, y1), (x2, y2), (x3, y3), (x4, y4), (x5, y5)}. +For any pair of (xi, yi) and (xj, yj), we define their lexicographical order: +(xi, yi) ≺ (xj, yj) +if +� xi < xj; or +xi = xj and yi < yj. +Theorem 21 With probability at least 1−2s(2+log2 z)η, Algorithm 6 returns a 2µ-coreset +E of k-center clustering with z outliers for the distributed input X = ⊔s +i=1Xi. The total +communication complexity is +� +4z + O(( 2 +µ)ρs(k + ln 1 +2η) ln 1 +2η) +� +B over 2 rounds, where B is +the communication cost for sending one point. The running time in each site is O(( 2 +µ)ρ(k + +ln 1 +2η)ni ln 1 +2η log2 z). +Remark 22 We can replace η by +η +2s(2+log2 z) in Algorithm 6 to achieve a success probability +1 − η. The communication complexity will be +� +4z + O +� +( 2 +µ)ρs(k + ln s log2 z +η +)(ln s log2 z +η +) +�� +B. +Before proving Theorem 21, we introduce Lemma 23 and Lemma 24 first. +Lemma 23 2ropt ≥ max1≤i≤s ropt(Xi, k, z∗ +i ). +Note that though Xi ⊂ X, the optimal radius ropt(Xi, k, z∗ +i ) of site i is not necessary to +be ≤ ropt, since the cluster centers of X may not belong to Xi (but for the problem in +Euclidean space, ropt(Xi, k, z∗ +i ) is always no larger than ropt since the cluster centers can +be any points in the space). Xi \ Oi is the set of inliers of site i. For each optimal cluster +Cj, 1 ≤ j ≤ k, we can arbitrarily take an inlier from (Xi \ Oi) ∩ Cj as the surrogate cluster +center (if (Xi \ Oi) ∩ Cj = ∅, we just ignore this cluster). From the triangle inequality, we +know ropt(Xi, k, z∗ +i ) ≤ 2ropt and thus obtain Lemma 23. +In the following analysis, we assume that the function hAi in Step 2(a) is non-increasing +for each i = 1, 2, · · · , s. Actually this assumption is easy to satisfy. If there exist a couple +q < q′ such that ˜ri,q > ˜ri,q′, we can simply replace the coreset Ei,q by the coreset Ei,q′ and +let ˜ri,q = ˜ri,q′ in Step 2 of Algorithm 6. The following lemma illustrates the key properties +of the obtained values z1, z2, · · · , zs in Algorithm 6. +20 + +3 +0 +(2) +(3) +(33) +()Randomized Greedy Algorithms for k-Center Clustering with Outliers +Algorithm 6 Distributed Coreset Construction +Input: An instance (X, d) of distributed metric k-center clustering with z outliers, and +X = ⊔s +i=1Xi; the parameters η ∈ (0, 1/2), µ ∈ (0, 1). +1. Let [z] = {0, 1, 2, · · · , z} and Γ = {2r : 1 ≤ r ≤ ⌊log2 z⌋ , r ∈ Z} ∪ {0, z}. Run the +following two-round communication between the sites and the central server. +2. (1st round) In each site i: run Algorithm 4 to obtain the radius ˜ri,q and the +coreset Ei,q for each q ∈ Γ, where ˜ri,q is the radius “˜r” in Algorithm 4 with setting +the number of outliers to be q. +(a) Define the set Ai = {(q, ˜ri,q) | q ∈ Γ}, and construct the corresponding +piecewise function hAi; +(b) Send the function hAi to the central server. +3. (1st round) In the central server: sort the s(z + 1) pairs {(hAi(q), i) | i = +1, 2, · · · , s; q ∈ [z]} with a lexicographical decreasing order; select the (2z + 1)-th +largest item, say “(hAi0(q0), i0)”, and broadcast it to all the sites. +4. (2nd round) In each site i: +(a) If i ̸= i0, let zi = min{q ∈ [z]: (hAi(q), i) ≺ (hAi0(q0), i0)} (if the set is ∅, let +zi = z); +(b) Else, i = i0, let zi0 = min{q ∈ Γ: hAi0(q) = hAi0(q0)}; +(c) Send Ei,zi to the central server. +5. (2nd round) In the central server: take the union E = ∪s +i=1Ei,zi as the final +coreset. +Lemma 24 The set {z1, . . . , zs} obtained in Step 4 of Algorithm 6 is the optimal solution +for the following minimax problem: +min +q1,...,qs max +1≤i≤s hAi(qi) +s.t. +s� +i=1 +qi ≤ 2z, +qi ∈ [z], +i = 1, . . . , s. +(14) +Proof We consider the following two cases. Recall that “i0” is the index obtained in Step 3. +Case (i): +hAi0(zi0) = maxi hAi(zi). In this case, by the definition of zi and the fact +that hAi(·) is non-increasing, we have (hAi0(q0), i0) ≺ (hAi(q), i) for each q = 0, . . . , zi − 1 +and each i = 1, . . . , s. Hence there are �s +i=1 zi pairs of (hAi(q), i)s that are larger than +(hAi0(q0), i0) in the lexicographical order . Therefore by the definition of (hAi0(q0), i0), we +21 + +Ding, Huang, Liu, Yu, and Wang +know that zi0 ≤ q0, which implies +s +� +i=1 +zi ≤ q0 + +� +i̸=i0 +zi. +The right-hand side “q0 + � +i̸=i0 zi” is exactly equal to 2z since it is the number of items +ranked ahead of (hAi0 (q0) , i0) in the sorted sequence in Step 3 of Algorithm 6. Suppose +Lemma 24 is not true, then there exists another solution {z′ +1, · · · , z′ +s} of the problem (14) +such that +max +i +hAi(z′ +i) < hAi0(zi0). +(15) +The right-hand side “hAi0(zi0)” is equal to hAi0(q0) by the definition of zi0. So it implies +hAi0(z′ +i0) < hAi0(q0); since hAi0(·) is non-increasing, we know z′ +i0 > q0. Without loss of +generality, we assume � +i z′ +i = 2z (again, because hAi(·) is non-increasing, we can always +enlarge the z′ +is until � +i z′ +i = 2z). Note that q0 + � +i̸=i0 zi = 2z. So there should exist an +index j ̸= i0, such that z′ +j < zj. By the definition of zj, we have (hAi0(q0), i0) ≺ (hAj(z′ +j), j). +Therefore hAj(z′ +j) ≥ hAi0(q0) = hAi0(zi0). Thus maxi hAi(z′ +i) ≥ hAj(z′ +j) ≥ hAi0(zi0), which +is contradictory to (15). +Case (ii): suppose hAi1(zi1) = maxi hAi(zi) and hAi1(zi1) > hAi0(zi0). In this case we +have zi1 = z in Step 4(a). Similar to the analysis for case (i), we have (hAi0(q0), i0) ≺ +(hAi(q), i) for q = 0, . . . , zi − 1, i ̸= i0, and meanwhile (hAi0(q0), i0) ≺ (hAi1(q), i1) for +q = 0, . . . , zi1. Hence similarly we have 1 + �s +i=1 zi ≤ q0 + 1 + � +i̸=i0 zi = 2z. For any +feasible solution {z′ +1, . . . , z′ +s}, since z′ +i1 ≤ z = zi1 and hAi1(·) is non-increasing, we have +max +i +hAi(z′ +i) ≥ hAi1(z′ +i1) ≥ hAi1(zi1) = max +i +hAi(zi), +which implies {z1, . . . , zs} is better than the solution {z′ +1, . . . , z′ +s}. So {z1, . . . , zs} should be +the optimal solution of the problem (14). +Proof (of Theorem 21) We define ˆzi = min{q ∈ Γ: q ≥ z∗ +i } for i = 1, 2, · · · , s. Then we +directly have maxi ropt(Xi, k, z∗ +i ) ≥ maxi ropt(Xi, k, ˆzi) since ˆzi ≥ z∗ +i . Because “ri,ˆzi” is +the radius of the coreset in Step 2, we have +˜ri,ˆzi ≤ µropt(Xi, k, ˆzi). +Also we know 2ropt ≥ maxi ropt(Xi, k, z∗ +i ) from Lemma 23. So we have 2ropt ≥ 1 +µ maxi ˜ri,ˆzi. +Note that hAi(ˆzi) = ˜ri,ˆzi as ˆzi ∈ Γ. Hence we have +2ropt ≥ 1 +µ max +i +hAi(ˆzi). +(16) +The definitions of ˆzi and Γ together imply that � +i ˆzi ≤ � +i 2z∗ +i = 2z. Therefore the set +{ˆz1, . . . , ˆzs} is a feasible solution for the problem (14). By Lemma 24, we have +max +i +hAi(ˆzi) ≥ max +i +hAi(zi). +(17) +22 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +For each i ̸= i0, we know zi ∈ Γ by the definition of the piecewise function hAi(·). By +Step 4(b) of Algorithm 6, we know zi0 ∈ Γ. Thus we have +hAi(zi) = ˜ri,zi = φ1(Xi, Ei,zi), i = 1, . . . , s. +(18) +Combining the inequalities (16), (17), and (18), we have +max +i +φ1(Xi, Ei,zi) ≤ 2µ · ropt. +Similarly to the inequality (12) in the proof of Theorem 17, we can define the bijective +mapping “f” from X to E (recall that E = ∪s +iEi,zi), such that +d(p, f(p)) ≤ 2µ · ropt, ∀p ∈ X. +(19) +The above inequality (19) implies that the set E is a qualified 2µ-coreset. +The success probability is at least 1 − s(2 + log2 z)2η since each site runs Algorithm 4 +no more than (2+ log2 z) times. Since �s +i zi ≤ 2z, the total number of points sent from the +sites to the central server is no larger than 4z + O(( 2 +µ)ρs(k + ln 1 +2η) ln 1 +2η). So we obtain the +communication complexity +� +4z + O(( 2 +µ)ρs(k + ln 1 +2η) ln 1 +2η) +� +B. The running time in each +site i is O(|Γ|( 2 +µ)ρ(k + ln 1 +2η)ni ln 1 +2η) = O(( 2 +µ)ρ(k + ln 1 +2η)ni ln 1 +2η log2 z). +6. Experiments +All the experiments were conducted on an Ubuntu workstation with 2.40GHz Intel(R) +Xeon(R) CPU E5-2680 and 256GB main memory. The algorithms were implemented in +MATLAB R2019b. Our code is available at https://github.com/OpsTreadstone/randomized-k-center. +Baselines. We compare our algorithms with two well known baselines, “CKM+” (Charikar et al., +2001) and “MK” (McCutchen and Khuller, 2008), as well as the recently proposed algo- +rithm “BVX” (Bhaskara et al., 2019). For the coreset construction problem, we compare +our algorithm with “CPP” (Ceccarello et al., 2019) and the uniform sampling method +“Uniform”. +For the distributed setting, we take “CPP”, “MKC+” (Malkomes et al., 2015), “GLZ” (Guha et al., +2019), and “LG” (Li and Guo, 2018) as the baselines. +All the experiments were repeated 10 times and we report the average results with the +standard deviations. +Data sets. We evaluate our algorithms on four real-world classification data sets from +the UCI KDD archive (Dua and Graff, 2017): Shuttle, Covertype, KDD Cup 1999 and Poker +Hand. The Shuttle data set (King et al., 1995) contains 43, 500 instances of 7 classes with 9 +numerical attributes. The Covertype data set contains 581, 012 instances of 7 classes. It has +54 attributes of continuous and categorical types. The KDD Cup 1999 data set contains +4, 898, 431 instances of 23 classes with 41 attributes. The Poker Hand data set contains +1, 025, 010 instances of 10 classes with 10 attributes. For each of the latter three data sets +Covertype, KDD Cup 1999, and Poker Hand, we randomly select 100, 000 instances and +run the algorithms on the selected instances. +23 + +Ding, Huang, Liu, Yu, and Wang +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +50 +100 +150 +ϕ +ϵ +(X, +ϵ) +ϵ += +0.2 +Alg 1 +Alg 3 +BVX +4 +6 +8 +10 +12 +14 +16 +18 +20 +50 +100 +150 +ϵ += +0.6 +4 +6 +8 +10 +12 +14 +16 +18 +20 +50 +100 +150 +ϵ += +1 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +5 +10 +15 +Running time (s) +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +2 +4 +6 +8 +10 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +5 +10 +k +Figure 3: The performance of Algorithm 1 and Algorithm 3 on Shuttle. +To generate the outliers, for each data set we compute the minimum enclosing ball of +the whole data set by using the algorithm of B˘adoiu and Clarkson (2003); let rmeb and cmeb +be the radius and the center, respectively. Then we randomly add 1% points as the outliers +inside the ball of radius 1.1 × rmeb centered at cmeb. +6.1 The Bi-criteria Algorithms +We compare Algorithm 1 and its sublinear version Algorithm 3 with BVX. For Algorithm 1, +we set ǫ = 0.2, 0.6, 1, and modify the parameters of Algorithm 3 and BVX accordingly so +that they can output the same number of centers. We vary k from 4 to 20. The experimental +results are shown in Figure 3, Figure 4, Figure 5, and Figure 6. Comparing with BVX, +Algorithm 1 and Algorithm 3 take significantly lower running time, and meanwhile achieve +similar or lower clustering cost φǫ(X, E). +To have a more clear comparison between Algorithm 1 and Algorithm 3, we zoom in on +the experimental results of ǫ = 1 without BVX (see Figure 7). We can see that the running +time of Algorithm 3 grows much slower than Algorithm 1 as k increases. This result also +agrees with our theoretical analysis since Algorithm 3 has only sublinear time complexity. +We also compare Algorithm 2 with CKM+, MK, and BVX for small k. We let k = 2, +3, 4, 5. +For Algorithm 2, we set ǫ = 1 and run it ln 10 +1−γ (1+ǫ +ǫ )k−1 times as Corollary 12 +suggests. The experimental results are shown in Figure 8. In general, Algorithm 2 achieves +comparable clustering cost with CKM+ and MK, but runs faster than these two baselines. +BVX is faster but has worse clustering cost than Algorithm 2. +24 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.5 +1.0 +1.5 +2.0 +ϕ +ϵ +(X, +ϵ) +(×10 +3 +) +ϵ += +0.2 +Alg 1 +Alg 3 +BVX +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.5 +1.0 +1.5 +2.0 +2.5 +ϵ += +0.6 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +ϵ += +1 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +100 +200 +300 +Running time (s) +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +50 +100 +150 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +50 +100 +150 +k +Figure 4: The performance of Algorithm 1 and Algorithm 3 on Covertype. +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +ϕ +ϵ +(X, +ϵ) +(×10 +5 +) +ϵ += +0.2 +Alg 1 +Alg 3 +BVX +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.0 +0.5 +1.0 +1.5 +2.0 +ϵ += +0.6 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0.0 +0.5 +1.0 +1.5 +ϵ += +1 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +50 +100 +150 +200 +250 +Running time (s) +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +50 +100 +150 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +20 +40 +60 +80 +100 +k +Figure 5: The performance of Algorithm 1 and Algorithm 3 on KDD Cup 1999. +25 + +Ding, Huang, Liu, Yu, and Wang +4 +6 +8 +10 +12 +14 +16 +18 +20 +4.5 +5.0 +5.5 +6.0 +ϕ +ϵ +(X, +ϵ) +ϵ += +0.2 +Alg 1 +Alg 3 +BVX +4 +6 +8 +10 +12 +14 +16 +18 +20 +5.0 +5.5 +6.0 +6.5 +7.0 +ϵ += +0.6 +4 +6 +8 +10 +12 +14 +16 +18 +20 +5.0 +5.5 +6.0 +6.5 +7.0 +ϵ += +1 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +20 +40 +60 +Running time (s) +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +10 +20 +30 +40 +4 +6 +8 +10 +12 +14 +16 +18 +20 +0 +10 +20 +30 +40 +k +Figure 6: The performance of Algorithm 1 and Algorithm 3 on Poker Hand. +4 +8 +12 +16 +20 +10 +20 +30 +ϕ +ϵ +(X, +ϵ) +Shuttle +Alg 1 +Alg 3 +4 +8 +12 +16 +20 +4 +6 +8 +×10 +2 +Covertype +4 +8 +12 +16 +20 +1 +2 +3 +×10 +2 +KDD Cup 1999 +4 +8 +12 +16 +20 +5.0 +5.5 +6.0 +Poker Hand +4 +8 +12 +16 +20 +0.00 +0.25 +0.50 +0.75 +Running time (s) +4 +8 +12 +16 +20 +0 +5 +10 +4 +8 +12 +16 +20 +0 +2 +4 +6 +4 +8 +12 +16 +20 +0 +1 +k +Figure 7: The comparison between Algorithm 1 and Algorithm 3. +26 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +2 +3 +4 +5 +2 +4 +ϕ +ϵ +(X, +ϵ) +×10 +2 +Shuttle +Alg 2 +BVX +CKM+ +MK +2 +3 +4 +5 +3 +4 +5 +×10 +3 +Covertype +2 +3 +4 +5 +2 +4 +×10 +4 +KDD Cup 1999 +2 +3 +4 +5 +12 +14 +16 +Poker Hand +2 +3 +4 +5 +0 +5 +10 +Running time (s) +×10 +2 +3 +4 +5 +0 +2 +4 +6 +×10 +2 +2 +3 +4 +5 +0 +2 +4 +×10 +2 +2 +3 +4 +5 +0 +2 +4 +6 +×10 +2 +2 +3 +4 +5 +0 +1 +2 +Running time (s) +2 +3 +4 +5 +0 +10 +2 +3 +4 +5 +0 +2 +2 +3 +4 +5 +0 +2 +4 +6 +k +Figure 8: The performance of Algorithm 2. The third row removes CKM+ to have a more +clear illustration on the running times of the other three algorithms. +6.2 The Coreset Algorithms +We compare Algorithm 5 with the coreset methods CPP and Uniform. We set the sizes +of coreset to be {4%n, 8%n, 12%n, 16%n, 20%n} for these three methods, where n is the +number of points (including the outliers). We run the algorithm Cluster proposed by +Malkomes et al. (2015), which is a modification of CKM+, as the “host” algorithm on +the obtained coresets constructed by Algorithm 5 and Uniform. We let RTcoreset denote +the coreset construction time, and let RTtotal denote the total running time (including the +coreset construction time and the time for running the k-center with outliers algorithm on +the coreset). To study the advantage of coreset, we also compare with CKM+ and MK; +we directly run these two algorithms on the whole data sets (without coreset) to compute +the clustering results. +The experimental results are shown in Figure 9. Note that we illustrate the clustering +cost φ0(X, E) (not φǫ(X, E)) in the first row of Figure 9 (and also Figure 10 in Section 6.3), +that is, we discard exactly z outliers rather than (1 + ǫ)z. Uniform is always the fastest +coreset method since it is only simple uniform sampling and does not need any construction +procedure; but its clustering cost is worse than Algorithm 5 and CPP for most cases. Both +of Algorithm 5 and CPP achieve lower clustering cost than CKM+ and MK. Comparing +with CPP, Algorithm 5 has lower clustering cost on Covertype and Poker Hand; Algorithm 5 +also has lower RTcoreset and RTtotal. The experimental results suggest that Algorithm 5 +27 + +Ding, Huang, Liu, Yu, and Wang +0.04 +0.08 +0.12 +0.16 +0.2 +1.0 +1.2 +1.4 +ϕ +0 +(X, +E) +×10 +3 +Shuttle +Alg 5 +UNIFORM +CPP +CKM+ +MK +0.04 +0.08 +0.12 +0.16 +0.2 +2.0 +2.5 +3.0 +×10 +3 +Covertype +0.04 +0.08 +0.12 +0.16 +0.2 +0 +5 +10 +×10 +4 +KDD Cup 1999 +0.04 +0.08 +0.12 +0.16 +0.2 +11 +12 +13 +Poker Hand +0.04 +0.08 +0.12 +0.16 +0.2 +0 +5 +10 +RT +coreset + (s) +0.04 +0.08 +0.12 +0.16 +0.2 +0.0 +2.5 +5.0 +7.5 +×10 +2 +0.04 +0.08 +0.12 +0.16 +0.2 +0 +2 +4 +×10 +2 +0.04 +0.08 +0.12 +0.16 +0.2 +0 +20 +40 +0.04 +0.08 +0.12 +0.16 +0.2 +0.0 +0.5 +1.0 +1.5 +RT +total + (s) +×10 +2 +0.04 +0.08 +0.12 +0.16 +0.2 +0.0 +0.5 +1.0 +1.5 +×10 +3 +0.04 +0.08 +0.12 +0.16 +0.2 +0.0 +0.5 +1.0 +1.5 +×10 +3 +0.04 +0.08 +0.12 +0.16 +0.2 +0 +5 +×10 +2 +Size of coreset (ratio) +Figure 9: The performance of the coreset method Algorithm 5. +can yield significant reduction on the running time (if setting the coreset size ≤ 12%) and +achieve good clustering quality as well. +6.3 The Distributed Algorithm +We compare Algorithm 6 with CPP, MKC+, GLZ and LG with varying the number of +sites s. For Algorithm 6, in Step 2 we run Algorithm 5 instead of Algorithm 4 since the +doubling dimensions of the four data sets are unknown. Similar with Section 6.2, we also run +the Cluster algorithm on the coresets constructed by Algorithm 6. For CPP, following +the setting of Ceccarello et al. (2019), each site sends a coreset of size λ(k+z) to the central +server with λ = 1, 2, 4. LG returns a (k, z)ǫ-center solution and we set ǫ = 0.1, 0.99 in the +algorithm as suggested in their paper (Li and Guo, 2018). +The experimental results of clustering cost and communication cost on the four data +sets are shown in Figure 10. The communication cost is measured by the total number of +floating numbers sent between the sites and the central server. GLZ and LG have lower +communication costs, but yield much higher clustering costs. Algorithm 6 can achieve quite +low clustering cost, but takes higher communication cost comparing with GLZ and LG. +7. Future Work +Following our work, several interesting problems deserve to be studied in future. For exam- +ple, can the coreset construction time of Algorithm 4 be improved, like the fast net construc- +28 + +Randomized Greedy Algorithms for k-Center Clustering with Outliers +Figure 10: The performance of Algorithm 6. In the second row, we remove GLZ and LG +(since they have much higher clustering costs than the others) and zoom in on +the comparison of other algorithms. +tion method proposed by Har-Peled and Mendel (2006) in doubling metrics? In theory, it +is interesting to study other optimization problems involving outliers by using greedy strat- +egy. Also, if we replace k-center clustering by k-center clustering with outliers, it may be +possible to improve the robustness for the applications in deep learning (Coleman et al., +2020), active learning (Sener and Savarese, 2018), and fairness (Kleindessner et al., 2019). +Appendix A. Proof of Claim 4 +Suppose H is an α-approximation of the instance (coreset) S. Let Hopt be the set of k +cluster centers yielding the optimal solution of X. Then we have +φ0(S, H) +≤ +αφ0(S, Hopt); +(20) +φ0(S, H) +∈ +(1 ± µ)φ0(X, H); +(21) +φ0(S, Hopt) +∈ +(1 ± µ)φ0(X, Hopt). +(22) +Combining the above inequalities, we directly have +φ0(X, H) ≤ +1 +1 − µφ0(S, H) ≤ +α +1 − µφ0(S, Hopt) ≤ α(1 + µ) +1 − µ +φ0(X, Hopt). +(23) +So H is an α(1+µ) +1−µ -approximation of X. +29 + +2 +Je +8 +Je +Je +4 +8 +4 +4 +8 +4 +8 +Je +0 +0 +FO +丁 +2 +S +3 +TO- +XT0e +X102 +XTOe +XJ02 +Je +S +4 +8 +Je +s +4 +8 +Te +4 +8 +Je +4 +8 +102 +T +8.2 +Fo.S +H +T +H +$o(x +H +T +0.a +JJ'O. +m +主 +s's- +8.e +0 +X103 +XTOs +×J03 +Je +S +8 +S +4 +4 +8 +e +4 +8 +Je +4 +8 +Je +fo.S +-00.1 +王 +n +rc(e = 0'aa) +rC(E= 0'J) +JJ +crs +5'2 +J'S2 +WKC +2 +Cbb (y = 4) +-0.5 +JS +Cbb (y = S) +02.1 +Cbb (y = J) +(ee.0 =u) a DlA -× +1O- +(e.0 = u) a plA +J3. +J"12. +×J03 +×J03 +×JO+ +KDD Cnb Jaaa +2nffI6 +bokGl HguqDing, Huang, Liu, Yu, and Wang +Appendix B. Proof of Claim 18 +We just need to prove the first inequality since the other one can be obtained by the same +manner. 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In Proceed- +ings of the thirty-sixth annual ACM symposium on Theory of computing, pages 281–290, +2004. +34 + diff --git a/xdE0T4oBgHgl3EQf-gIr/content/tmp_files/load_file.txt b/xdE0T4oBgHgl3EQf-gIr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e0574830b3450f93368a42b686ee98a9b5cab95 --- /dev/null +++ b/xdE0T4oBgHgl3EQf-gIr/content/tmp_files/load_file.txt @@ -0,0 +1,1455 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf,len=1454 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='02814v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='LG] 7 Jan 2023 Randomized Greedy Algorithms for k-Center Clustering with Outliers Randomized Greedy Algorithms and Composable Coreset for k-Center Clustering with Outliers∗ Hu Ding huding@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='cn School of Computer Science and Technology University of Science and Technology of China Anhui, China Ruomin Huang hrm@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='cn School of Data Science University of Science and Technology of China Anhui, China Kai Liu liukai0010@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='cn School of Computer Science and Technology University of Science and Technology of China Anhui, China Haikuo Yu yhk7786@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='cn School of Computer Science and Technology University of Science and Technology of China Anhui, China Zixiu Wang wzx2014@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='cn School of Computer Science and Technology University of Science and Technology of China Anhui, China Abstract In this paper, we study the problem of k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The problem has many important applications in real world, but the presence of outliers can significantly increase the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Though a number of methods have been developed in the past decades, it is still quite challenging to design quality guaranteed algorithm with low complexity for this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Our idea is inspired by the greedy method, Gonzalez’s algorithm, that was developed for solving the ordinary k-center clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Based on some novel observations, we show that a simple randomized version of this greedy strat- egy actually can handle outliers efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We further show that this randomized greedy approach also yields small coreset for the problem in doubling metrics (even if the doubling dimension is not given), which can greatly reduce the computational complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Moreover, together with the partial clustering framework proposed by Guha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019), we prove that our coreset method can be applied to distributed data with a low communication complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The experimental results suggest that our algorithms can achieve near optimal solutions and yield lower complexities comparing with the existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Keywords: k-center clustering, outliers, coreset, doubling metrics, distributed algorithms ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' This work was supported in part by National Key R&D program of China through grant 2021YFA1000900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' A preliminary version of this paper has appeared in 27th Annual European Sym- posium on Algorithms (ESA2019) (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1 Ding, Huang, Liu, Yu, and Wang 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Introduction Clustering is one of the most fundamental problems that has been widely applied in the fields of machine learning and data mining (Jain, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Given a set of elements, the goal of clustering is to partition the input set into several groups based on their similarities or dissimilarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Several clustering models have been extensively studied, such as the k-center, k-median, and k-means clusterings (Awasthi and Balcan, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In practice, the data sets often contain outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In particular, the outliers can be arbitrarily located in the space, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', an adversarial attacker can inject a small number of specially crafted samples into the data (Biggio and Roli, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Even a small number of outliers could seriously destroy the final clustering result (Chandola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The clustering with outliers problem is also closely related to the topics like robust statistics (Diakonikolas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019) and outliers removal (Schubert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The key difference with these topics is that the focus of clustering with outliers is to optimize the clustering objective function via excluding a small number of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this paper, we focus on the problem of k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Given a metric space with n vertices and a pre-specified number of outliers z < n, the problem is to find k balls to cover at least n − z vertices and minimize the maximum radius of the balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The problem can be also defined in Euclidean space so that the cluster centers can be any points in the space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', not restricted to be selected from the input points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The k-center clustering with outliers problem can be viewed as a generalization of the ordinary k-center clustering problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', the number of outliers z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The ordinary k-center clustering has many important applications in machine learning, such as deep learning (Coleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2020), active learning (Sener and Savarese, 2018), and fairness (Kleindessner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The 2-approximation algorithms for ordinary k-center clustering (without outliers) were given by Gonzalez (1985) and Hochbaum and Shmoys (1985), where the “approximation ratio” is the ratio of the obtained radius to the optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' It was also proved that any approximation ratio lower than “2” implies P = NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Comparing with the ordinary k-center clustering problem, the challenge for solving the case with outliers can be greatly increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For example, there are �n z � different cases that need to consider for optimizing the objective if we do not know who are the outliers in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The number �n z � can be quite large even if z is a constant num- ber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So existing algorithms often suffer from the issue of high computational complex- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' A 3-approximation algorithm for k-center clustering with outliers in arbitrary met- rics was proposed by Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The time complexity of their algorithm is O(kn2 log n) (or O(kn2D log n) in a D-dimensional Euclidean space) which is quadratic in the input size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' A following streaming (4 + ǫ)-approximation algorithm was proposed by McCutchen and Khuller (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The time complexity is O �1 ǫ (kzn + (kz)2 log Φ) � , where Φ is the ratio of the optimal radius to the smallest pairwise distance among the vertices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', if z = 5%n, the complexity is quadratic in the input size n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' de Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2021) proposed the first streaming algorithm in the sliding-window model based on the static approxima- tion algorithm of Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Recently, Chakrabarty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2016) proposed a 2-approximation algorithm for metric k-center clustering with outliers, but the algorithm needs to solve a complicated model of linear programming and the exact time complexity is not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2 Randomized Greedy Algorithms for k-Center Clustering with Outliers Obviously, when the input data size is large, these existing algorithms cannot be effi- ciently implemented in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore, from both the theoretical and practical perspec- tives, an interesting question is that whether we can reduce the computational complexity of k-center clustering with outliers with preserving the clustering quality guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1 Our Contributions In this paper, our contributions are threefold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (1) First, we show that a simple randomized greedy data selection strategy can yield a quality guaranteed solution with linear time complexity (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Our idea is inspired by the greedy method from Gonzalez (1985) which was developed for solving the ordinary k- center clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The Gonzalez’s algorithm greedily selects k points iteratively, where each iteration takes the point that has the largest distance to the set of already selected points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Based on some novel insights, we show that a randomized version of this greedy method also works for the problem with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Roughly speaking, we replace each greedy selection step by a bi-level “greedy selection+random sampling” step: select the farthest (1+ǫ)z points (rather than the farthest single point) with a small parameter ǫ ∈ (0, 1), and then take a random sample from this selected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Our approach can achieve the approximation ratio “2” with respect to the clustering cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', the radius), if (1 + ǫ)z (slightly more than the pre-specified number z) outliers are allowed to be discarded;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' moreover, the time complexity is linear in the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Another advantage of our method is that it can be further improved to be sublinear time, that is, the time complexity can be independent of the input data size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus our result is a significantly improvement upon the previous approximation algorithms on time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Being independent of our preliminary work (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019), Bhaskara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) proposed a similar greedy algorithm for k-center clustering with outliers, but their clustering approximation ratio is 2 + δ (δ ∈ (0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also, it is unclear that whether their runtime can be improved to be sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2) We then study the coreset construction problem for k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Given a large data set X, the technique of “coreset” is to generate a much smaller set S that can approximately preserve the structure of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' therefore we can run any existing algorithm on S so as to reduce the total complexity (Feldman, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We consider the uniform sampling approach for coreset construction first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2003) showed that the uniform random sampling technique can be applied to reduce the data size for metric k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Recently, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018) showed a similar result for the problem in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1, we revisit the result of Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018) and provide a more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In particular, we show that the sample size can be reduced by a factor of 1 γ where γ = z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' This improvement could be important for the case z ≪ n, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', z = √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Although the uniform sampling approach is very easy to implement, it is not a standard coreset since it always incurs an inevitable error on the number of discarded outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So we further consider to build a coreset that can remedy this issue, but we need to add some mild assumption first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Many real-world data sets have low intrinsic dimensions (Belkin, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For example, image sets usually can be represented in low dimensional manifold though the Euclidean dimension of the image vectors can be very high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The “doubling 3 Ding, Huang, Liu, Yu, and Wang Methods Size Construction Time Uniform sampling Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018) ˜O � n2 ǫ2z2kD � This paper (Theorem 14) ˜O � n ǫ2zkD � µ-Coreset Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) O � (k + z) � 24 µ �ρ� O � (k + z) � 24 µ �ρ n � This paper (Theorem 17) 2z + ˜O �� 2 µ �ρ k � ˜O �� 2 µ �ρ kn � Table 1: Existing and our data compressing method for k-center clustering with z outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' “D” and “ρ” stand for the dimension of the Euclidean space and doubling dimen- sion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' dimension” is widely used for measuring the intrinsic dimensions of data sets (Talwar, 2004) (the formal definition is given in Section 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' With the “low doubling dimension” assumption, we show that our aforementioned randomized greedy approach can be used to construct a coreset that incurs no error on the number of outliers (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The size of our coreset is 2z + ˜O � ( 2 µ)ρk � , where ρ is the doubling dimension and µ ∈ (0, 1) is the small parameter measuring the quality of the coreset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' the construction time is ˜O(( 2 µ)ρkn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Recently, Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) also provided a coreset for k-center clustering with z outliers in doubling metrics, where their coreset size is T = O((k + z)(24 µ )ρ) with O(nT) construction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So our result is a significant improvement upon their result in terms of both coreset size and construction time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Please see Table 1 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Comparing with the results of Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019), another advantage of our approach is that we only assume that the inliers of the given data have a low doubling dimension ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We do not have any assumption on the outliers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' namely, the outliers can scatter arbitrarily in the space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', the outliers may be added by an adversarial attacker (Biggio and Roli, 2018)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We believe that this assumption captures a large range of high dimensional instances in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (3) Due to the rapid increase of real-world data volume, the study on distributed com- puting has received a great amount of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Several distributed algorithms for k-center clustering with outliers were proposed recently (Malkomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Guha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Li and Guo, 2018), but most of them have large approximation ra- tios, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', the algorithm of Li and Guo (2018) has the approximation ratio > 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore, it is necessary to develop a communication-efficient composable coreset (Indyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2014) so that one can compute an approximate solution with higher accuracy in the central cen- tral server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Namely, the input data is partitioned to be stored in s sites, and each site can compute an individual coreset and send it to the central central server;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' finally, the central server computes an approximation result on the union of the collected coresets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let B be the information encoding a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' A straightforward implementation of our proposed core- set of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 yields a communication cost s � 2z + O � ( 2 µ)ρk �� B, which can be too high if z is large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', if z = 5%n and s = 10, the cost can be larger than nB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In Section 5, we prove that the communication cost can be reduced to be (roughly) � 4z +s·O � ( 2 µ)ρk �� B by using the partial clustering framework of Guha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' so we reduce the item “2sz” to be “4z”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To the best of our knowledge, this is the first communication-efficient composable 4 Randomized Greedy Algorithms for k-Center Clustering with Outliers Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Total Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (B) Rounds Local Time Malkomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2015) 3α + 2 s(k + z) 1 O ((k + z) ni) Guha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) 5α + 4 sk + z 2 O((k + z)ni) Li and Guo (2018) ((5α + 4)(1 + µ), 1 + µ) O � sk µ · log ∆ µ � 2 O � n2 i · log ∆ µ � Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) Deterministic 3 + µ s(k + z)(24 µ )ρ 1 O((k + z)ni(24 µ )ρ) Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) Randomized (sk + 6z + s log n)(24 µ )ρ 1 ˜O((k + z/s)ni(24 µ )ρ) This paper (Theorem 21) α × 1+2µ 1−2µ = α × � 1 + O(µ) � 4z + ˜O((sk)( 2 µ)ρ) 2 ˜O � k � 2 µ �ρ ni log2 z � Table 2: Existing and our results for distributed k-center clustering with z outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The “local time” column illustrates the running time on each site, where ni is the data size in site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' “∆” and “ρ” stand for the aspect ratio and the doubling dimension, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' “α” is the approximation ratio of the algorithm run on the union of the collected coresets in the central server (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', if we run the 3-approximation algorithm of Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2001, α = 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The result of Li and Guo (2018) is a bi-criteria approximation that discards (1 + µ)z outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' coreset for k-center clustering with outliers that guarantees a (1 + O(µ))-approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Please see Table 2 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 Other Related Works Clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Besides the aforementioned prior works for k-center clus- tering with outliers, a number of results for other clustering with outliers problems were also proposed in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For example, the k-means/median clustering with outliers algorithms with provable guarantees have been proposed by Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Chen (2008);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Krishnaswamy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Friggstad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018), but they are difficult to im- plement due to their high complexities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The heuristic but practical algorithms without provable guarantees have also been studied, such as Chawla and Gionis (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' By us- ing the local search method, Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2017) provided a constant factor approximation algorithm for k-means clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Furthermore, Bhaskara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) and Deshpande et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2020) respectively showed that the quality can be improved by modifying the k-means++ seeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Other recent clustering with outliers algorithms include Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Im et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Chakrabarty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Coresets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The study on coresets was initiated by Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2004), and the technique has been extensively applied for dealing with large-scale data sets in many different areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For example, it can be used to reduce the computational complexities for clustering and regression problems in machine learning (Cohen-Addad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Munteanu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To handle the problems with distributed data, the techniques like “mergeable summaries ”(Agarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2013) and “composable coresets” (Indyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Mirrokni and Zadimoghaddam, 2015) were introduced recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Aghamolaei and Ghodsi (2018) also considered the composable coreset in doubling metrics but their method is only for the ordinary k-center clustering problem (without outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 5 Ding, Huang, Liu, Yu, and Wang 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Preliminaries We consider the problem of k-center with outliers in arbitrary metrics and Euclidean space RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let (X, d) be an abstract metric, where X contains n vertices and d(·, ·) is the distance function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' with a slight abuse of notation, we also use the function d to denote the shortest distance between two subsets X1, X2 ⊆ X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', d(X1, X2) = minp∈X1,q∈X2 d(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In RD, we use ||p−q|| to denote the Euclidean distance between any two points p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For simplicity, we assume that the distance between any pair of vertices in X can be obtained in O(1) time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' for the problem in Euclidean space, it takes O(D) time to compute the distance between any pair of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Below, we introduce several important definitions that are used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Definition 1 (k-Center Clustering with Outliers) Given a metric (X, d) with two pos- itive integers k and z < n, the k-center clustering with outliers problem is to find a subset X′ ⊆ X, where |X′| ≥ n − z, and k centers {c1, · · · , ck} ⊆ X, such that max p∈X′ min 1≤j≤k d(p, cj) is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If given a set P of n points in RD, the problem is to find a subset P ′ ⊆ P, where |P ′| ≥ n − z, and k centers {c1, · · · , ck} ⊂ RD, such that maxp∈P ′ min1≤j≤k ||p − cj|| is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this paper, we always use Xopt, a subset of X with size n − z, to denote the subset yielding the optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also, let {C1, · · · , Ck} be the k clusters forming Xopt, and the resulting clustering cost be ropt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' that is, each Cj is covered by an individual ball with radius ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Usually, the optimization problems with outliers are challenging to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus we often relax our goal and allow to remove slightly more than the pre-specified number of outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Actually the same relaxation idea has been adopted by a number of works on clustering with outliers problems before (Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Li and Guo, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So we introduce Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For the sake of convenience, we describe the following Definition 2 and Definition 3 only for metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In fact, the definitions can be easily modified for the problem in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Definition 2 ((k, z)ǫ-Center Clustering) Let (X, d) be an instance of k-center clustering with z outliers, and ǫ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (k, z)ǫ-center clustering is to find a subset X′ of X, where |X′| ≥ n − (1 + ǫ)z, such that the corresponding clustering cost of Definition 1 on X′ is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (i) Given a set A of cluster centers (|A| could be larger than k), we define the clustering cost φǫ(X, A) := min � max p∈X′ min c∈A d(p, c) | X′ ⊆ X, |X′| ≥ n − (1 + ǫ)z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (ii) If |A| = k and φǫ(X, A) ≤ αropt with α > 01, the set A is called an α-approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' if |A| = βk with β > 1, the set A is called an (α, β)-approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since we discard more than z outliers, it is possible to have an approximation ratio α < 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', φǫ(X, A) < ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 6 Randomized Greedy Algorithms for k-Center Clustering with Outliers Obviously, the problem in Definition 1 is a special case of (k, z)ǫ-center clustering with ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also, Definition 1 and Definition 2 can be naturally extended to the weighted case: each vertex p has a non-negative weight wp and the total weight of outliers should be equal to z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we have the following definition for coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Definition 3 (Coreset) Given a small parameter µ ∈ (0, 1) and an instance (X, d) of k-center clustering with z outliers, a set S ⊆ X is called a µ-coreset of X, if each vertex of S is assigned a non-negative weight and φ0(S, H) ∈ (1 ± µ)φ0(X, H) for any set H ⊆ X of k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Given a large-scale instance (X, d), we can run an existing algorithm on its coreset S to compute an approximate solution for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If |S| ≪ n, the running time can be significantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Formally, we have the following claim (see the proof in Section A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Claim 4 If the set H yields an α-approximation of the µ-coreset S, it yields an α × 1+µ 1−µ- approximation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' As mentioned before, we also consider the case with low doubling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Roughly speaking, the doubling dimension describes the expansion rate of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For any p ∈ X and r ≥ 0, we use Ball(p, r) to denote the ball centered at p with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Definition 5 (Doubling Dimension) The doubling dimension of a metric (X, d) is the smallest number ρ > 0, such that for any p ∈ X and r ≥ 0, X ∩ Ball(p, 2r) is always covered by the union of at most 2ρ balls with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In Section 3, we present our constant factor approximations for k-center clustering with outliers, where the main idea is a randomized greedy approach based on the Gonzalez’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In Section 4, we study the coreset for k-center clustering with outliers in Euclidean space and doubling metrics, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In Section 5, we show that our proposed coreset in Section 4 can be constructed by a communication-efficient way for distributed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In Section 6, we conduct the experiments to evaluate the proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Randomized Greedy Algorithms for (k, z)ǫ-Center Clustering For the sake of completeness, we briefly introduce the algorithm of Gonzalez (1985) for ordinary k-center clustering first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Initially, it arbitrarily selects a vertex from X, and iteratively selects the following k − 1 vertices, where each j-th step (2 ≤ j ≤ k) chooses the vertex having the largest minimum distance to the already selected j − 1 vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' finally, each input vertex is assigned to its nearest neighbor of these selected k vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' This greedy strategy yields a 2-approximation of k-center clustering;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' the algorithm also works for the problem in Euclidean space and yields the same approximation ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this section, we show that a randomized version of the Gonzalez’s algorithm can solve the (k, z)ǫ-center clustering problem with quality guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 7 Ding, Huang, Liu, Yu, and Wang Algorithm 1 Bi-criteria Approximation Algorithm Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' parameters ǫ > 0, η ∈ (0, 1/2), and t ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let γ = z/n and initialize a set E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Initially, j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' randomly select 1 1−γ log 1 η vertices from X and add them to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Run the following steps until j = t: (a) Update j = j + 1 and let Qj be the subset of X that are the farthest (1 + ǫ)z vertices to E (for each vertex p ∈ X, its distance to E is defined as minq∈E d(p, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (b) Randomly select 1+ǫ ǫ log 1 η vertices from Qj and add them to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Output E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1 (2, O(1 ǫ ))-Approximation We consider the bi-criteria approximation that returns more than k cluster centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Our high-level idea is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The main challenge for implementing the Gonzalez’s algorithm is that the outliers and inliers are mixed in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For example, the selected vertex, which has the largest minimum distance to the already selected vertices, is very likely to be an outlier, and then the cluster- ing quality could be arbitrarily bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To resolve this issue, we replace each greedy selection step by a bi-level “greedy selection+random sampling” step: select the farthest (1 + ǫ)z points (rather than the farthest single point) with a small parameter ǫ ∈ (0, 1), and then take a random sample from this selected set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Such a combined strategy can guarantee us to successfully sample a sufficient number of inliers from the k optimal clusters, and mean- while restrict the number of sampled outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We implement our idea in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For simplicity, let γ denote z/n in the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 6 Let ǫ > 0 and η ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If we set t = ck 1−η with c = 2 + 2 k(1−η) ln 1 η in Algorithm 1, with probability at least 1 − 2η, φǫ(X, E) ≤ 2ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If 1 η and 1 1−γ are constant numbers, the size |E| = 1 1−γ log 1 η + 1+ǫ ǫ log 1 η × t = O(k ǫ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So Theorem 6 implies that E is a � 2, O(1 ǫ ) � approximation for (k, z)ǫ-center clustering of X with probability at least 1 − 2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To prove Theorem 6, we need Lemma 7 and Lemma 10 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Lemma 7 With probability at least 1 − η, the set E in Step 2 of Algorithm 1 contains at least one point from Xopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since |Xopt|/|X| = 1 − γ, Lemma 7 can be easily obtained by the following claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Claim 8 Let U be a set of elements and V ⊆ U with |V | |U| = τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Given η ∈ (0, 1), if one randomly samples 1 τ log 1 η elements from U, with probability at least 1 − η, the sample contains at least one element from V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 8 Randomized Greedy Algorithms for k-Center Clustering with Outliers Actually Claim 8 is a folklore result that has been presented in several papers before (such as Ding and Xu 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since each sampled element falls in V with probability τ, we know that the sample S contains at least one element from V with probability 1 − (1 − τ)|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore, if we want 1 − (1 − τ)|S| ≥ 1 − η, |S| should be at least log 1/η log 1/(1−τ) ≤ 1 τ log 1 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Recall that {C1, C2, · · · , Ck} are the k clusters forming Xopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Denote by λj(E) the number of the clusters which have non-empty intersection with E at the beginning of j-th round in Step 3 of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For example, through Lemma 7 we know that λ1(E) should be at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Obviously, if λj(E) = k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', Cl ∩ E ̸= ∅ for any 1 ≤ l ≤ k, E will yield a 2-approximate solution by using the triangle inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Claim 9 If λj(E) = k, then φ0(X, E) ≤ 2ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Lemma 10 In each round of Step 3 of Algorithm 1, either the event (1) d(Qj, E) ≤ 2ropt happens, or with probability at least 1 − η, the event (2) λj(E) ≥ λj−1(E) + 1 happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Proof Suppose that the event (1) does not happen, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', d(Qj, E) > 2ropt, and then we prove that the event (2) should happen with probability at least 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let J include all the indices l ∈ {1, 2, · · · , k} with E ∩ Cl ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We claim that Qj ∩ Cl = ∅ for each l ∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Otherwise, we arbitrarily select p ∈ Qj ∩Cl and p′ ∈ E ∩Cl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' by using the triangle inequality, we know that d(p, p′) ≤ 2ropt which is in contradiction to the assumption d(Qj, E) > 2ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus, Qj ∩ Xopt only contains the vertices from Cl with l /∈ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that the number of outliers is z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So we have |Qj \\ Xopt| ≤ z and |Qj∩Xopt| |Qj| ≥ ǫ 1+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' By Claim 8, if randomly selecting 1+ǫ ǫ log 1 η vertices from Qj, with probability at least 1 − η, the sample contains at least one vertex from Qj ∩ Xopt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' also, the vertex must come from ∪l/∈J Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' That is, the event (2) λj(E) ≥ λj−1(E) + 1 happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If the event (1) of Lemma 10 happens, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', d(Qj, E) ≤ 2ropt, then it implies that max p∈X\\Qj d(p, E) ≤ 2ropt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' moreover, since |Qj| = (1 + ǫ)z, we have φǫ(X, E) ≤ 2ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Next, we assume that the event (1) in Lemma 10 never happens, and prove that λj(E) = k with constant probability when j = Θ(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The following idea is inspired from Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2009) which achieves a bi-criteria approximation for k-means clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We define a random variable xj: xj = 1 if λj(E) = λj−1(E), or xj = 0 if λj(E) ≥ λj−1(E) + 1, for j = 1, 2, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So E[xj] ≤ η by Lemma 10 and � 1≤s≤j (1 − xs) ≤ λj(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (1) Also, let Jj = � 1≤s≤j(xs −η) and J0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then, {J0, J1, J2, · · · } is a super-martingale with Jj+1 −Jj < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Through the Azuma-Hoeffding inequality (Alon and Spencer, 2004), we have Prob(Jt ≥ J0 + h) ≤ e− h2 2t for any t ∈ Z+ and h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let t = ck 1−η with c = 2 + 2 k(1−η) ln 1 η 9 Ding, Huang, Liu, Yu, and Wang Algorithm 2 2-Approximation Algorithm Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' a parameter ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Initialize a set E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' randomly select one vertex from X and add it to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Run the following steps until j = k: (a) Update j = j + 1 and let Qj be the subset of X that are the farthest (1 + ǫ)z vertices to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (b) Randomly select one vertex from Qj and add it to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Output E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' and h = (c − 1)k, the inequality implies Prob( � 1≤s≤t (1 − xs) ≥ t(1 − η) − h) ≥ 1 − e− h2 2t =⇒ Prob( � 1≤s≤t (1 − xs) ≥ k) ≥ 1 − e− k(c−1)2(1−η) 2c ≥ 1 − e−(c/2−1)k(1−η) =⇒ Prob( � 1≤s≤t (1 − xs) ≥ k) ≥ 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2) Combining (1) and (2), we know that λt(E) ≥ k with probability at least 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Moreover, when λt(E) = k, it is easy to know that E is a 2-approximate solution by Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Together with Lemma 7, we immediately have Theorem 6 where the overall success probability is at least (1 − η)2 > 1 − 2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In each round of Step 3, there are O(1 ǫ) new vertices added to E, thus it takes O(1 ǫn) time to update the distances from the vertices of X to E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' to select the set Qj, we can apply the linear time selection algorithm of Blum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Overall, the running time of Algorithm 1 is O(k ǫ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If the given instance is in RD, the running time will be O(k ǫ nD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 2-Approximation for Constant k If k is a constant number, we show that a single-criterion 2-approximation can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Actually, we use the same strategy as Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1, but only run k rounds with each round sampling only one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' See Algorithm 2 for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Denote by {v1, · · · , vk} the k sampled vertices of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Actually, the proof of Theorem 11 is similar to the analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The only difference is that the probability that the event (2) λj(E) ≥ λj−1(E) + 1 in Lemma 10 happens is changed to be at least ǫ 1+ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also note that v1 ∈ Xopt with probability 1 − γ (because γ = z/n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If all of these events happen, either we obtain a 2-approximation before k steps (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', d(E, X \\ Qj) ≤ 2ropt for some j < k), or {v1, · · · , vk} fall into the k optimal clusters C1, C2, · · · , Ck separately (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 10 Randomized Greedy Algorithms for k-Center Clustering with Outliers Algorithm 3 Sublinear Time Implementation of Algorithm 1 Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' parameters ǫ > 0, η ∈ (0, 1/2), and t ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let γ = z/n, σ = 2 1+ � 1+ 4(1+ǫ) 3ǫ , n′ = 3 σ2(1+ǫ)γ log 4 η and initialize a set E = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Initially, j = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' randomly select 1 1−γ log 1 η vertices from X and add them to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Run the following steps until j = t: (a) Update j = j+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' uniformly sample n′ vertices from X and denote the sampled set as Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (b) Let ˆrj be the (1 + σ)(1 + ǫ)γn′-th farthest distance from Aj to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let ˆAj = {p ∈ Aj | d(p, E) ≥ ˆrj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (c) Add ˆAj to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Output E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' λk(E) = k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' No matter which case happens, we always obtain a 2-approximation with respect to the (k, z)ǫ-center clustering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So we have the following Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 11 Algorithm 2 returns a 2-approximation for the problem of (k, z)ǫ-center clus- tering on X, with probability at least (1− γ)( ǫ 1+ǫ)k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The time complexity is O(kn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If the given instance is in RD, the time complexity will be O(knD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To boost the probability of Theorem 11, we just need to repeatedly run the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The success probability is easy to calculate by taking the union bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Corollary 12 If we run Algorithm 2 O � 1 1−γ (1+ǫ ǫ )k−1� times, with constant probability, at least one time the algorithm returns a 2-approximation for the problem of (k, z)ǫ-center clustering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='3 Sublinear Time Implementation of Algorithm 1 The input data size n can be quite large in practice, so in this section we consider to implement Algorithm 1 with a lower time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We present a modified version of Algorithm 1 that only needs an O( k2 γǫ2) time complexity, where γ = z n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' When the data contains heavy noise and the outliers takes a constant factor of n (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', z = 5%n and 1 γ = 20), the algorithm has a sublinear time complexity that is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Moreover, the quality of Algorithm 1 presented in Theorem 6 can be guaranteed exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The key observation and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Recall that in each round of Algorithm 1, we need to scan the whole data set and select the farthest (1 + ǫ)z vertices to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus it takes linear time in each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Our key observation is that we can actually avoid this step by simple random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The new idea is shown in Algorithm 3 (Step 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We still let Qj be the set of (1 + ǫ)z farthest vertices to E from X in the j-th round (as Step 3(a) in 11 Ding, Huang, Liu, Yu, and Wang Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We randomly sample n′ vertices from X and use Aj to denote this sampled set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We can view each sampled vertex of Aj as an independent random variable ∈ {0, 1}: each sampled vertex is labeled by “1” if it belongs to Qj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' otherwise, it is labeled by “0”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let σ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Through the Chernoff bound, we have Prob � |Aj ∩ Qj| ∈ (1 ± σ)(1 + ǫ)γn′� ≥ 1 − 2e− 1 3(σ2(1+ǫ)γn′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (3) We select the farthest (1 + σ)(1 + ǫ)γn′ vertices to E from Aj, where the selected set is denoted by ˆAj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We set the parameter σ = 2 1+ � 1+ 4(1+ǫ) 3ǫ , and then the right hand-side of (3) becomes 1 − η 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So with probability at least 1 − η 2, |Aj ∩ Qj| ≤ (1 + σ)(1 + ǫ)γn′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Because | ˆAj| = (1 + σ)(1 + ǫ)γn′, we have | ˆAj| ≥ |Aj ∩ Qj|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (4) Denote by rj the distance d(Aj ∩ Qj, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since Qj is the set of (1 + ǫ)z farthest vertices to E from X, we have {p ∈ Aj | d(p, E) ≥ rj} = Aj ∩ Qj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (5) {p ∈ Aj | d(p, E) > rj} ⫋ Aj ∩ Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (6) Now we claim that ˆrj ≤ rj where ˆrj is the (1 + σ)(1 + ǫ)γn′-th farthest distance from Aj to E (as defined in Step 3(b) of Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Otherwise, if ˆrj > rj, we can deduce that ˆAj ⊆ {p ∈ Aj | d(p, E) > rj} ⫋ Aj ∩ Qj from (6) the fact ˆAj = {p ∈ Aj | d(p, E) ≥ ˆrj};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' so we have | ˆAj| < |Aj ∩ Qj| which is contradictory to (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore we have ˆrj ≤ rj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' together with (5), it implies Aj ∩ Qj ⊆ {p ∈ Aj | d(p, E) ≥ ˆrj} = ˆAj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Hence Aj ∩ Qj ⊆ ˆAj ∩ Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' On the other hand, since ˆAj ⊆ Aj, it is easy to know ˆAj ∩ Qj ⊆ Aj ∩ Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore, Aj ∩ Qj = ˆAj ∩ Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (7) Moreover, from (3) again we know that |Aj ∩ Qj| ≥ (1 − σ)(1 + ǫ)γn′ = 1 + ǫ ǫ log 2 η (8) with probability at least 1− η 2, where we set n′ = 3 σ2(1+ǫ)γ log 4 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' From (7) and (8), we know that | ˆAj ∩ Qj| = |Aj ∩ Qj| ≥ 1 + ǫ ǫ log 2 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore, ˆAj contains at least 1+ǫ ǫ log 2 η vertices from Qj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we can obtain the similar result as Lemma 10: in each round, the event “ either (1) d(Qj, E) ≤ 2ropt or (2) λj(E) ≥ λj−1(E) + 1” happens with probability at least 1 − η 2 − η 2 = 1 − η (recall the probability in (3) is 1 − η 2, so the overall probability is at least 1 − η 2 − η 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Together with the same super-martingale argument of Theorem 6, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 12 Randomized Greedy Algorithms for k-Center Clustering with Outliers Theorem 13 Let ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If we set t = ck 1−η with c = 2 + 2 k(1−η) ln 1 η for Algorithm 3, with probability at least 1 − 2η, φǫ(X, E) ≤ 2ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Quality and time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We assume that γ and 1/η are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Algorithm 3 adds O( 1 σ2 ) vertices to E at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that σ = Θ(√ǫ), which implies the number of vertices added to E at each iteration is O(1 ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So |E| = O(k ǫ ) at the end of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then Theorem 13 implies that E is a � 2, O(1 ǫ ) � approximation for (k, z)ǫ-center clustering of X with constant probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Each round we compute the distances from the vertices of Aj to E and select ˆAj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since |Aj| = O( 1 γǫ) and |E| = O(k ǫ ), we have the time complexity O( k γǫ2 ) for computing the distances in each round of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The selection of ˆAj takes O(|Aj|) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Overall, the time complexity of Algorithm 3 is O( k2 γǫ2), which is independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If the given distance is in RD, the time complexity will be O( k2 γǫ2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If the input data size n is large, Algorithm 3 can significantly reduce the time complexity and meanwhile preserve the same clustering quality of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Coresets for k-Center Clustering with Outliers In this section, we consider the coreset construction problem for k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' First, we show that the simple uniform sampling approach can yield a slightly weaker coreset for (k, z)ǫ-center clustering in Euclidean space, where the number of dis- carded outliers is amplified from (1 + ǫ)z to be (1 + O � ǫ) � z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we consider the coreset construction in doubling metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We show that the idea of Algorithm 1 can be extended for building the coreset efficiently, even if the doubling dimension is not given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1 Uniform Sampling in Euclidean Space Given a metric (X, d), Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2003) showed that we can use a random sample S to replace X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Recall γ = z/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let |S| = O( k ǫ2γ ln n) and E be an α-approximate solution of (k, z)ǫ-center clustering on (S, d), then E is an α-approximate solution of (k, z)O(ǫ)- center clustering on (X, d) with constant probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In a D-dimensional Euclidean space, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018) showed a similar result, where the sample size |S| = ˜O( 1 ǫ2γ2 kD)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this section, we show that the sample size of Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2018) can be further improved by a factor 1 γ and the new sample size is ˜O( 1 ǫ2γ kD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' This improvement could be important for the case z ≪ n, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', z = √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Below We revisit their idea first, and then provide a more careful analysis to achieve the improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let P be a set of n points in RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Consider the range space Σ = (P, Π) where each range π ∈ Π is the complement of union of k balls in RD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We know that the VC dimension of balls is O(D) (Alon and Spencer, 2004), and therefore the VC dimension of union of k balls is O(kD log k) (Blumer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' That is, the VC dimension of the range space Σ is O(kD log k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let ǫ ∈ (0, 1), and an “ǫ-sample” S of P is defined as follows: ∀π ∈ Π, ��|π ∩ P| |P| − |π ∩ S| |S| �� ≤ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The asymptotic notation ˜O(f) = O � f · polylog( kD ǫγ ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 13 Ding, Huang, Liu, Yu, and Wang Roughly speaking, S is an approximation of P with an additive error within each range π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Given a range space with the VC dimension dvc, an ǫ-sample can be easily obtained via uniform sampling (Alon and Spencer, 2004), where the success probability is 1 − λ and the sample size is O � 1 ǫ2(dvc log dvc ǫ + log 1 λ) � for any 0 < λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For our problem, we need to replace the “ǫ” of the “ǫ-sample” by ǫγ to guarantee that the number of uncovered points is bounded by � 1+O(ǫ) � γn (we show the details below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since dvc = O(kD log k), the sample size is ˜O( 1 ǫ2γ2 kD) (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Actually, the front factor 1 ǫ2γ2 of the sample size can be further reduced to be 1 ǫ2γ by a more careful analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We observe that there is no need to guarantee the additive error for each range π (as the definition of ǫ-sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Instead, only a multiplicative error for the ranges covering at least γn points should be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that when a range covers more points, the multiplicative error is weaker than the additive error and thus the sample size is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For this purpose, we use the relative approximation (Har-Peled and Sharir, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2001): let S ⊆ P be a subset of size ˜O( 1 ǫ2γ kD) chosen uniformly at random, then with constant probability, ∀π ∈ Π, ���|π ∩ P| |P| − |π ∩ S| |S| ��� ≤ ǫ × max �|π ∩ P| |P| , γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (9) We formally state our result below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 14 shows that if we have an α-approximation algorithm, we can run it on the sample S to obtain a solution E, which is also an α- approximate solution for (k, z)O(ǫ)-center clustering on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Because |S| ≪ |P|, we can reduce a great amount of runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 14 Let P be an instance for the problem of k-center clustering with outliers in RD as described in Definition 1, and S ⊆ P be a subset of size ˜O( 1 ǫ2γ kD) chosen uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Suppose ǫ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let S be a new instance for the problem of k-center clustering with outliers where the number of outliers is set to be z′ = (1 + ǫ)γ|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If E is an α- approximate solution of (k, z′)ǫ-center clustering on S, then E is an α-approximate solution of (k, z)O(ǫ)-center clustering on P, with constant probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Proof We assume that S is a relative approximation of P and (9) is true (this happens with constant probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let Bopt be the set of k balls covering (1 − γ)n points induced by the optimal solution for P, and BS be the set of k balls induced by an α-approximate solution of (k, z′)ǫ-center clustering on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Suppose the radius of each ball in Bopt (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', BS) is ropt (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', rS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We denote the complements of Bopt and BS as πopt and πS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' First, since Bopt covers (1 − γ)n points of P and S is a relative approximation of P, we have ��πopt ∩ S �� |S| ≤ ��πopt ∩ P �� |P| + ǫ × max �|πopt ∩ P| |P| , γ � = (1 + ǫ)γ by (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' That is, the set balls Bopt cover at least � 1 − (1 + ǫ)γ � |S| points of S, and therefore it is a feasible solution for the instance S with respect to the problem of k-center clustering with z′ outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since BS is an α-approximate solution of (k, z′)ǫ-center clustering on S, we have rS ≤ αropt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' |πS ∩ S| ≤ (1 + ǫ)z′ = (1 + ǫ)2γ|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (10) 14 Randomized Greedy Algorithms for k-Center Clustering with Outliers Now, we claim that ��πS ∩ P �� ≤ (1 + ǫ)2 1 − ǫ γ|P|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (11) Assume that (11) is not true, then (9) implies ���|πS ∩ P| |P| − |πS ∩ S| |S| ��� ≤ ǫ × max �|πS ∩ P| |P| , γ � = ǫ|πS ∩ P| |P| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So |πS∩S| |S| ≥ (1−ǫ)|πS∩P | |P | > (1+ǫ)2γ, which is in contradiction with the second inequality of (10), and thus (11) is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We assume ǫ ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5, so 1 1−ǫ ≤ 1+2ǫ and (1+ǫ)2 1−ǫ = 1+O(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Con- sequently (11) and the first inequality of (10) together imply that BS is an α-approximate solution of (k, z)O(ǫ)-center clustering on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 Coreset Construction in Doubling Metrics Actually the sample obtained in Theorem 14 is not a standard coreset as Definition 3, since it always incurs an error on the number of discarded outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this section, we consider constructing the coreset that strictly satisfies Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We introduce the following assumption first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Assumption 15 Given an instance (X, d) of k-center clustering with outliers, the metric (Xopt, d), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', the metric formed by the set of inliers, has a constant doubling dimension ρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We do not have any restriction on the outliers X \\ Xopt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus the above assumption is more relaxed and practical than assuming the whole (X, d) has a constant doubling dimen- sion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', the previous coreset construction algorithm of Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) assumed that the whole (X, d) has a constant doubling dimension ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' From Definition 5, we directly know that each optimal cluster Cj of Xopt can be covered by 2ρ balls with radius ropt/2 (see the left figure in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So we can imagine that the instance (X, d) has 2ρk clusters, where the optimal radius is at most ropt/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore, we can just replace k by 2ρk in Algorithm 1, so as to reduce the approximation ratio (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', the ratio of the obtained radius to ropt) from 2 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 16 If we set t = 2ρck 1−η with c = 2+ 2 k(1−η) ln 1 η for Algorithm 1, with probability at least 1 − 2η, φǫ(X, E) ≤ ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So the set E is a � 1, O(2ρ ǫ ) � approximation for the problem of (k, z)ǫ-center clustering, and the time complexity is O((k + ln 1 2η)2ρ ǫ n ln 1 2η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 16 is a warm-up, and we can further construct the coreset for k-center clustering with outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let µ ∈ (0, 1), and for simplicity we assume that log 2 µ is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If applying Definition 5 recursively, we know that each Cj is covered by 2ρ log 2/µ = ( 2 µ)ρ balls with radius µ 2 ropt, and Xopt is covered by ( 2 µ)ρk such balls in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' See the right figure in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we have Algorithm 4 based on this observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 15 Ding, Huang, Liu, Yu, and Wang ✦ �✧ ★ ✦ �✧ ★ ✁ ✂ ✄☎ �✧ ★ ✁ ✂ ✄☎ �✧ ★ ✁ ✂ ✦ � ✧ ★ ✁ ✂ ✦ � ✧ ★ Figure 1: Illustrations for Theorem 16 and Theorem 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Algorithm 4 Coreset Construction in Doubling Metrics Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' parameters η ∈ (0, 1/2) and µ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let l = ( 2 µ)ρk, c = 2 + 2 k(1−η) ln 1 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Set ǫ = 1 and run Algorithm 1 with t = cl 1−η rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Denote by ˜r = φ1(X, E) the maximum distance between E and X by excluding the farthest 2z vertices, after the final round of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let X˜r = {p | p ∈ X and d(x, E) ≤ ˜r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For each vertex p ∈ X˜r, assign it to its nearest neighbor in E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' for each vertex q ∈ E, let its weight be the number of vertices assigning to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Add X \\ X˜r to E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' each vertex of X \\ X˜r has weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Output E as the coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 17 Let η ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' With probability at least 1 − 2η, Algorithm 4 returns a µ-coreset E of k-center clustering with z outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The size of E is at most 2z +O � ( 2 µ)ρ(k + ln 1 2η) ln 1 2η � , and the construction time is O(n( 2 µ)ρ(k + ln 1 2η) ln 1 2η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Proof Similar to Theorem 16, we know that |X˜r| = n − 2z and ˜r ≤ 2× µ 2ropt = µropt with probability at least 1 − 2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The size of E is |X \\ X˜r| + O � ( 2 µ)ρ(k + ln 1 2η ) ln 1 2η � = 2z + O � ( 2 µ)ρ(k + ln 1 2η ) ln 1 2η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Moreover, it is easy to see that the running time of Algorithm 4 is O � ( 2 µ)ρ(k+ln 1 2η)n ln 1 2η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Next, we show that E is a qualified µ-coreset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For each vertex q ∈ E, denote by w(q) the weight of q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' for the sake of convenience in our proof, we view each q as a set of w(q) overlapping unit weight vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus, from the construction of E, we can see that there is a bijective mapping f between X and E, where d (p, f(p)) ≤ ˜r ≤ µropt, ∀p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (12) 16 Randomized Greedy Algorithms for k-Center Clustering with Outliers Let H = {c1, c2, · · · , ck} be any k vertices of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Suppose that H induces k clusters {A1, A2, · · · , Ak} (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', {B1, B2, · · · , Bk}) with respect to the problem of k-center cluster- ing with z outliers on E (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', X), where each Aj (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', Bj) has the cluster center cj for 1 ≤ j ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let rE = φ0(E, H) and rX = φ0(X, H), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also, let r′ E (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', r′ X) be the smallest value r, such that for any 1 ≤ j ≤ k, f(Bj) ⊆ Ball(cj, r) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', f −1(Aj) ⊆ Ball(cj, r)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We need the following claim (see the proof in Section B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Claim 18 |r′ E − rX| ≤ µropt and |r′ X − rE| ≤ µropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In addition, since {f(B1), · · · , f(Bk)} also form k clusters for the instance E with the fixed k cluster centers of H, we know that r′ E ≥ φ0(E, H) = rE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Similarly, we have r′ X ≥ rX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Combining Claim 18, we have rX − µropt ≤ r′ X − µropt ≤ rE � �� � by Claim 18 ≤ r′ E ≤ rX + µropt � �� � by Claim 18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So |rX − rE| ≤ µropt, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', φ0(E, H) ∈ φ0(X, H) ± µropt ⊆ (1 ± µ)φ0(X, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore E is a µ-coreset of (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Remark 19 (1) It is worth emphasizing that the uniform sampling idea in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1 cannot avoid the error on the number of excluded outliers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' the sample size will become infinity if not allowing to remove more than z outliers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 1 ǫ = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' But our proposed coreset method in Theorem 17 can guarantee the clustering quality for excluding exactly z outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2) The coefficient “2” of z in the coreset size actually can be further reduced by modi- fying the value of ǫ in Step 2 of Algorithm 4 (we set ǫ = 1 just for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In general, the size of E is (1 + ǫ)z + O �1 ǫ ( 2 µ)ρ(k + ln 1 2η ) ln 1 2η � and the construction time is O(n 1 ǫ( 2 µ)ρ(k + ln 1 2η) ln 1 2η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='3 When the Doubling Dimension ρ Is Not Given In Algorithm 4, we run Algorithm 1 t = cl 1−η rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' But when the doubling dimension ρ is not given, we cannot determine the values of l and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We are aware of several techniques for estimating the doubling dimension of a given data set (Har-Peled and Mendel, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) also mentioned that their coreset construction method can be ap- plied to the case that even ρ is not given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' These ideas mainly rely on the fact that if one runs the Gonzalez’s k-center clustering algorithm on the data, the obtained radius can be significantly reduced due to the property of doubling metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' However, we need to empha- size that these doubling dimension estimation techniques cannot be applied to our problem under Assumption 15, since the outliers and inliers are mixed and only the inliers have the nice property of doubling metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We perform the following modification for Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Roughly speaking, we decompose Step 2 of Algorithm 4 into two substeps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 17 Ding, Huang, Liu, Yu, and Wang (1) First, we run Algorithm 1 ˜k = ck 1−η rounds and then obtain the radius ˜r = φ1(E, X) ≤ 2ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Now X is partitioned into ˜k clusters H1, H2, · · · , H˜k with excluding 2z outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Each Hj ∩ Xopt has a constant doubling dimension ρ (note that each Hj may also contain some points from X \\ Xopt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also, the size ��� � ∪˜k j=1 Hj � \\ Xopt ��� ≤ z, and it implies ��� � ∪ ˜k j=1 Hj � ∩ Xopt ��� = ��� ∪ ˜k j=1 Hj ��� − ��� � ∪ ˜k j=1 Hj � \\ Xopt ��� ≥ n − 2z − z = n − 3z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore, if we view the instance (X, d) as an instance of ˜k-center clustering with 3z outliers, the optimal radius (denote by r(−3z) opt ) should be at most ˜r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Overall, we have the upper and lower bounds for ˜r: r(−3z) opt ≤ ˜r ≤ 2ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2) Then, if we run Step 3 of Algorithm 1 (replacing “z” by “3z”) with at most t = cl′ 1−η rounds where l′ = �r(−3z) opt 1 4µ˜r �ρ˜k ≤ � r(−3z) opt 1 4µr(−3z) opt �ρ˜k = O � ( 4 µ)ρk � , the obtained radius (excluding the farthest 6z vertices) should be at most 2 × 1 4µ˜r ≤ 2 × 1 2µropt = µropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we can use the similar idea of the proof of Theorem 17 to show that the obtained set E is a qualified µ-coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Overall, we have Algorithm 5 for the case that the doubling dimension ρ is not given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The time complexity is O( cl′ 1−ηn) = O � n( 4 µ)ρ(k + ln 1 2η) ln 1 2η � , and the coreset size is 6z + O � ( 4 µ)ρ(k + ln 1 2η) ln 1 2η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 20 With probability at least 1−2η, Algorithm 5 outputs a µ-coreset E of k-center clustering with z outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The size of E is at most 6z + O � ( 4 µ)ρ(k + ln 1 2η) ln 1 2η � , and the construction time is O(( 4 µ)ρ(k + ln 1 2η)n ln 1 2η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We can apply the same idea of Remark 19 (2) to reduce the coreset size to be 3(1 + ǫ)z + O �1 ǫ( 4 µ)ρ(k + ln 1 2η) ln 1 2η � , and meanwhile, the time complexity becomes O � n ǫ ( 4 µ)ρ(k + ln 1 2η) ln 1 2η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Coreset for Distributed Data In this section, we consider the coreset for distributed clustering in the coordinator model (ˇDuriˇs and Rolim, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Suppose the data X = ⊔s i=1Xi are distributed disjointly among s ≥ 2 sites;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' all the sites can communicate with a central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let O = ⊔s i=1Oi, where Oi ⊂ Xi, be the set of outliers in the optimal solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' also suppose each |Oi| = z∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let X \\ O = ⊔k j=1Cj be the k optimal clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that the value of each z∗ i is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 18 Randomized Greedy Algorithms for k-Center Clustering with Outliers Algorithm 5 Coreset Construction in Doubling Metrics with Unknown ρ Input: An instance (X, d) of metric k-center clustering with z outliers, and |X| = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' parameters µ ∈ (0, 1) and η ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Set ǫ = 1 and run Algorithm 1 t = ck 1−η rounds where c = 2+ 2 k(1−η) ln 1 η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Denote by ˜r = φ1(X, E) the maximum distance between E and X by excluding the farthest 2z vertices, after the final round of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Continue to run Step 3 of Algorithm 1 (but replacing “z” by “3z”) until φ5(X, E) ≤ 1 2µ˜r (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', excluding 6z outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Set ˜r′ = φ5(X, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let X˜r′ = {p | p ∈ X and d(p, E) ≤ ˜r′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For each vertex p ∈ X˜r′, assign it to its nearest neighbor in E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' for each vertex q ∈ E, let its weight be the number of vertices assigning to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Add X \\ X˜r′ to E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' each vertex of X \\ X˜r′ has weight 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Output E as the coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus a straightforward approach is to compute a coreset for the k-center clustering with z outliers on each Xi, and directly send the obtained coresets to the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let B be the information encoding a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Obviously this approach takes a communication cost � 2sz + s · O � ( 2 µ)ρ(k + log 1 2η ) ln 1 2η �� B, (13) which can be to too high if z is large (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', if z = 5%n and s = 10, the cost can be larger than nB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this section, we show that the framework for distributed k-median/means clustering with outliers developed by Guha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) can also be applied to the k-center clustering with z outliers problem with our proposed coreset method in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' in particular, the term “2sz” of (13) can be reduced to be “4z”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The high level idea of Guha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' First, we need to design a set of numbers {z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zs}, where each zi is an upper bound of z∗ i for 1 ≤ i ≤ s and their sum �s i=1 zi ≤ 2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that the requirement “�s i=1 zi ≤ 2z” is important for bounding the total communication cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Each site i runs the coreset algorithm for k-center clustering with zi outliers on Xi to construct a local coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then each site sends the weighted points of the local coreset to the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Finally the central server aggregates the weighted points to form a global coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The key challenge is to compute the set {z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zs} that are suitable for our coreset method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Below we introduce some notations first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' ropt(Xi, k, z∗ i ): the optimal radius of k-center clustering with z∗ i outliers on Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Given a set of 2-dimensional points A = {(x1, y1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , (xl, yl)} ⊂ R2 where x1 < x2 < · · · < xl, we define the corresponding piecewise function hA(·) from [x1, ∞) to R: hA(x) = yi if xi ≤ x < xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Here we define xl+1 = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' See Figure 2 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 19 Ding, Huang, Liu, Yu, and Wang Figure 2: An illustration for a non-increasing piecewise function hA(·) with A = {(x1, y1), (x2, y2), (x3, y3), (x4, y4), (x5, y5)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For any pair of (xi, yi) and (xj, yj), we define their lexicographical order: (xi, yi) ≺ (xj, yj) if � xi < xj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' or xi = xj and yi < yj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Theorem 21 With probability at least 1−2s(2+log2 z)η, Algorithm 6 returns a 2µ-coreset E of k-center clustering with z outliers for the distributed input X = ⊔s i=1Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The total communication complexity is � 4z + O(( 2 µ)ρs(k + ln 1 2η) ln 1 2η) � B over 2 rounds, where B is the communication cost for sending one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The running time in each site is O(( 2 µ)ρ(k + ln 1 2η)ni ln 1 2η log2 z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Remark 22 We can replace η by η 2s(2+log2 z) in Algorithm 6 to achieve a success probability 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The communication complexity will be � 4z + O � ( 2 µ)ρs(k + ln s log2 z η )(ln s log2 z η ) �� B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Before proving Theorem 21, we introduce Lemma 23 and Lemma 24 first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Lemma 23 2ropt ≥ max1≤i≤s ropt(Xi, k, z∗ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that though Xi ⊂ X, the optimal radius ropt(Xi, k, z∗ i ) of site i is not necessary to be ≤ ropt, since the cluster centers of X may not belong to Xi (but for the problem in Euclidean space, ropt(Xi, k, z∗ i ) is always no larger than ropt since the cluster centers can be any points in the space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Xi \\ Oi is the set of inliers of site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For each optimal cluster Cj, 1 ≤ j ≤ k, we can arbitrarily take an inlier from (Xi \\ Oi) ∩ Cj as the surrogate cluster center (if (Xi \\ Oi) ∩ Cj = ∅, we just ignore this cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' From the triangle inequality, we know ropt(Xi, k, z∗ i ) ≤ 2ropt and thus obtain Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In the following analysis, we assume that the function hAi in Step 2(a) is non-increasing for each i = 1, 2, · · · , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Actually this assumption is easy to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' If there exist a couple q < q′ such that ˜ri,q > ˜ri,q′, we can simply replace the coreset Ei,q by the coreset Ei,q′ and let ˜ri,q = ˜ri,q′ in Step 2 of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The following lemma illustrates the key properties of the obtained values z1, z2, · · · , zs in Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 20 3 0 (2) (3) (33) ()Randomized Greedy Algorithms for k-Center Clustering with Outliers Algorithm 6 Distributed Coreset Construction Input: An instance (X, d) of distributed metric k-center clustering with z outliers, and X = ⊔s i=1Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' the parameters η ∈ (0, 1/2), µ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let [z] = {0, 1, 2, · · · , z} and Γ = {2r : 1 ≤ r ≤ ⌊log2 z⌋ , r ∈ Z} ∪ {0, z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Run the following two-round communication between the sites and the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (1st round) In each site i: run Algorithm 4 to obtain the radius ˜ri,q and the coreset Ei,q for each q ∈ Γ, where ˜ri,q is the radius “˜r” in Algorithm 4 with setting the number of outliers to be q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (a) Define the set Ai = {(q, ˜ri,q) | q ∈ Γ}, and construct the corresponding piecewise function hAi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (b) Send the function hAi to the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (1st round) In the central server: sort the s(z + 1) pairs {(hAi(q), i) | i = 1, 2, · · · , s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' q ∈ [z]} with a lexicographical decreasing order;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' select the (2z + 1)-th largest item, say “(hAi0(q0), i0)”, and broadcast it to all the sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2nd round) In each site i: (a) If i ̸= i0, let zi = min{q ∈ [z]: (hAi(q), i) ≺ (hAi0(q0), i0)} (if the set is ∅, let zi = z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (b) Else, i = i0, let zi0 = min{q ∈ Γ: hAi0(q) = hAi0(q0)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (c) Send Ei,zi to the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2nd round) In the central server: take the union E = ∪s i=1Ei,zi as the final coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Lemma 24 The set {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zs} obtained in Step 4 of Algorithm 6 is the optimal solution for the following minimax problem: min q1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=',qs max 1≤i≤s hAi(qi) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' s� i=1 qi ≤ 2z, qi ∈ [z], i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (14) Proof We consider the following two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Recall that “i0” is the index obtained in Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Case (i): hAi0(zi0) = maxi hAi(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this case, by the definition of zi and the fact that hAi(·) is non-increasing, we have (hAi0(q0), i0) ≺ (hAi(q), i) for each q = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zi − 1 and each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Hence there are �s i=1 zi pairs of (hAi(q), i)s that are larger than (hAi0(q0), i0) in the lexicographical order .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore by the definition of (hAi0(q0), i0), we 21 Ding, Huang, Liu, Yu, and Wang know that zi0 ≤ q0, which implies s � i=1 zi ≤ q0 + � i̸=i0 zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The right-hand side “q0 + � i̸=i0 zi” is exactly equal to 2z since it is the number of items ranked ahead of (hAi0 (q0) , i0) in the sorted sequence in Step 3 of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Suppose Lemma 24 is not true, then there exists another solution {z′ 1, · · · , z′ s} of the problem (14) such that max i hAi(z′ i) < hAi0(zi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (15) The right-hand side “hAi0(zi0)” is equal to hAi0(q0) by the definition of zi0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So it implies hAi0(z′ i0) < hAi0(q0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' since hAi0(·) is non-increasing, we know z′ i0 > q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Without loss of generality, we assume � i z′ i = 2z (again, because hAi(·) is non-increasing, we can always enlarge the z′ is until � i z′ i = 2z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that q0 + � i̸=i0 zi = 2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So there should exist an index j ̸= i0, such that z′ j < zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' By the definition of zj, we have (hAi0(q0), i0) ≺ (hAj(z′ j), j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore hAj(z′ j) ≥ hAi0(q0) = hAi0(zi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus maxi hAi(z′ i) ≥ hAj(z′ j) ≥ hAi0(zi0), which is contradictory to (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Case (ii): suppose hAi1(zi1) = maxi hAi(zi) and hAi1(zi1) > hAi0(zi0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In this case we have zi1 = z in Step 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Similar to the analysis for case (i), we have (hAi0(q0), i0) ≺ (hAi(q), i) for q = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zi − 1, i ̸= i0, and meanwhile (hAi0(q0), i0) ≺ (hAi1(q), i1) for q = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Hence similarly we have 1 + �s i=1 zi ≤ q0 + 1 + � i̸=i0 zi = 2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For any feasible solution {z′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , z′ s}, since z′ i1 ≤ z = zi1 and hAi1(·) is non-increasing, we have max i hAi(z′ i) ≥ hAi1(z′ i1) ≥ hAi1(zi1) = max i hAi(zi), which implies {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zs} is better than the solution {z′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , z′ s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So {z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , zs} should be the optimal solution of the problem (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Proof (of Theorem 21) We define ˆzi = min{q ∈ Γ: q ≥ z∗ i } for i = 1, 2, · · · , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we directly have maxi ropt(Xi, k, z∗ i ) ≥ maxi ropt(Xi, k, ˆzi) since ˆzi ≥ z∗ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Because “ri,ˆzi” is the radius of the coreset in Step 2, we have ˜ri,ˆzi ≤ µropt(Xi, k, ˆzi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also we know 2ropt ≥ maxi ropt(Xi, k, z∗ i ) from Lemma 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So we have 2ropt ≥ 1 µ maxi ˜ri,ˆzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that hAi(ˆzi) = ˜ri,ˆzi as ˆzi ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Hence we have 2ropt ≥ 1 µ max i hAi(ˆzi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (16) The definitions of ˆzi and Γ together imply that � i ˆzi ≤ � i 2z∗ i = 2z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Therefore the set {ˆz1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , ˆzs} is a feasible solution for the problem (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' By Lemma 24, we have max i hAi(ˆzi) ≥ max i hAi(zi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (17) 22 Randomized Greedy Algorithms for k-Center Clustering with Outliers For each i ̸= i0, we know zi ∈ Γ by the definition of the piecewise function hAi(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' By Step 4(b) of Algorithm 6, we know zi0 ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Thus we have hAi(zi) = ˜ri,zi = φ1(Xi, Ei,zi), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' , s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (18) Combining the inequalities (16), (17), and (18), we have max i φ1(Xi, Ei,zi) ≤ 2µ · ropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Similarly to the inequality (12) in the proof of Theorem 17, we can define the bijective mapping “f” from X to E (recall that E = ∪s iEi,zi), such that d(p, f(p)) ≤ 2µ · ropt, ∀p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (19) The above inequality (19) implies that the set E is a qualified 2µ-coreset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The success probability is at least 1 − s(2 + log2 z)2η since each site runs Algorithm 4 no more than (2+ log2 z) times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Since �s i zi ≤ 2z, the total number of points sent from the sites to the central server is no larger than 4z + O(( 2 µ)ρs(k + ln 1 2η) ln 1 2η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' So we obtain the communication complexity � 4z + O(( 2 µ)ρs(k + ln 1 2η) ln 1 2η) � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The running time in each site i is O(|Γ|( 2 µ)ρ(k + ln 1 2η)ni ln 1 2η) = O(( 2 µ)ρ(k + ln 1 2η)ni ln 1 2η log2 z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Experiments All the experiments were conducted on an Ubuntu workstation with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='40GHz Intel(R) Xeon(R) CPU E5-2680 and 256GB main memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The algorithms were implemented in MATLAB R2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Our code is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='com/OpsTreadstone/randomized-k-center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We compare our algorithms with two well known baselines, “CKM+” (Charikar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2001) and “MK” (McCutchen and Khuller, 2008), as well as the recently proposed algo- rithm “BVX” (Bhaskara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For the coreset construction problem, we compare our algorithm with “CPP” (Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019) and the uniform sampling method “Uniform”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For the distributed setting, we take “CPP”, “MKC+” (Malkomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2015), “GLZ” (Guha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019), and “LG” (Li and Guo, 2018) as the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' All the experiments were repeated 10 times and we report the average results with the standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We evaluate our algorithms on four real-world classification data sets from the UCI KDD archive (Dua and Graff, 2017): Shuttle, Covertype, KDD Cup 1999 and Poker Hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The Shuttle data set (King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 1995) contains 43, 500 instances of 7 classes with 9 numerical attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The Covertype data set contains 581, 012 instances of 7 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' It has 54 attributes of continuous and categorical types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The KDD Cup 1999 data set contains 4, 898, 431 instances of 23 classes with 41 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The Poker Hand data set contains 1, 025, 010 instances of 10 classes with 10 attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For each of the latter three data sets Covertype, KDD Cup 1999, and Poker Hand, we randomly select 100, 000 instances and run the algorithms on the selected instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 23 Ding, Huang, Liu, Yu, and Wang 4 6 8 10 12 14 16 18 20 0 50 100 150 ϕ ϵ (X, ϵ) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 Alg 1 Alg 3 BVX 4 6 8 10 12 14 16 18 20 50 100 150 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='6 4 6 8 10 12 14 16 18 20 50 100 150 ϵ = 1 4 6 8 10 12 14 16 18 20 0 5 10 15 Running time (s) 4 6 8 10 12 14 16 18 20 0 2 4 6 8 10 4 6 8 10 12 14 16 18 20 0 5 10 k Figure 3: The performance of Algorithm 1 and Algorithm 3 on Shuttle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To generate the outliers, for each data set we compute the minimum enclosing ball of the whole data set by using the algorithm of B˘adoiu and Clarkson (2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' let rmeb and cmeb be the radius and the center, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we randomly add 1% points as the outliers inside the ball of radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1 × rmeb centered at cmeb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1 The Bi-criteria Algorithms We compare Algorithm 1 and its sublinear version Algorithm 3 with BVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For Algorithm 1, we set ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='6, 1, and modify the parameters of Algorithm 3 and BVX accordingly so that they can output the same number of centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We vary k from 4 to 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The experimental results are shown in Figure 3, Figure 4, Figure 5, and Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Comparing with BVX, Algorithm 1 and Algorithm 3 take significantly lower running time, and meanwhile achieve similar or lower clustering cost φǫ(X, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To have a more clear comparison between Algorithm 1 and Algorithm 3, we zoom in on the experimental results of ǫ = 1 without BVX (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We can see that the running time of Algorithm 3 grows much slower than Algorithm 1 as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' This result also agrees with our theoretical analysis since Algorithm 3 has only sublinear time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We also compare Algorithm 2 with CKM+, MK, and BVX for small k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We let k = 2, 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For Algorithm 2, we set ǫ = 1 and run it ln 10 1−γ (1+ǫ ǫ )k−1 times as Corollary 12 suggests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The experimental results are shown in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In general, Algorithm 2 achieves comparable clustering cost with CKM+ and MK, but runs faster than these two baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' BVX is faster but has worse clustering cost than Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 24 Randomized Greedy Algorithms for k-Center Clustering with Outliers 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 ϕ ϵ (X, ϵ) (×10 3 ) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 Alg 1 Alg 3 BVX 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='6 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 ϵ = 1 4 6 8 10 12 14 16 18 20 0 100 200 300 Running time (s) 4 6 8 10 12 14 16 18 20 0 50 100 150 4 6 8 10 12 14 16 18 20 0 50 100 150 k Figure 4: The performance of Algorithm 1 and Algorithm 3 on Covertype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 ϕ ϵ (X, ϵ) (×10 5 ) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 Alg 1 Alg 3 BVX 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='6 4 6 8 10 12 14 16 18 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 ϵ = 1 4 6 8 10 12 14 16 18 20 0 50 100 150 200 250 Running time (s) 4 6 8 10 12 14 16 18 20 0 50 100 150 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 k Figure 5: The performance of Algorithm 1 and Algorithm 3 on KDD Cup 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 25 Ding, Huang, Liu, Yu, and Wang 4 6 8 10 12 14 16 18 20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 ϕ ϵ (X, ϵ) ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 Alg 1 Alg 3 BVX 4 6 8 10 12 14 16 18 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='6 4 6 8 10 12 14 16 18 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 ϵ = 1 4 6 8 10 12 14 16 18 20 0 20 40 60 Running time (s) 4 6 8 10 12 14 16 18 20 0 10 20 30 40 4 6 8 10 12 14 16 18 20 0 10 20 30 40 k Figure 6: The performance of Algorithm 1 and Algorithm 3 on Poker Hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 4 8 12 16 20 10 20 30 ϕ ϵ (X, ϵ) Shuttle Alg 1 Alg 3 4 8 12 16 20 4 6 8 ×10 2 Covertype 4 8 12 16 20 1 2 3 ×10 2 KDD Cup 1999 4 8 12 16 20 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 Poker Hand 4 8 12 16 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='75 Running time (s) 4 8 12 16 20 0 5 10 4 8 12 16 20 0 2 4 6 4 8 12 16 20 0 1 k Figure 7: The comparison between Algorithm 1 and Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 26 Randomized Greedy Algorithms for k-Center Clustering with Outliers 2 3 4 5 2 4 ϕ ϵ (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' ϵ) ×10 2 Shuttle Alg 2 BVX CKM+ MK 2 3 4 5 3 4 5 ×10 3 Covertype 2 3 4 5 2 4 ×10 4 KDD Cup 1999 2 3 4 5 12 14 16 Poker Hand 2 3 4 5 0 5 10 Running time (s) ×10 2 3 4 5 0 2 4 6 ×10 2 2 3 4 5 0 2 4 ×10 2 2 3 4 5 0 2 4 6 ×10 2 2 3 4 5 0 1 2 Running time (s) 2 3 4 5 0 10 2 3 4 5 0 2 2 3 4 5 0 2 4 6 k Figure 8: The performance of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The third row removes CKM+ to have a more clear illustration on the running times of the other three algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 The Coreset Algorithms We compare Algorithm 5 with the coreset methods CPP and Uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We set the sizes of coreset to be {4%n, 8%n, 12%n, 16%n, 20%n} for these three methods, where n is the number of points (including the outliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We run the algorithm Cluster proposed by Malkomes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2015), which is a modification of CKM+, as the “host” algorithm on the obtained coresets constructed by Algorithm 5 and Uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' We let RTcoreset denote the coreset construction time, and let RTtotal denote the total running time (including the coreset construction time and the time for running the k-center with outliers algorithm on the coreset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' To study the advantage of coreset, we also compare with CKM+ and MK;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' we directly run these two algorithms on the whole data sets (without coreset) to compute the clustering results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The experimental results are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Note that we illustrate the clustering cost φ0(X, E) (not φǫ(X, E)) in the first row of Figure 9 (and also Figure 10 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='3), that is, we discard exactly z outliers rather than (1 + ǫ)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Uniform is always the fastest coreset method since it is only simple uniform sampling and does not need any construction procedure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' but its clustering cost is worse than Algorithm 5 and CPP for most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Both of Algorithm 5 and CPP achieve lower clustering cost than CKM+ and MK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Comparing with CPP, Algorithm 5 has lower clustering cost on Covertype and Poker Hand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Algorithm 5 also has lower RTcoreset and RTtotal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The experimental results suggest that Algorithm 5 27 Ding, Huang, Liu, Yu, and Wang 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='4 ϕ 0 (X, E) ×10 3 Shuttle Alg 5 UNIFORM CPP CKM+ MK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 ×10 3 Covertype 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0 5 10 ×10 4 KDD Cup 1999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 11 12 13 Poker Hand 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0 5 10 RT coreset (s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 ×10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0 2 4 ×10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 RT total (s) ×10 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 ×10 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 0 5 ×10 2 Size of coreset (ratio) Figure 9: The performance of the coreset method Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' can yield significant reduction on the running time (if setting the coreset size ≤ 12%) and achieve good clustering quality as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='3 The Distributed Algorithm We compare Algorithm 6 with CPP, MKC+, GLZ and LG with varying the number of sites s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For Algorithm 6, in Step 2 we run Algorithm 5 instead of Algorithm 4 since the doubling dimensions of the four data sets are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Similar with Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2, we also run the Cluster algorithm on the coresets constructed by Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For CPP, following the setting of Ceccarello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (2019), each site sends a coreset of size λ(k+z) to the central server with λ = 1, 2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' LG returns a (k, z)ǫ-center solution and we set ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='99 in the algorithm as suggested in their paper (Li and Guo, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The experimental results of clustering cost and communication cost on the four data sets are shown in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The communication cost is measured by the total number of floating numbers sent between the sites and the central server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' GLZ and LG have lower communication costs, but yield much higher clustering costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Algorithm 6 can achieve quite low clustering cost, but takes higher communication cost comparing with GLZ and LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Future Work Following our work, several interesting problems deserve to be studied in future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' For exam- ple, can the coreset construction time of Algorithm 4 be improved, like the fast net construc- 28 Randomized Greedy Algorithms for k-Center Clustering with Outliers Figure 10: The performance of Algorithm 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In the second row, we remove GLZ and LG (since they have much higher clustering costs than the others) and zoom in on the comparison of other algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' tion method proposed by Har-Peled and Mendel (2006) in doubling metrics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In theory, it is interesting to study other optimization problems involving outliers by using greedy strat- egy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Also, if we replace k-center clustering by k-center clustering with outliers, it may be possible to improve the robustness for the applications in deep learning (Coleman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2020), active learning (Sener and Savarese, 2018), and fairness (Kleindessner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Proof of Claim 4 Suppose H is an α-approximation of the instance (coreset) S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let Hopt be the set of k cluster centers yielding the optimal solution of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Then we have φ0(S, H) ≤ αφ0(S, Hopt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (20) φ0(S, H) ∈ (1 ± µ)φ0(X, H);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (21) φ0(S, Hopt) ∈ (1 ± µ)φ0(X, Hopt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (22) Combining the above inequalities, we directly have φ0(X, H) ≤ 1 1 − µφ0(S, H) ≤ α 1 − µφ0(S, Hopt) ≤ α(1 + µ) 1 − µ φ0(X, Hopt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' (23) So H is an α(1+µ) 1−µ -approximation of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 29 2 Je 8 Je Je 4 8 4 4 8 4 8 Je 0 0 FO 丁 2 S 3 TO- XT0e X102 XTOe XJ02 Je S 4 8 Je s 4 8 Te 4 8 Je 4 8 102 T 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='2 Fo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='S H T H $o(x H T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content="a JJ'O." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=" m 主 s's- 8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e 0 X103 XTOs ×J03 Je S 8 S 4 4 8 e 4 8 Je 4 8 Je fo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='S 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content="1 王 n rc(e = 0'aa) rC(E= 0'J) JJ crs 5'2 J'S2 WKC 2 Cbb (y = 4) 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='5 JS Cbb (y = S) 02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='1 Cbb (y = J) (ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 =u) a DlA -× 1O- (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='0 = u) a plA J3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' J"12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' ×J03 ×J03 ×JO+ KDD Cnb Jaaa 2nffI6 bokGl HguqDing, Huang, Liu, Yu, and Wang Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Proof of Claim 18 We just need to prove the first inequality since the other one can be obtained by the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Because each Bj ⊆ Ball(cj, rX) and each vertex p is moved by a distance at most µropt based on (12), we know that f(Bj) ⊆ Ball(cj, rX + µropt), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=', r′ E ≤ rX + µropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Let p0 be the vertex realizing rX = φ0(X, H), that is, there exists some 1 ≤ j0 ≤ k such that d(cj0, p0) = rX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' The triangle inequality and (12) together imply d(cj0, f(p0)) ≥ rX − µropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Hence r′ E ≥ rX − µropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' Overall, we have |r′ E − rX| ≤ µropt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' References P.' metadata={'source': 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+page_content=' Bypassing the embedding: algorithms for low dimensional metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' In Proceed- ings of the thirty-sixth annual ACM symposium on Theory of computing, pages 281–290, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} +page_content=' 34' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/xdE0T4oBgHgl3EQf-gIr/content/2301.02814v1.pdf'} diff --git a/xtE3T4oBgHgl3EQfPAlS/content/2301.04398v1.pdf b/xtE3T4oBgHgl3EQfPAlS/content/2301.04398v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..45312565bcd02066d41b3c1152c62fe644d436ad --- /dev/null +++ b/xtE3T4oBgHgl3EQfPAlS/content/2301.04398v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:057da9dca1aae2a2687fba90785a2ccb30fd34c97b46fa6b56329fb97f57d4a9 +size 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∗ Alexander Fedorov,2, 3, 4, † Lingxiao Zhao,5, ‡ Alexander Yaresko,6 Bernd B¨uchner,2, 7 and Sergey Borisenko2, § +1School of Physical Science and Technology, Lanzhou University, Lanzhou 730000, China +2Leibniz Institute for Solid State and Materials Research, IFW Dresden, 01069 Dresden, Germany +3Helmholtz-Zentrum Berlin f¨ur Materialien und Energie, Albert-Einstein-Straße 15, 12489 Berlin, Germany +4Joint Laboratory “Functional Quantum Materials” at BESSY II, 12489 Berlin, Germany +5Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China +6Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany +7Institute for Solid State and Materials Physics, TU Dresden, 01062 Dresden, Germany +Magnetic topological materials are a class of compounds with the underlying interplay of nontrivial band +topology and magnetic spin configuration. Extensive interests have been aroused due to their application po- +tential involved with an array of exotic quantum states. With angle-resolved photoemission spectroscopy and +first-principles calculations, here we study the electronic properties of two magnetic Weyl semimetal candidates +PrAlSi and SmAlSi. Though the two compounds harbor distinct magnetic ground states (ferromagnetic and +antiferromagnetic for PrAlSi and SmAlSi, respectively) and 4f shell fillings, we find that they share quite anal- +ogous low-energy band structure. By the measurements across the magnetic transitions, we further reveal that +there is no evident evolution of the band structure in both compounds and the experimental spectra can be well +reproduced by the nonmagnetic calculations, together suggesting a negligible effect of the magnetism on their +electronic structures and a possibly weak coupling between the localized 4f electrons and the itinerant conduc- +tion electrons. Our results offer essential insights into the interactions between magnetism, electron correlations, +and topological orders in the RAlX (R = light rare earth and X = Si or Ge) family. +The last decade has witnessed the discovery of a rich va- +riety of novel quantum states in topological semimetals [1– +31], such as those exhibiting Dirac-fermion excitations near +the points of linear band crossings close to the Fermi level +(EF) [8, 9, 23, 26–28]. The breaking of either spatial inversion +symmetry or time-reversal symmetry splits the degeneracy of +the Dirac point, leading to a pair of topologically protected +Weyl points [2, 4, 5]. The nonmagnetic (NM) Weyl fermions +have been observed and intensively studied in TaAs family +[10–13, 25, 32], WTe2 [33, 34], MoTe2 [35–37], etc., while +the realization of magnetic Weyl semimetal state is still rare +[38–40]. Compared with the NM ones, the magnetic Weyl +semimetals offer a fertile playground for the interplay be- +tween magnetism, electron correlations, and nontrivial topol- +ogy, which could give rise to diverse exotic quantum phenom- +ena, like quantum anomalous Hall effect, Majorana fermions, +pair density wave, and topological axion states [41]. +It has recently been proposed that the RAlX (R = light rare +earth and X = Si or Ge) family with the non-centrosymmetric +LaPtSi-type structure would be an ideal candidate of mag- +netic Weyl semimetal, where various magnetic ground states +and Weyl semimetal states have been suggested by varying +the rare-earth ions [42–56]. Due to the formation of Weyl +fermions by the inversion symmetry breaking prior to the +magnetic transitions, the Weyl nodes are predicted to be ro- +bust and less dependent on the details of the magnetism, +which acts as a simple Zeeman coupling that shifts the Weyl +nodes in momentum space [55]. +Many intriguing proper- +ties have been observed among the RAlX family of com- +pounds, including the topological Hall effect in CeAlGe [47], +novel anisotropic anomalous Hall effect in CeAlSi [48], large +anomalous Hall conductivity and field-induced Lifshitz transi- +tion in PrAlSi [44, 45], Weyl fermions driven collective mag- +netism in NdAlSi [49], and topological spiral magnetic or- +der and non-saturated magnetoresistance in SmAlSi [46, 50]. +Despite these substantial findings, the experiments to directly +disclose the effect of f-electrons-induced magnetism and cor- +relations on the nontrivial band topology are still lacking. +In order to explore the underlying Weyl physics and gain in- +sights into its intricate interaction with the spin configuration +of 4f states, we present here combined angle-resolved pho- +toemission spectroscopy (ARPES) measurements and first- +principles calculations of PrAlSi and SmAlSi crystals. We +observe that their low-energy electronic structures are very +similar and less sensitive to the respective magnetic transi- +tion. +The band structure calculations in the paramagnetic +(PM) phase can well describe the ARPES spectra. Our ob- +servations indicate a possibly negligible coupling between the +well-localized 4 f states and the itinerant electronic states in +PrAlSi and SmAlSi. +High-quality single crystals of PrAlSi and SmAlSi were +synthesized using the flux method [45, 46]. ARPES measure- +ments were performed using the 13-ARPES end station of UE- +112-PGM2 beamline at Helmholtz Zentrum Berlin BESSY- +II light source. The energy and angular resolutions were set +to better than 5 meV and 0.1◦, respectively. Samples were +cleaved in situ, yielding flat mirrorlike (001) surfaces. During +the experiments, the sample temperature was kept at 1.5 K +if not specified otherwise, and the vacuum conditions were +maintained better than 8 × 10−11 Torr. +Density functional +based calculations were performed for the experimental crys- +tal structure of PrAlSi [44] using the PY linear muffin-tin +orbital (LMTO) computer code [57]. We used the Perdew- +Burke-Ernzerhof revised for solid parameterization of the ex- +arXiv:2301.13768v1 [cond-mat.mtrl-sci] 31 Jan 2023 + +2 +FIG. 1: Single crystals of PrAlSi and SmAlSi. (a) Schematic crystal structures of PrAlSi and SmAlSi. (b) Angle-integrated photoemission +spectra of PrAlSi with Pr N edge on-resonant (124 eV) and off-resonant (120 eV) photons, respectively. (c) Core-level photoemission spectra +of PrAlSi and SmAlSi recorded at hν = 200 eV, respectively. (d),(e) Temperature dependence of the resistivity ρxx of PrAlSi and SmAlSi at +µ0H = 0 T, respectively. Insets show the temperature ranges close to Tc and TN, respectively. (f) Sketches of 3D BZ and (001)-surface BZ for +the noncentrosymmetric I41/md space group structure. (g) Calculated bulk band structure along the high-symmetry lines in the PM phase of +PrAlSi including the SOC effect. (h) Constant-energy ARPES image of PrAlSi (hν = 100 eV, linear horizontal polarization, T = 2 K) obtained +by integrating the photoemission intensity within EF ± 40 meV. The red solid curve represents the (001)-projected BZ. a (= 4.22 Å) is the +in-plane lattice constant of PrAlSi. +change correlation potential [58]. Spin-orbit coupling (SOC) +was included in the LMTO Hamiltonian at the variational step. +Pr 4f 2 electrons were treated as semi-core states. +As shown in Fig. +1(a), the RAlX family crystallizes in +a tetragonal structure with the non-centrosymmetric space +group I41/md (No. 109), where each stacking layer along the +c axis consists of one type of element. Figure 1(b) displays +the resonant ARPES measurements of PrAlSi at N edge of Pr +element. At the Pr 4d → 4f resonant photon energy of 124 +eV, the resonant enhancement of Pr 4 f (EF to ∼–6 eV) and 5p +(∼–17 eV to ∼–20 eV) states are clearly observed compared to +the off-resonant spectra with 120-eV photons. The Al 2p and +Si 2p core-level states of PrAlSi and SmAlSi are presented +in Fig. 1(c). Their line shapes are composed of both main +peaks and shoulders, indicating the existence of several sites +and components of Al and Si atoms. As illustrated in Figs. +1(d) and 1(e), the underlying ferromagnetic (FM) and antifer- +romagnetic (AFM) transitions of PrAlSi and SmAlSi at Tc ∼ +18 K and TN ∼ 11 K, respectively, are validated by the zero- +field resistivity measurements, consistent with previous stud- +ies [44, 59]. The onset of magnetic orderings below Tc and TN +leads to the loss of spin-disorder scattering and the decrease +of the electrical resistivity, which manifests as an anomaly in +the ρxx-T curves. +Figure 1(f) presents the bulk and (001)-projected surface +Brillouin zones (BZs) of the RAlX family. Along the high- + +(a) +(b) +(d) +100 +Pr +Pr +oH= O T +4d -→4f +Intensity (a.u.) +80 +(μQ cm) +I ll ab +On-resonant +Off-resonant × 2.5 +60 +Tc~ 18 K +(uo n) +Pr/Sm +xxd +32 +40 +10 +20 +30 +20 +-20.0 +-15.0 +-10.0 +-5.0 +0.0 +0 +50 +100 +150 +200 +250 +Al +C +E- E-(eV) +T (K) +Si +(c) +(e) +200 eV +Sm +(f) +MoH= O T +(n'e) +09 +Pxx (μQ cm) +I ll ab +Al 2p +Si 2p +(001) +Pr +Intensity +V +40 +(o n) ×xd +IM(Z +16 +20- +Sm +N +M +8 +0 +10 +20 +-104.0 -100.0 +-96.0 +76.0 +-72.0 +-68.0 +0 +50 +150 +200 +250 +100 +E- E(eV) +T (K) +(g) +1.2 +(h) +with so +2.0 - +Pr +Pr +2 K +0.6 +1.0- +(eV) +(e /) +0.0- +0.0 +ky +-1.0- +-0.6 +T>Tc +-2.0 +100eVLH +-1.2 +MNM_ +-1.0 +0.0 +M' +M(Z) +X +1.0 +kx (π/a)3 +FIG. 2: Electronic structures of PrAlSi and SmAlSi in the PM states. +(a),(b) ARPES intensity plots at EF of PrAlSi (hν = 40 eV, CR+ po- +larization, T = 25 K) and SmAlSi (hν = 40 eV, CR− polarization, T = +20 K), respectively. The red solid curves are the (001)-surface BZs. +The in-plane lattice constant of SmAlSi (= 4.16 Å) is also denoted +as a. (c),(d) Intensity plots of PrAlSi (hν = 40 eV, CR+ polariza- +tion) recorded along the ¯Γ- ¯M and ¯Γ- ¯X directions, respectively. (e),(f) +Same as (c),(d) of SmAlSi (hν = 40 eV, CR− polarization). (g),(h) +Bulk band calculations integrated over kz from 0 to π in the PM phase +of PrAlSi taken along the ¯Γ- ¯M and ¯Γ- ¯X directions, respectively. +symmetry lines marked out in Fig. 1(f) (red curves), we plot +the overall bulk band structure calculations in the PM state +of PrAlSi with the SOC effect in Fig. 1(g). The semimetal- +lic ground state is demonstrated by the crossing of conduction +and valance bands along the Γ-Σ-N-Σ1 path. As shown in Fig. +1(h), the experimental constant-energy map at EF of PrAlSi +recorded by 100-eV photons, which covers also some parts +of the second BZ, can confirm the tetragonal crystalline sym- +metry with correct in-plane lattice parameter. The measured +Fermi surface (FS) consists of the two square-like pockets +around ¯Γ, the dumbbell-like pockets around ¯X, and the ripple- +shaped FS contours across the BZ boundaries. +In order to uncover the detailed electronic properties of +the Weyl semimetal phases in PrAlSi and SmAlSi, we carry +out high-resolution ARPES measurements in their PM states +at first, where the Weyl physics is determined by the in- +version symmetry breaking as suggested by earlier studies +[42, 55, 60]. As shown in Figs. 2(a) and 2(b), apart from +the small differences in the size of certain pockets, the anal- +ogous FS topologies are shared between PrAlSi and SmAlSi +in the first BZs (see Fig. S1 of Supplemental Material [61] +for their similar FSs under the same photon polarization). In +comparison to that in Fig. 1(h), one can clearly observe an ad- +ditional FS sheet (indicated by red arrows) close to the corner +of the outer pocket around ¯Γ in both PrAlSi and SmAlSi. This +contour has also been previously reported in LaAlGe [42], +PrAlGe [43], and CeAlSi [60], where it is proposed as the un- +closed “Fermi arc” connecting the topological Weyl fermions. +The “arc”-like feature is further suggested to have distinct +winding types between the FM and NM phases of PrAlGe +[43], with the asymmetric and symmetric forms across the +¯Γ- ¯M line, respectively. Nevertheless, our measurements of +the NM PrAlSi [Fig. 2(a)] reveal an evident asymmetry of +the “arc”, similar to that in the NM phase of CeAlSi, which +also hosts the FM ground state [60]. These results may raise +concern about the previous assignment of the Fermi arcs in +experiments and will be discussed in more detail later. +In Figs. 2(c) and 2(d) [Figs. 2(e) and 2(f)], we present the +near-EF band structure in the PM phase of PrAlSi (SmAlSi) +recorded along the ¯Γ- ¯M and ¯Γ- ¯X directions, respectively. It +can be seen that the outer faint band (β) around ¯Γ and the +neighbouring hole band (red arrow), which defines the “arc”- +like FS, are clearly separated from each other along the ¯Γ- ¯M +direction [Figs. 2(c) and 2(e)]. As shown in Figs. 2(d) and +2(f), due to the effect of ARPES matrix element, the Dirac- +like hole band (γ) and multiple electron bands (δ) are observ- +able at counter ¯X points of the BZ respectively. On switching +the photon polarization from left-handed circular (CR+) [Fig. +2(d)] to right-handed circular (CR−) [Fig. 2(f)], the obser- +vations are exchanged accordingly. Since the kz component +is not strictly conserved in ARPES measurements, and thus +cause the kz broadening effect, which is found to be significant +in the vacuum ultraviolet (VUV) regime, the ARPES spectra +reflect the electronic states integrated over a certain kz region +of the bulk BZ [21, 62]. Therefore, we perform the bulk band +calculations by considering the kz integration from 0 to π. The +corresponding results in the PM phase of PrAlSi are shown in +Figs. 2(g) and 2(h). The simulations can well reproduce most +of the experimental band dispersions including the “arc”-like +feature along the ¯Γ- ¯M direction, where the β band and the +“arc”-related band could be degenerate near EF in the calcula- +tions [Fig. 2(g)]. The good consistency between experiments +and theory suggests the bulk origin of these ARPES spectra +and the presence of intrinsic kz projections. The intense elec- +tron band (α) at ¯Γ is not captured in the bulk calculations and +thus could be the surface state, consistent with the previous +observation and assignment in LaAlGe [42], PrAlGe [43], and + +(a) +(b) +1.0- +P +1.0 +Sm +25 K +20 K +X +0.0 +0.0 +X +ky +"arc" +"arc" +M +M +-1.0 +-1.0. +40 eV,CR+ +40 eV, ER- +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +kx (π/a) +kx (π/a) +(c) +(d) +x +r +X +M +M +0.0 +0.0 +(eV) +(eV) +"arc' +出 +-0.2 +-0.2 +E +Pr +25 K +Pr +25 K +-0.4 - +-0.4 +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +kil (π/a) +k(元/a) +(f) +(e) +x +I +x +M +M +0.0 +0.0 +(eV) +(eV) +α +α +"arc" +-0.2 +-0.2 +E +E +20 K +Sm +20 K +Sm +-0.4 +-0.4 +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +k (π/a) +k (元/a) +(g) +(h) +x +x +M +M +0.0 +0.0 +(eV) +(eV) +"arc +-0.4 +-0.4 +-0.8 +-0.8 +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +H +k(π/a) +k (π/a)4 +FIG. 3: Electronic structures of PrAlSi and SmAlSi in the magnetic states. (a),(b) Constant-energy maps at EF of PrAlSi (hν = 40 eV, CR+ +polarization, T = 1.5 K) and SmAlSi (hν = 40 eV, CR− polarization, T = 1.5 K), respectively. (c) Comparison of the MDCs along the ¯Γ- ¯M +direction above and below Tc of PrAlSi. (d) Same as (c) of SmAlSi by going across TN. (e) Kz-integrated FS mapping from 0 to π in the PM +state of PrAlSi by bulk calculations. (f) Same as (e) for kz ∼ π/2 [T-N-P plane in Fig. 1(f)] without kz integration. The red solid curves in +(a),(b) and (e),(f) show the (001)-projected BZs. Cuts #a and #b in (a) and (e) indicate the locations of the band dispersions in (g) (also Fig. S2 +[61]) and (h), respectively. (g) Intensity plot of PrAlSi (hν = 40 eV, CR+ polarization) measured along cut #a in (a). (h) Calculated bulk band +structure of the NM PrAlSi integrated over kz from 0 to π along cut #b in (e). +CeAlSi [60]. +To investigate the influence of FM and AFM orderings on +the electronic structures of PrAlSi and SmAlSi, we now con- +duct ARPES measurements deep within their magnetic states +respectively, as shown in Fig. 3. Figures 3(a) and 3(b) il- +lustrate the corresponding FS mappings at T = 1.5 K with +the same experimental setups as in Figs. 2(a) and 2(b), re- +spectively. No noticeable change is observed across Tc and +TN, especially the “arc”-like FSs, where the likely asymme- +try of PrAlSi and symmetry of SmAlSi across the ¯Γ- ¯M line +still persist. We further focus on the temperature evolution +of the “arc”-like feature by plotting the momentum distribu- +tion curves (MDCs) along the ¯Γ- ¯M direction in Figs. 3(c) and +3(d). It can be clearly seen that the MDC peak positions above +and below Tc and TN coincide with each other in the studied +energy range, showing the temperature independence of the +ARPES spectra. These observations unambiguously demon- +strate a negligible effect of the long-range magnetic orders on +the conduction electrons in PrAlSi and SmAlSi. +Now we discuss the possible origin of the “unclosed” FS +sheet which is early assigned as the topological Fermi arc +[43]. Based on the present results that this “arc”-like struc- +ture can be well described by the kz-integrated bulk band cal- +culations, we further simulate the bulk FS topologies of the +(001) surface in the full 3D BZ of the PM PrAlSi. As the +good agreement illustrated in Fig. 2, the theoretical FS inte- +grated over kz from 0 to π in Fig. 3(e) reproduces the outer +square-like pocket around ¯Γ, the ripple-shaped FS across the +BZ boundary, and particularly the “arc”-like FS contour. The +dumbbell-like pockets around ¯X can be seen in the calculated +FS for kz = π/2 [T-N-P plane in Fig. 1(f)] without kz inte- +gration in Fig. 3(f). The surface state nature of the inner +pocket at ¯Γ can be further confirmed by its absence in Figs. +3(e) and 3(f). +To solely study the “arc”-related hole band +without any possible interference from the β band, we record +the ARPES spectra along cut #a [green line in Fig. 3(a)], as +shown in Fig. 3(g). By comparing with the bulk calculations +fully integrated over kz taken along cut #b in Fig. 3(e) [Fig. +3(h)], one can obtain a high consistency again. Although our +photon-energy-dependent measurements find that the “arc”- +like feature shows similar ∆kF ∼ 0.4 Å−1 from 30 to 60 eV +(see Fig. S2 of Supplemental Material [61]) as reported in +earlier studies [43, 60], due to the concern that the observed +“arcs” may be actually the bulk states, it is inferred that the +ARPES intensity suffers from the large kz-broadening effect. +This is judged from not only the good agreement between ex- +periments and bulk calculations, but also the observation of +similar band structure in a wide VUV-photon energy range, +as seen from Figs. 1(h) (hν = 100 eV), 3(a)-3(b) (hν = 40 +eV), and S3 (hν = 50 eV) [61]. We suspect that the theoret- +ically predicted topological Fermi arc is likely hidden by the +“arc”-like feature revealed here. Future investigations to dif- + +(a) +(b) +(c) +(d) +1.0- +1.0- +Pr +Sm +Pr +1.5 K +Sm +1.5 K +"arc" +"arc +23 +#a +Intensity (a.u.) +25 K +(e /μ) +0.0. +0.0 +20 K +ky +1.5 K +"arc" +'arc" +M +M +-1.0- +-1.0- +40eV,CR+ +M +-0.4 eV +V +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +kx (元/a) +kx (π/a) +k (π/a) +k (π/a) +(e) +(f) +(g) +(h) +kz integ. +kz～元/2 +"arc" +"arc" +e# +#b +1.0- +1.0 +Y +0.0 +0.0. +人 +#b +(eV) +(e/) +r +0.0 +-0.2 +X +0.0- +-0.4 - +X +E +-0.4 +M +M +-0.8- +-1.0 +1.5 K +-1.0 +Pr +-0.5 +0.0 +0.5 +-0.5 +0.0 +0.5 +-1.0 +0.0 +1.0 +-1.0 +0.0 +1.0 +kx(π/a) +L +kx (π/a) +k (π/a) +k (π/a)5 +ferentiate between the surface and bulk contributions will be +required. +Last but not least, we now turn to the potential implica- +tions of the temperature-independent electronic structures in +PrAlSi and SmAlSi. +It has been previously proposed that +the occupied f-electron states in the RAlX family (exclud- +ing the La-based compounds) are spin polarized and the mag- +netic interactions between the local moments of f electrons +can lead to the long-range magnetic ordering, which, in turn, +serves as an effective Zeeman field to make the near-EF con- +duction bands also spin polarized [46, 55]. Whereas, our re- +sults uncover a minor impact of the magnetic orders on the +low-energy electronic states of PrAlSi and SmAlSi. This de- +coupling relationship implies that the 4 f electrons in these +two compounds could be well localized and their interactions +with the itinerant conduction electrons are likely negligible or +in the weak-coupling regime. This weak hybridization phe- +nomenon could be further revealed by the resonant ARPES +measurements. As shown in Fig. S4 [61], no resonant en- +hancement is observed for the conduction states near EF, the +overall intensities of photoemission signals measured under +on-resonant and off-resonant photons are comparable to each +other. In a recent ARPES study of the AFM Kondo lattice +CeRh2Si2, the surface- and bulk-type Ce-4 f electrons have +been demonstrated to show distinct hybridization phenom- +ena with the itinerant electrons, representing the weakly and +strongly hybridized 4f states, respectively [63]. Possibly sim- +ilar to CeRh2Si2, the well-localized 4 f states uncovered here +may originate from the surface-type Pr/Sm ions, where the ef- +fect of bulk magnetism is negligible in the surface-sensitive +VUV-ARPES spectra. Further bulk-sensitive photoemission +studies will be required. +On the other side, it has been recently reported that the mag- +netic properties of these two compounds could be more com- +plicated [44, 50]. In PrAlSi, the reentrant spin glassy phases +are suggested to emerge just below Tc [44]; in SmAlSi, the +Weyl-mediated Ruderman-Kittel-Kasuya-Yosida interactions +are proposed to induce a spiral magnetic order in the ground +state [50]. These random and/or competing exchange inter- +actions may cause the instability of the long-range magnetic +orders, leading to another challenge of observing the magnetic +impact on the electronic states as well. +In summary, we have systematically studied the electronic +structures in the NM and magnetic phases of PrAlSi and +SmAlSi. The low-energy band structure has almost no change +when the respective long-range FM and AFM order develops +in them and the measured spectra show good agreement with +the NM band calculations. These facts can be attributed to +the possibly negligible coupling between the well-localized +4 f electrons and the itinerant conduction electrons. Our ob- +servations shed light on the further research of the interplay +between magnetism, correlations, and topology, which will +facilitate realizing more topological quantum phenomena in +the RAlX family. +We thank Denis Vyalikh for helpful discussions. This work +was supported by the National Natural Science Foundation +of China (Grants No. +11904144 and No. +12004123) and +the Deutsche Forschungsgemeinschaft under Grant SFB 1143 +(project C04). B. B. and S. B. acknowledge the support from +BMBF via project UKRATOP. R. L., A. F., B. B., and S. B. +acknowledge the support from W¨urzburg-Dresden Cluster of +Excellence on Complexity and Topology in Quantum Matter- +ct.qmat (EXC 2147, project-id 390858490). +∗ lourui@lzu.edu.cn +† a.fedorov@ifw-dresden.de +‡ zhaolx@mail.sustech.edu.cn +§ s.borisenko@ifw-dresden.de +[1] H. M. Weng, X. Dai, and Z. Fang, J. Phys. Condens. Matter 28, +303001 (2016). +[2] X. G. Wan, A. M. Turner, A. Vishwanath, and S. Y. Savrasov, +Phys. Rev. B 83, 205101 (2011). +[3] Z. Wang, Y. Sun, X.-Q. Chen, C. Franchini, G. Xu, H. M. Weng, +X. Dai, and Z. Fang, Phys. Rev. B 85, 195320 (2012). +[4] H. M. Weng, C. Fang, Z. Fang, B. A. Bernevig, and X. Dai, +Phys. Rev. X 5, 011029 (2015). +[5] S.-M. Huang, S. Xu, I. 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Commun. 7, +11029 (2016). + diff --git a/yNFST4oBgHgl3EQfTDh5/content/tmp_files/load_file.txt b/yNFST4oBgHgl3EQfTDh5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7807cc492296b03f1b745852b676a3964f00e766 --- /dev/null +++ b/yNFST4oBgHgl3EQfTDh5/content/tmp_files/load_file.txt @@ -0,0 +1,1511 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf,len=1510 +page_content='Signature of weakly coupled f electrons and conduction electrons in magnetic Weyl semimetal candidates PrAlSi and SmAlSi Rui Lou,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' ∗ Alexander Fedorov,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' † Lingxiao Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' ‡ Alexander Yaresko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='6 Bernd B¨uchner,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 7 and Sergey Borisenko2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' § 1School of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lanzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lanzhou 730000,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' China 2Leibniz Institute for Solid State and Materials Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' IFW Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 01069 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Germany 3Helmholtz-Zentrum Berlin f¨ur Materialien und Energie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Albert-Einstein-Straße 15,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 12489 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Germany 4Joint Laboratory “Functional Quantum Materials” at BESSY II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 12489 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Germany 5Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Southern University of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Shenzhen 518055,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' China 6Max Planck Institute for Solid State Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 70569 Stuttgart,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Germany 7Institute for Solid State and Materials Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' TU Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 01062 Dresden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Germany Magnetic topological materials are a class of compounds with the underlying interplay of nontrivial band topology and magnetic spin configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Extensive interests have been aroused due to their application po- tential involved with an array of exotic quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' With angle-resolved photoemission spectroscopy and first-principles calculations, here we study the electronic properties of two magnetic Weyl semimetal candidates PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Though the two compounds harbor distinct magnetic ground states (ferromagnetic and antiferromagnetic for PrAlSi and SmAlSi, respectively) and 4f shell fillings, we find that they share quite anal- ogous low-energy band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' By the measurements across the magnetic transitions, we further reveal that there is no evident evolution of the band structure in both compounds and the experimental spectra can be well reproduced by the nonmagnetic calculations, together suggesting a negligible effect of the magnetism on their electronic structures and a possibly weak coupling between the localized 4f electrons and the itinerant conduc- tion electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Our results offer essential insights into the interactions between magnetism, electron correlations, and topological orders in the RAlX (R = light rare earth and X = Si or Ge) family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The last decade has witnessed the discovery of a rich va- riety of novel quantum states in topological semimetals [1– 31], such as those exhibiting Dirac-fermion excitations near the points of linear band crossings close to the Fermi level (EF) [8, 9, 23, 26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The breaking of either spatial inversion symmetry or time-reversal symmetry splits the degeneracy of the Dirac point, leading to a pair of topologically protected Weyl points [2, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The nonmagnetic (NM) Weyl fermions have been observed and intensively studied in TaAs family [10–13, 25, 32], WTe2 [33, 34], MoTe2 [35–37], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=', while the realization of magnetic Weyl semimetal state is still rare [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Compared with the NM ones, the magnetic Weyl semimetals offer a fertile playground for the interplay be- tween magnetism, electron correlations, and nontrivial topol- ogy, which could give rise to diverse exotic quantum phenom- ena, like quantum anomalous Hall effect, Majorana fermions, pair density wave, and topological axion states [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' It has recently been proposed that the RAlX (R = light rare earth and X = Si or Ge) family with the non-centrosymmetric LaPtSi-type structure would be an ideal candidate of mag- netic Weyl semimetal, where various magnetic ground states and Weyl semimetal states have been suggested by varying the rare-earth ions [42–56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Due to the formation of Weyl fermions by the inversion symmetry breaking prior to the magnetic transitions, the Weyl nodes are predicted to be ro- bust and less dependent on the details of the magnetism, which acts as a simple Zeeman coupling that shifts the Weyl nodes in momentum space [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Many intriguing proper- ties have been observed among the RAlX family of com- pounds, including the topological Hall effect in CeAlGe [47], novel anisotropic anomalous Hall effect in CeAlSi [48], large anomalous Hall conductivity and field-induced Lifshitz transi- tion in PrAlSi [44, 45], Weyl fermions driven collective mag- netism in NdAlSi [49], and topological spiral magnetic or- der and non-saturated magnetoresistance in SmAlSi [46, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Despite these substantial findings, the experiments to directly disclose the effect of f-electrons-induced magnetism and cor- relations on the nontrivial band topology are still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' In order to explore the underlying Weyl physics and gain in- sights into its intricate interaction with the spin configuration of 4f states, we present here combined angle-resolved pho- toemission spectroscopy (ARPES) measurements and first- principles calculations of PrAlSi and SmAlSi crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' We observe that their low-energy electronic structures are very similar and less sensitive to the respective magnetic transi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The band structure calculations in the paramagnetic (PM) phase can well describe the ARPES spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Our ob- servations indicate a possibly negligible coupling between the well-localized 4 f states and the itinerant electronic states in PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' High-quality single crystals of PrAlSi and SmAlSi were synthesized using the flux method [45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' ARPES measure- ments were performed using the 13-ARPES end station of UE- 112-PGM2 beamline at Helmholtz Zentrum Berlin BESSY- II light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The energy and angular resolutions were set to better than 5 meV and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='1◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Samples were cleaved in situ, yielding flat mirrorlike (001) surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' During the experiments, the sample temperature was kept at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 K if not specified otherwise, and the vacuum conditions were maintained better than 8 × 10−11 Torr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Density functional based calculations were performed for the experimental crys- tal structure of PrAlSi [44] using the PY linear muffin-tin orbital (LMTO) computer code [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' We used the Perdew- Burke-Ernzerhof revised for solid parameterization of the ex- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='13768v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='mtrl-sci] 31 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1: Single crystals of PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (a) Schematic crystal structures of PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (b) Angle-integrated photoemission spectra of PrAlSi with Pr N edge on-resonant (124 eV) and off-resonant (120 eV) photons, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (c) Core-level photoemission spectra of PrAlSi and SmAlSi recorded at hν = 200 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (d),(e) Temperature dependence of the resistivity ρxx of PrAlSi and SmAlSi at µ0H = 0 T, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Insets show the temperature ranges close to Tc and TN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (f) Sketches of 3D BZ and (001)-surface BZ for the noncentrosymmetric I41/md space group structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (g) Calculated bulk band structure along the high-symmetry lines in the PM phase of PrAlSi including the SOC effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (h) Constant-energy ARPES image of PrAlSi (hν = 100 eV, linear horizontal polarization, T = 2 K) obtained by integrating the photoemission intensity within EF ± 40 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The red solid curve represents the (001)-projected BZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' a (= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='22 Å) is the in-plane lattice constant of PrAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' change correlation potential [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Spin-orbit coupling (SOC) was included in the LMTO Hamiltonian at the variational step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Pr 4f 2 electrons were treated as semi-core states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(a), the RAlX family crystallizes in a tetragonal structure with the non-centrosymmetric space group I41/md (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 109), where each stacking layer along the c axis consists of one type of element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Figure 1(b) displays the resonant ARPES measurements of PrAlSi at N edge of Pr element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' At the Pr 4d → 4f resonant photon energy of 124 eV, the resonant enhancement of Pr 4 f (EF to ∼–6 eV) and 5p (∼–17 eV to ∼–20 eV) states are clearly observed compared to the off-resonant spectra with 120-eV photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The Al 2p and Si 2p core-level states of PrAlSi and SmAlSi are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Their line shapes are composed of both main peaks and shoulders, indicating the existence of several sites and components of Al and Si atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' As illustrated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(d) and 1(e), the underlying ferromagnetic (FM) and antifer- romagnetic (AFM) transitions of PrAlSi and SmAlSi at Tc ∼ 18 K and TN ∼ 11 K, respectively, are validated by the zero- field resistivity measurements, consistent with previous stud- ies [44, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The onset of magnetic orderings below Tc and TN leads to the loss of spin-disorder scattering and the decrease of the electrical resistivity, which manifests as an anomaly in the ρxx-T curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Figure 1(f) presents the bulk and (001)-projected surface Brillouin zones (BZs) of the RAlX family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Along the high- (a) (b) (d) 100 Pr Pr oH= O T 4d -→4f Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=') 80 (μQ cm) I ll ab On-resonant Off-resonant × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 60 Tc~ 18 K (uo n) Pr/Sm xxd 32 40 10 20 30 20 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content="0 0 50 100 150 200 250 Al C E- E-(eV) T (K) Si (c) (e) 200 eV Sm (f) MoH= O T (n'e) 09 Pxx (μQ cm) I ll ab Al 2p Si 2p (001) Pr Intensity V 40 (o n) ×xd IM(Z 16 20- Sm N M 8 0 10 20 104." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 -100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0 50 150 200 250 100 E- E(eV) T (K) (g) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2 (h) with so 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 - Pr Pr 2 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- (eV) (e /) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 ky 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='6 T>Tc 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 100eVLH 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2 MNM_ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content="0 M' M(Z) X 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 kx (π/a)3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2: Electronic structures of PrAlSi and SmAlSi in the PM states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (a),(b) ARPES intensity plots at EF of PrAlSi (hν = 40 eV, CR+ po- larization, T = 25 K) and SmAlSi (hν = 40 eV, CR− polarization, T = 20 K), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The red solid curves are the (001)-surface BZs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The in-plane lattice constant of SmAlSi (= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='16 Å) is also denoted as a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (c),(d) Intensity plots of PrAlSi (hν = 40 eV, CR+ polariza- tion) recorded along the ¯Γ- ¯M and ¯Γ- ¯X directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (e),(f) Same as (c),(d) of SmAlSi (hν = 40 eV, CR− polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (g),(h) Bulk band calculations integrated over kz from 0 to π in the PM phase of PrAlSi taken along the ¯Γ- ¯M and ¯Γ- ¯X directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' symmetry lines marked out in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(f) (red curves), we plot the overall bulk band structure calculations in the PM state of PrAlSi with the SOC effect in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The semimetal- lic ground state is demonstrated by the crossing of conduction and valance bands along the Γ-Σ-N-Σ1 path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(h), the experimental constant-energy map at EF of PrAlSi recorded by 100-eV photons, which covers also some parts of the second BZ, can confirm the tetragonal crystalline sym- metry with correct in-plane lattice parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The measured Fermi surface (FS) consists of the two square-like pockets around ¯Γ, the dumbbell-like pockets around ¯X, and the ripple- shaped FS contours across the BZ boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' In order to uncover the detailed electronic properties of the Weyl semimetal phases in PrAlSi and SmAlSi, we carry out high-resolution ARPES measurements in their PM states at first, where the Weyl physics is determined by the in- version symmetry breaking as suggested by earlier studies [42, 55, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(a) and 2(b), apart from the small differences in the size of certain pockets, the anal- ogous FS topologies are shared between PrAlSi and SmAlSi in the first BZs (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' S1 of Supplemental Material [61] for their similar FSs under the same photon polarization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' In comparison to that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(h), one can clearly observe an ad- ditional FS sheet (indicated by red arrows) close to the corner of the outer pocket around ¯Γ in both PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' This contour has also been previously reported in LaAlGe [42], PrAlGe [43], and CeAlSi [60], where it is proposed as the un- closed “Fermi arc” connecting the topological Weyl fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The “arc”-like feature is further suggested to have distinct winding types between the FM and NM phases of PrAlGe [43], with the asymmetric and symmetric forms across the ¯Γ- ¯M line, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Nevertheless, our measurements of the NM PrAlSi [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(a)] reveal an evident asymmetry of the “arc”, similar to that in the NM phase of CeAlSi, which also hosts the FM ground state [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' These results may raise concern about the previous assignment of the Fermi arcs in experiments and will be discussed in more detail later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(c) and 2(d) [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(e) and 2(f)], we present the near-EF band structure in the PM phase of PrAlSi (SmAlSi) recorded along the ¯Γ- ¯M and ¯Γ- ¯X directions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' It can be seen that the outer faint band (β) around ¯Γ and the neighbouring hole band (red arrow), which defines the “arc”- like FS, are clearly separated from each other along the ¯Γ- ¯M direction [Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(c) and 2(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' As shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(d) and 2(f), due to the effect of ARPES matrix element, the Dirac- like hole band (γ) and multiple electron bands (δ) are observ- able at counter ¯X points of the BZ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' On switching the photon polarization from left-handed circular (CR+) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(d)] to right-handed circular (CR−) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(f)], the obser- vations are exchanged accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Since the kz component is not strictly conserved in ARPES measurements, and thus cause the kz broadening effect, which is found to be significant in the vacuum ultraviolet (VUV) regime, the ARPES spectra reflect the electronic states integrated over a certain kz region of the bulk BZ [21, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Therefore, we perform the bulk band calculations by considering the kz integration from 0 to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The corresponding results in the PM phase of PrAlSi are shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(g) and 2(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The simulations can well reproduce most of the experimental band dispersions including the “arc”-like feature along the ¯Γ- ¯M direction, where the β band and the “arc”-related band could be degenerate near EF in the calcula- tions [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(g)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The good consistency between experiments and theory suggests the bulk origin of these ARPES spectra and the presence of intrinsic kz projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The intense elec- tron band (α) at ¯Γ is not captured in the bulk calculations and thus could be the surface state, consistent with the previous observation and assignment in LaAlGe [42], PrAlGe [43], and (a) (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 Sm 25 K 20 K X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 X ky "arc" "arc" M M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 40 eV,CR+ 40 eV, ER- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 kx (π/a) kx (π/a) (c) (d) x r X M M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 (eV) (eV) "arc\' 出 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2 E Pr 25 K Pr 25 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 kil (π/a) k(元/a) (f) (e) x I x M M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 (eV) (eV) α α "arc" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2 E E 20 K Sm 20 K Sm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 k (π/a) k (元/a) (g) (h) x x M M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 (eV) (eV) "arc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 H k(π/a) k (π/a)4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3: Electronic structures of PrAlSi and SmAlSi in the magnetic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (a),(b) Constant-energy maps at EF of PrAlSi (hν = 40 eV, CR+ polarization, T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 K) and SmAlSi (hν = 40 eV, CR− polarization, T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 K), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (c) Comparison of the MDCs along the ¯Γ- ¯M direction above and below Tc of PrAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (d) Same as (c) of SmAlSi by going across TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (e) Kz-integrated FS mapping from 0 to π in the PM state of PrAlSi by bulk calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (f) Same as (e) for kz ∼ π/2 [T-N-P plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(f)] without kz integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The red solid curves in (a),(b) and (e),(f) show the (001)-projected BZs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Cuts #a and #b in (a) and (e) indicate the locations of the band dispersions in (g) (also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' S2 [61]) and (h), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (g) Intensity plot of PrAlSi (hν = 40 eV, CR+ polarization) measured along cut #a in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' (h) Calculated bulk band structure of the NM PrAlSi integrated over kz from 0 to π along cut #b in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' CeAlSi [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' To investigate the influence of FM and AFM orderings on the electronic structures of PrAlSi and SmAlSi, we now con- duct ARPES measurements deep within their magnetic states respectively, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Figures 3(a) and 3(b) il- lustrate the corresponding FS mappings at T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 K with the same experimental setups as in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2(a) and 2(b), re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' No noticeable change is observed across Tc and TN, especially the “arc”-like FSs, where the likely asymme- try of PrAlSi and symmetry of SmAlSi across the ¯Γ- ¯M line still persist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' We further focus on the temperature evolution of the “arc”-like feature by plotting the momentum distribu- tion curves (MDCs) along the ¯Γ- ¯M direction in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(c) and 3(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' It can be clearly seen that the MDC peak positions above and below Tc and TN coincide with each other in the studied energy range, showing the temperature independence of the ARPES spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' These observations unambiguously demon- strate a negligible effect of the long-range magnetic orders on the conduction electrons in PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Now we discuss the possible origin of the “unclosed” FS sheet which is early assigned as the topological Fermi arc [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Based on the present results that this “arc”-like struc- ture can be well described by the kz-integrated bulk band cal- culations, we further simulate the bulk FS topologies of the (001) surface in the full 3D BZ of the PM PrAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' As the good agreement illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2, the theoretical FS inte- grated over kz from 0 to π in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(e) reproduces the outer square-like pocket around ¯Γ, the ripple-shaped FS across the BZ boundary, and particularly the “arc”-like FS contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The dumbbell-like pockets around ¯X can be seen in the calculated FS for kz = π/2 [T-N-P plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(f)] without kz inte- gration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The surface state nature of the inner pocket at ¯Γ can be further confirmed by its absence in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(e) and 3(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' To solely study the “arc”-related hole band without any possible interference from the β band, we record the ARPES spectra along cut #a [green line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(a)], as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' By comparing with the bulk calculations fully integrated over kz taken along cut #b in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(e) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3(h)], one can obtain a high consistency again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Although our photon-energy-dependent measurements find that the “arc”- like feature shows similar ∆kF ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 Å−1 from 30 to 60 eV (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' S2 of Supplemental Material [61]) as reported in earlier studies [43, 60], due to the concern that the observed “arcs” may be actually the bulk states, it is inferred that the ARPES intensity suffers from the large kz-broadening effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' This is judged from not only the good agreement between ex- periments and bulk calculations, but also the observation of similar band structure in a wide VUV-photon energy range, as seen from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 1(h) (hν = 100 eV), 3(a)-3(b) (hν = 40 eV), and S3 (hν = 50 eV) [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' We suspect that the theoret- ically predicted topological Fermi arc is likely hidden by the “arc”-like feature revealed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Future investigations to dif- (a) (b) (c) (d) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- Pr Sm Pr 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 K Sm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 K "arc" "arc 23 #a Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=') 25 K (e /μ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 20 K ky 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='5 K "arc" \'arc" M M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- 40eV,CR+ M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 eV V 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 kx (元/a) kx (π/a) k (π/a) k (π/a) (e) (f) (g) (h) kz integ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' kz～元/2 "arc" "arc" e# #b 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 Y 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 人 #b (eV) (e/) r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='2 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 - X E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='4 M M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='8- 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='0 kx(π/a) L kx (π/a) k (π/a) k (π/a)5 ferentiate between the surface and bulk contributions will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Last but not least, we now turn to the potential implica- tions of the temperature-independent electronic structures in PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' It has been previously proposed that the occupied f-electron states in the RAlX family (exclud- ing the La-based compounds) are spin polarized and the mag- netic interactions between the local moments of f electrons can lead to the long-range magnetic ordering, which, in turn, serves as an effective Zeeman field to make the near-EF con- duction bands also spin polarized [46, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Whereas, our re- sults uncover a minor impact of the magnetic orders on the low-energy electronic states of PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' This de- coupling relationship implies that the 4 f electrons in these two compounds could be well localized and their interactions with the itinerant conduction electrons are likely negligible or in the weak-coupling regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' This weak hybridization phe- nomenon could be further revealed by the resonant ARPES measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' S4 [61], no resonant en- hancement is observed for the conduction states near EF, the overall intensities of photoemission signals measured under on-resonant and off-resonant photons are comparable to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' In a recent ARPES study of the AFM Kondo lattice CeRh2Si2, the surface- and bulk-type Ce-4 f electrons have been demonstrated to show distinct hybridization phenom- ena with the itinerant electrons, representing the weakly and strongly hybridized 4f states, respectively [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Possibly sim- ilar to CeRh2Si2, the well-localized 4 f states uncovered here may originate from the surface-type Pr/Sm ions, where the ef- fect of bulk magnetism is negligible in the surface-sensitive VUV-ARPES spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Further bulk-sensitive photoemission studies will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' On the other side, it has been recently reported that the mag- netic properties of these two compounds could be more com- plicated [44, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' In PrAlSi, the reentrant spin glassy phases are suggested to emerge just below Tc [44];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' in SmAlSi, the Weyl-mediated Ruderman-Kittel-Kasuya-Yosida interactions are proposed to induce a spiral magnetic order in the ground state [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' These random and/or competing exchange inter- actions may cause the instability of the long-range magnetic orders, leading to another challenge of observing the magnetic impact on the electronic states as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' In summary, we have systematically studied the electronic structures in the NM and magnetic phases of PrAlSi and SmAlSi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' The low-energy band structure has almost no change when the respective long-range FM and AFM order develops in them and the measured spectra show good agreement with the NM band calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' These facts can be attributed to the possibly negligible coupling between the well-localized 4 f electrons and the itinerant conduction electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Our ob- servations shed light on the further research of the interplay between magnetism, correlations, and topology, which will facilitate realizing more topological quantum phenomena in the RAlX family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' We thank Denis Vyalikh for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' This work was supported by the National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 11904144 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 12004123) and the Deutsche Forschungsgemeinschaft under Grant SFB 1143 (project C04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' acknowledge the support from BMBF via project UKRATOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=', A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=', B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=', and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' acknowledge the support from W¨urzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter- ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='qmat (EXC 2147, project-id 390858490).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' ∗ lourui@lzu.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='borisenko@ifw-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='de [1] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Weng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Dai, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Fang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Sun, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='-Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Franchini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Xu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Weng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Fang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Fang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Bernevig, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Dai, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' X 5, 011029 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Xu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Belopolski, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lee, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Alidoust, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Bian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Neupane, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Cava, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 113, 027603 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lv, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Weng, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Fu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Miao, J.' metadata={'source': 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Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Sankar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Yuan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Huang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Zheng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Jia, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Hasan, Science 349, 613 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [12] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Sun, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Peng, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Yang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Guo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Rahn, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Prabhakaran, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Hussain, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Mo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Felser, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Yan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chen, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 11, 728 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Cao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Ma, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Kong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Richard, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Shi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Qian, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Ding, Nature (London) 546, 627 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [16] Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wu, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 2, 014202 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [21] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lou, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Guo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Sun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Thirupatha- 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Quan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 3, 43 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [23] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Liu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Kushwaha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Sankar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Krizan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Belopolski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Neupane, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Bian, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Huang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chang, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Belopolski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Sanchez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Neupane, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Alidoust, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Jeng, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Yuan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} 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F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Matt, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Bisti, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Strokov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Mesot, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Fang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Dai, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Qian, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Shi, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Ding, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 11, 724 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Ricc`o, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Kim, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Hoesch, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Barreteau, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Giannini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Besnard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Soluyanov, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Baumberger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' X 6, 031021 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [36] J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Naamneh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Plumb, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Keller, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Cubitt, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Pomjakushina, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 117, 222410 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [52] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Yang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Singh, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Huang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Bahrami, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chiu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Graf, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Huang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Lin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Torchinsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Bansil, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' B 104, 014412 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' [55] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Chang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' Xu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFST4oBgHgl3EQfTDh5/content/2301.13768v1.pdf'} +page_content=' 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new file mode 100644 index 0000000000000000000000000000000000000000..ae391dba0ecd10b12abec811e099f9e699b7da70 --- /dev/null +++ b/ytFAT4oBgHgl3EQfBBwz/content/tmp_files/2301.08401v1.pdf.txt @@ -0,0 +1,1888 @@ +1 +On the Relationship Between +Information-Theoretic Privacy Metrics And +Probabilistic Information Privacy +Chong Xiao Wang and Wee Peng Tay, Senior Member, IEEE +Abstract +Information-theoretic (IT) measures based on f-divergences have recently gained interest as a mea- +sure of privacy leakage as they allow for trading off privacy against utility using only a single-value +characterization. However, their operational interpretations in the privacy context are unclear. In this +paper, we relate the notion of probabilistic information privacy (IP) to several IT privacy metrics based +on f-divergences. We interpret probabilistic IP under both the detection and estimation frameworks and +link it to differential privacy, thus allowing a precise operational interpretation of these IT privacy metrics. +We show that the χ2-divergence privacy metric is stronger than those based on total variation distance +and Kullback-Leibler divergence. Therefore, we further develop a data-driven empirical risk framework +based on the χ2-divergence privacy metric and realized using deep neural networks. This framework is +agnostic to the adversarial attack model. Empirical experiments demonstrate the efficacy of our approach. +Index Terms +Inference privacy, privacy measure, f-divergence, differential privacy, χ2-divergence. +I. INTRODUCTION +The past decades have witnessed the proliferation of digital services such as cloud computing, which +necessitates the collection of prodigious amounts of data from a myriad of sources. The concomitant risk +of exposing sensitive information arouses the antipathy of data owners towards external access to their +data. For example, studies have shown that users’ personal information such as sexual orientation and +The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. +E-mails: {wangcx, wptay}@ntu.edu.sg +January 23, 2023 +DRAFT +arXiv:2301.08401v1 [cs.IT] 20 Jan 2023 + +2 +political affiliation can be accurately inferred from their activities on social networking platforms [1]. +Data providers must privatize or sanitize the data to mitigate the tension between the need to share data +and the need to protect sensitive information [2]. +Data privacy involves the proper collection and dissemination of data in ways that conceal the identity +or attribute of any individual datum while inference privacy [3]–[8] seeks to prevent the disclosure +of sensitive information that is statistically dependent on the original data. The key distinction is that +inference privacy is completely built upon a statistical inference framework, while ingredients of data +privacy can be partially or totally non-stochastic. For both cases, a major challenge in developing privacy- +preserving methodologies is to formally quantify the amount of privacy leakage, given all possible +auxiliary information the adversary may have. This quantification plays a crucial role in designing +privatization schemes as an indicator of the necessary amount of perturbation needed for a desirable +level of privacy protection. +A. Privacy Metrics +Privacy notions that have gained wide visibility trace back to the concept of group-based anonymization, +which hides individual records by reducing the granularity of data in a database. A popular technique +is k-anonymity [9], which guarantees that the identity of an individual whose data is contained in a +database is indistinguishable from at least k−1 other individual participants when projected on the quasi- +identifiers. However, attackers can still make inferences about sensitive values that exhibit homogeneity +within an anonymized group. Subsequently, ℓ-diversity [10] is proposed to overcome the weakness of the +anonymized database by additionally requiring the sensitive fields in an equivalence class to have at least +ℓ well-represented values to maintain diversity. One problem with ℓ-diversity is that it does not consider +semantic meanings of sensitive values and hence is not immune to attacks with global knowledge about +the sensitive attributes. The definition of t-closeness [11] refines ℓ-diversity by taking into account the +distributions of the sensitive attributes in an equivalence class and the whole database. +Over the past decade, differential privacy (DP) [12]–[15] has emerged out of attempts to withhold +individual information when releasing aggregate information about a database. Owing to its rigorous +approach and formal privacy guarantees, DP has become the mainstream data privacy metric. It formalizes +the idea that the presence or absence of an individual in a database does not appreciably affect the +distribution of a randomized inquiry. Compared to k-anonymity and ℓ-diversity, which are semantic, DP +is algorithmic and provides semantic privacy guarantees [16], [17]. One of the extraordinary characteristics +of DP is that it abstracts away the attacker’s auxiliary information about the data, and DP is thus proof +against an attacker with arbitrary side information [18]. However, enforcing this strict guarantee comes +January 23, 2023 +DRAFT + +3 +with a price. A differentially private algorithm in practice can significantly distort data, thus diminishing +the overall utility of the privatized results [5], [19], [20]. +It should be noted that DP is independent of the data distribution. Going beyond this, many privacy +works leverage the distribution of the data to obtain interesting results. For instance, references [21], [22] +relate t-closeness to DP by making assumptions about the prior and posterior views of the data. The work +[23] demonstrates that under proper choices of the prior, responding to queries using samples from the +posterior is sufficient to guarantee DP, and the work [24] generalizes DP by choosing prior distribution +families. +Because the data distribution is often available to the attacker as side information, privacy mechanisms +can take advantage of the uncertainty of the data in a probabilistic manner. For example, Bayesian DP +proposed by [25] calibrates noise perturbation to the data distribution to provide practical DP guarantees. +Quantifiers from information theory [26] that measure the uncertainty of a random variable from observing +another random variable become a natural choice to formalize the measure of privacy leakage as well +as utility. The reader is referred to the survey [27] for a detailed history of the field. Works like [3], +[28]–[30] cast the privacy-utility trade-off as a modified rate-distortion problem [31] or the opposite of +the information bottleneck problem [32], in which finding the privatization scheme is formulated as an +optimization over a privacy-assuring probabilistic mapping. The most well-known information-theoretic +(IT) privacy metrics include mutual information, total variation distance [33], chi-square information and +maximal correlation [34]–[38], which are the subjects of our study. +There is a growing interest in IT privacy metrics as each typically uses a single-value characterization +of privacy leakage (e.g., mutual information), whereas the number of constraints to formulate DP is +contingent on the size of data, thus making it unwieldy in optimization frameworks. Due to their concise +formulations, IT privacy metrics can be combined with a utility measure as a loss function for finding +an optimal sanitizer while maintaining computational tractability. Therefore, IT privacy metrics are more +accessible to many application domains that emphasize optimal privacy-utility trade-off. On the other +hand, DP suffers from several practical problems and limitations [39]. For example, employing DP as +a privacy measure for learning an arbitrary sanitizer [40] requires the data distribution to be known. +The differentially private mechanism of adding Laplacian noise can significantly decrease the utility. +In contrast, in practical cases where the data is continuous and high-dimensional and its distribution is +unavailable, it is possible to derive an estimate of an IT privacy metric from a finite number of samples. +On the downside [41], IT privacy metrics do not come with a cogent operational interpretation. Although +operational interpretations of some IT privacy metrics like mutual information do arise in transmission +and compression settings and are related to statistical dependency between variables, they are not explicit +January 23, 2023 +DRAFT + +4 +operational interpretations like those provided by privacy notions like DP and information privacy (IP) +[3]–[5]. This paper aims to bridge this gap. +B. Contributions +The goal of this paper is to provide an interpretation of IT privacy metrics formed by f-divergences. +This is achieved by relating to the notion of probabilistic IP [6], which confines an adversary’s posterior +belief about the private variable with high probability. While it has been shown that DP can bound IT +privacy metrics (e.g., ϵ-DP ensures ϵ-mutual information privacy) [42], how IT privacy metrics can imply +(weak) DP has not been identified yet. The authors in [43], [44] investigated the relationship between +mutual information and DP based on their impact on data distortion. To the best of our knowledge, +our work is the first paper that examines the connections between f-divergence IT privacy metrics and +probabilistic IP (cf. Corollary 1) and thus weak DP (cf. Lemma 1). Our contributions are summarized as +follows: +• We review the probabilistic IP concept, which is consistent with an axiomatic view of a leakage +measure. We show that probabilistic IP implies weak DP. Probabilistic IP is premised on a Bayesian +model, which allows us to exploit the adversary’s uncertainty about data. The key to probabilistic +IP is restricting the coverage of privacy protection to typical scenarios (which contain the events +that are likely to happen). We show how probabilistic IP is related to the decision error under the +detection framework and the mean square estimation error under the estimation framework. +• We derive the relationship of several IT privacy metrics formed by f-divergences to probabilistic IP. +The f-divergences we study are the total variation (TV) distance, Kullback-Leibler (KL) divergence +and χ2-divergence. We show that the IT privacy metric that is strongest amongst them is the χ2- +divergence privacy metric. +• We consider practical cases where data distribution is not available and propose a statistically +consistent estimator of the χ2-divergence. Based on that, we develop a data-driven framework for +learning a neural network sanitizer, which can be instantiated appropriately depending on the problem +domain. +The focus of this paper is on the interpretation of IT privacy metrics via their relationships to probabilistic +IP. It is expected that some of our results are useful in studying privacy-utility trade-offs. The latter study +is interesting future work and beyond the scope of the current paper. +The rest of the paper is organized as follows. In Section II, we bring in the notion of probabilistic +IP and derive its properties. In Section III, we characterize IT privacy metrics using probabilistic IP. +In Section IV, we present an estimate of the χ2-divergence which converges in the large sample size +January 23, 2023 +DRAFT + +5 +regime and propose a data-driven privacy-preserving framework using the χ2-divergence privacy metric. +In Section V, we conduct experiments for privacy-utility trade-off. Finally, we make conclusions in +Section VI. +Notations: We use capital letters like X to denote random variables or vectors, and lowercase letters +like x for deterministic scalars or vectors. Throughout this paper, all random variables are defined on the +same probability space with probability measure P. We use E[X] := +� +X dP to denote the expectation of +X and E[X | Y ] is the conditional expectation. We assume that every random variable has a (generalized) +probability density function (pdf) (for discrete random variables, this specializes to a probability mass +function). We use pX(·) to denote the pdf of X, and pX|Y (· | ·) to denote the conditional pdf of X given +Y . We use X ∼ p to say that the random variable X follows a pdf p. We use EX∼p[X] := +� +xp(x) dx +to emphasize that the expectation is with respect to (w.r.t.) X with pdf p. We use ◦ to denote function +composition. The Cartesian product of two sets A and B are denoted as A × B. The indicator function +1A(x) takes value 1 if and only if x belongs to set A. Γc denotes the complement of the set Γ. We +denote |a| as the absolute value of a. The inverse function of a function f is f−1. The logarithm log is +the natural logarithm. +II. PROBABILISTIC INFORMATION PRIVACY +In this section, we review the probabilistic IP definition and concept [3], [6]. We characterize the +properties of probabilistic IP under a statistical framework and show that probabilistic IP implies DP +with high probability. Our goal is to relate probabilistic IP with IT privacy metrics that are based on +f-divergences. These are introduced in Section III. +Consider a probability space (Ω, F, P), where Ω is the sample space, F is a σ-algebra of events and +P is a probability measure. A random element X is a measurable function from (Ω, F) to (X, B(X)), +where X is a topological space X takes values in and B(X) denotes the Borel σ-algebra generated by +the open sets of X. Recall that for any subset A ⊂ Ω, X(A) = {X(ω) ∈ X : ω ∈ A} is the image of +A under X. For any subset B ⊂ X, the inverse set map X−1(B) = {ω ∈ Ω : X(ω) ∈ B}. +We use a random element S taking values in some set S to typify the private variable to be protected. +A random element X ∈ X denotes the raw data, which is supposed to be released but is correlated with +S. Releasing X will inevitably disclose information about S. To preserve the privacy of S, we let X pass +through a noisy channel pY |X. This generates a sanitized variable Y ∈ Y to replace X as the released +data. The process of generating Y from X is called the privatization mechanism or data sanitization. +Note that S, X and Y form a Markov chain S − X − Y . +January 23, 2023 +DRAFT + +6 +When sanitizing X to produce Y , the utility of Y should also be taken into consideration. However, +measuring utility is not the focus of this paper, and we simply quantify it by the empirical risk in +Section V. The discussion of privacy definitions involves the random elements S and Y only. +In this paper, for simplicity, we assume that all random elements have probability density functions +or probability mass functions (i.e., there exists a dominating probability measure w.r.t. we can take +Radon-Nikodym derivatives). Accordingly, pS(·) and pY (·) denote the marginal distributions of S and +Y , respectively. We assume that pS(s) > 0 for all s ∈ S. +A. Definition of probabilistic IP +A privacy metric provides a formal measure of the amount of privacy “leakage” when publishing the +sanitized variable. In a general statistical framework, the prior distribution (before the release of any +information) of the private variable S is known to an adversary, which constitutes the adversary’s side +information. For each (s, y) ∈ (S, Y), the relative disparity between the posterior belief (after observing +Y = y) and the prior about S = s is defined as +d(s, y) = pS|Y (s | y) +pS(s) +. +For ϵ > 0, S given Y achieves ϵ-information privacy (ϵ-IP) [3], [6] if for almost surely all (s, y), we +have +e−ϵ ≤ d(s, y) ≤ eϵ, +(1) +where ϵ is called the privacy budget. The privacy budget limits the adversary’s posterior belief about S +when observing Y . We note that ϵ-IP provides the worst-case privacy guarantee in at least two senses. +First, inequality (1) must hold for every s ∈ S, meaning that the privacy for almost surely every s ∈ S is +protected. Second, inequality (1) requires that the bounds hold for almost surely every possible sanitization +outcome y ∈ Y, even if y occurs only with very low probability. This can be unwieldy in many practical +learning settings. For example, the privatization mechanism designer may not have global knowledge +about the population of S or Y but has access to only data samples. Moreover, an excessive utility +trade-off may be needed to account for the rare cases of (s, y). +Probabilistic IP is a relaxation of ϵ-IP by imposing the privacy constraint (1) on the most probable +occurrences (which are referred to as typical scenarios). As a consequence, it is possible but unlikely for +an adversary to gain information about the private variable S from observing the sanitized variable Y . +We give the formal definition of probabilistic IP, or, equivalently, (ϵ, δ)-IP as follows. +January 23, 2023 +DRAFT + +7 +Definition 1 ((ϵ, δ)-IP; [6]). For ϵ > 0 and 0 ≤ δ ≤ 1, we say S given Y achieves (ϵ, δ)-IP if +P +�� +ω ∈ Ω : e−ϵ ≤ d(S(ω), Y (ω)) ≤ eϵ�� +≥ 1 − δ, +(2) +and achieves strong (ϵ, δ)-IP if +P +�� +s∈S +� +ω ∈ Ω : e−ϵ ≤ d(s, Y (ω)) ≤ eϵ� +� +≥ 1 − δ. +(3) +There is a subtle but non-trivial difference between (2) and (3) in Definition 1. The event in (2) includes +the randomness of both S and Y , whereas, in (3), the event of interest is w.r.t. the randomness of Y only +(i.e., the former is a union of events while the latter is an intersection of events). The motivation behind +(3) is the observation that in a majority of practical problems we desire that a sanitized variable Y does +not disclose information about S, regardless of the realization of S. In this case, we only require that +this happens with high probability. +By taking δ → 0, (ϵ, δ)-IP degenerates to ϵ-IP. Either decreasing ϵ or δ yields a stronger privacy +guarantee. +To facilitate our analysis, we define two useful “tail” events in which the sanitized variable Y leaks +information about S: +Lϵ = +� +ω ∈ Ω : d (S(ω), Y (ω)) < e−ϵ� +, +(4) +Rϵ = {ω ∈ Ω : d (S(ω), Y (ω)) > eϵ}. +(5) +Note that (ϵ, δ)-IP is equivalent to P(Lϵ ∪ Rϵ) ≤ δ. Since Y (Lϵ ∪ Rϵ) is the set of values Y (ω) in Y +for some ω ∈ Lϵ ∪ Rϵ, we have +Y −1 ◦ Y (Lϵ ∪ Rϵ) = +� +s∈S +� +ω ∈ Ω : e−ϵ ≤ d(s, Y (ω)) ≤ eϵ�c, +(6) +and strong (ϵ, δ)-IP in (3) is equivalent to +P +� +Y −1 ◦ Y (Lϵ ∪ Rϵ) +� +≤ δ. +It is obvious that strong (ϵ, δ)-IP implies (ϵ, δ)-IP because +P +� +Y −1 ◦ Y (Lϵ ∪ Rϵ) +� +≥ P(Lϵ ∪ Rϵ). +If S given Y achieves (ϵ, δ)-IP, it also achieves (ϵ′, δ′)-IP for any ϵ′ > ϵ and δ′ > δ because Lϵ′ (resp. +Rϵ′) is a subset of Lϵ (resp. Rϵ) for ϵ′ ≥ ϵ. +We wish to make connections between probabilistic IP and weak DP or (ϵ, δ)-DP since DP is deemed +a gold standard within the privacy research community. We recall the concept of (ϵ, δ)-DP, whose goal is +to simultaneously withhold information about an individual record in a database when releasing aggregate +January 23, 2023 +DRAFT + +8 +information about the database. A randomized query is differentially private if it is almost equally likely +to be from any two databases that differ in a single individual data record. In the following, we adopt a +stronger notion of neighbors in our inference framework. +Definition 2 ((ϵ, δ)-Differential privacy). Any s ∈ S and s′ ∈ S are said to be neighbors if they take +distinct values. We say S given Y achieves (ϵ, δ)-DP if for every pair of neighbors s, s′ ∈ S and all +A ∈ B(Y), we have +P(Y ∈ A | S = s) ≤ eϵP +� +Y ∈ A +�� S = s′� ++ δ. +If δ = 0, we say that S given Y achieves ϵ-DP. +Remark 1. If S ⊂ Rn (e.g., in a database), the typical definition of neighbors s = (s1, s2, . . . , sn) and +s′ = (s′ +1, s′ +2, . . . , s′ +n) in the DP framework require that s and s′ differ only in one component, i.e., si ̸= s′ +i +for some i and sj = s′ +j for all j ̸= i. In our inference framework, S is not necessarily embedded in an +n-dimensional vector space. Hence, we consider any distinct s and s′ to be neighbors. Nevertheless, our +framework can also accommodate database privacy using the usual definition of neighbors in DP. +For every run of the privatization algorithm pY |X, ϵ-DP ensures that Y is almost equally likely to be +observed on every pair of neighboring private data, simultaneously. In practice, ϵ-DP can be too strong +to satisfy in some scenarios. A commonly used relaxation is to allow a small error probability such that +it is possible but unlikely that ex post facto an observation of Y will be much more or much less likely +to be generated when S = s than when S = s′ (cf. [15, Lemma 3.17]). +As opposed to DP, which is independent of the prior distribution of S, probabilistic IP makes use +of pS to model the side information of an adversary [23], [25]. In addition, the interpretation of δ in +DP is somewhat problematic due to taking the probability space over the privatization mechanism. As +pointed out by [16], the probability that a privacy breach occurs is not bounded by δ in DP. In contrast, +δ in probabilistic IP explicitly amounts to the probability over the “tail” scenarios out of the coverage of +privacy protection. +It is easy to see that ϵ-IP immediately leads to 2ϵ-DP [5]. In what follows, we show that strong (ϵ, δ)-IP +can guarantee a certain level of (ϵ, δ)-DP. +Lemma 1. Suppose α = infs∈S pS(s) > 0. If S given Y achieves strong (ϵ, δ)-IP, it is also (2ϵ, δ/α)-DP. +Proof: Let Ψ = Y −1 ◦ Y (Lϵ ∪ Rϵ). For y ∈ Y (Ψc) and any neighbors s, s′ ∈ S with s ̸= s′, we +January 23, 2023 +DRAFT + +9 +have +pY |S(y | s) +pY |S(y | s′) = pS|Y (s | y) +pS(s) +pS(s′) +pS|Y (s′ | y) ≤ e2ϵ. +Therefore, for any B ⊂ Ψc, we have +P(B | S = s) ≤ e2ϵP +� +B +�� S = s′� +. +(7) +On the other hand, we have +P(Ψ | S = s) ≤ P +� +Ψ ∩ S−1(s) +� +P(S = s) +≤ +P(Ψ) +P(S = s) ≤ δ/α. +(8) +Finally, for any A ∈ B(Y), we have +P +� +Y −1(A) +�� S = s +� += P +� +Y −1(A) ∩ Ψc �� S = s +� ++ P +� +Y −1(A) ∩ Ψ +�� S = s +� +≤ e2ϵP +� +Y −1(A) ∩ Ψc �� S = s′� ++ P(Ψ | S = s) +≤ e2ϵP +� +Y −1(A) +�� S = s′� ++ P(Ψ | S = s), +where the last equality follows from (7). From (8), the proof is complete. +From the proof of Lemma 1, we also have that (ϵ, δ)-IP ensures 2ϵ-DP with probability 1 − δ (w.r.t. +the randomness over S and Y ). +B. Error Bounds +The goal of invoking a privacy definition is to limit an adversary’s capability of inferring S based on +Y . Therefore, a quantitative characterization of this capability is important to justify the appropriateness +of the privacy definition. We show that (ϵ, δ)-IP indeed lower-bounds the detection error and estimation +error of S. The following Lemma 2 provides a non-trivial bound to the probability of error under the +detection framework when enforcing (ϵ, δ)-IP. +Lemma 2. Suppose S and Y are finite alphabets, and S given Y achieves (ϵ, δ)-IP. Then, for any decision +rule γ : Y → S, we have +P(γ(Y ) ̸= S) ≥ 1 − δ − eϵ max +s∈S pS(s). +Proof: It is known that the maximum a posteriori rule minimizes P(γ(Y ) ̸= S), i.e., the optimal +decision rule γ is given by +γ(y) = arg max +s∈S +pS|Y (s | y), ∀ y ∈ Y. +January 23, 2023 +DRAFT + +10 +Let Γy = S(Y −1(y) ∩ Rϵ) for y ∈ Y. Firstly, we have +� +y∈Y +pY (y) max +s∈Γy pS|Y (s | y) +(9) +≤ +� +y∈Y +� +s∈Γy +pS,Y (s, y) += +� +y∈Y +� +s∈Γy +P +� +S−1(s) ∩ Y −1(y) +� += P(Rϵ) ≤ δ, +where the last inequality is due to P(Rϵ) ≤ P(Lϵ ∪ Rϵ) ≤ δ. Secondly, we have +� +y∈Y +pY (y) max +s∈Γc +y +pS|Y (s | y) +(10) +≤ +� +y∈Y +pY (y) max +s∈Γc +y +{eϵpS(s)} ≤ eϵ max +s∈S pS(s). +Finally, the proof is completed by noting that +sup +γ P(γ(Y ) = S) = +� +y∈Y +pY (y) max +s∈S pS|Y (s | y) ≤ (9) + (10). +Either decreasing ϵ or δ elevates the lower bound of the error probability, which suggests a lower +accuracy for the Bayes classifier. This observation is consistent with the claim that a smaller ϵ or δ +provides stronger privacy protection. In the extreme case where ϵ = δ = 0, it is no surprise that the +bound reaches the largest Bayes error of 1 − maxs∈S pS(s). +Next, we provide a bound for the estimation error when enforcing (ϵ, δ)-IP for continuous S and Y . +Note that estimation error is defined w.r.t. the variable range while (ϵ, δ)-IP is not. To relate them, we +need to assume a regularity condition. +Lemma 3. Suppose S ⊂ R≥0 and Y ⊂ R. Let M = (Lϵ ∪ Rϵ)c and Γy = S(Y −1(y) ∩ M). Suppose +for y ∈ Y (M) and α ∈ {1, 2}, the following regularity condition holds: +E[Sα] = E +� +Sα �� S−1(Γy) +� +. +(11) +If S given Y achieves (ϵ, δ)-IP, then for any estimator γ : Y → S, we have +E +� +(S − γ(Y ))2� +≥ (1 − δ)e−2ϵE +� +S2� +− e2ϵE[S]2. +January 23, 2023 +DRAFT + +11 +Proof: Firstly, we have +E +� +E[S1M | Y ]2� += +� +Y (M) +�� +Γy +spS|Y (s | y) ds +�2 +pY (y) dy +≤ +� +Y (M) +� +eϵ +� +Γy +spS(s) ds +�2 +pY (y) dy +≤ e2ϵ +� +Y (M) +E[S | Γy]2pY (y) dy += e2ϵE[S]2 +� +Y (M) +pY (y) dy +≤ e2ϵE[S]2. +(12) +Secondly, we have +E +� +(S1M)2� += +� +M +s2pS,Y (s, y) ds dy += +� +Y (M) +pY (y) +� +Γy +s2pS|Y (s | y) ds dy +≥ e−ϵ +� +Y (M) +pY (y) +� +Γy +s2pS(s) ds dy += e−ϵE +� +S2� � +Y (M) +� +Γy +pY (y)pS(s) ds dy +≥ e−2ϵE +� +S2� � +M +pS,Y (s, y) ds dy +≥ (1 − δ)e−2ϵE +� +S2� +. +(13) +Finally, we have +E +� +(S − γ(Y ))2� +≥ E +� +(S1M − γ(Y )1M)2� +≥ E +� +(S1M − E[S1M | Y ])2� += E +� +(S1M)2� +− E +� +E[S1M | Y ]2� +. +(14) +The proof is completed by substituting (12) and (13) into (14). +To interpret the regularity condition in Lemma 3, note for y ∈ Y (M), +Γy = +� +s : e−ϵ ≤ d(s, y) ≤ eϵ� +⊂ S. +Thus, Γy contains all points in S that are protected by (ϵ, δ)-IP when conditioned on Y = y. The +regularity condition (11) ensures that the first and second moments of S on Γy are consistent with that +over Γc +y. The regularity condition is always satisfied for strong (ϵ, δ)-IP because Γy = S for y ∈ Y (M) +by Definition 1 and hence S−1(Γy) = Ω. When S is independent of Y , the estimation error bound reaches +January 23, 2023 +DRAFT + +12 +its maximum value (which equals the variance of S). One can enlarge this error bound by decreasing δ +or ϵ to provide stronger privacy protection. +III. FROM IT PRIVACY METRICS TO PROBABILISTIC IP +In this section, we present the relationship of several well-known IT privacy metrics with probabilistic +IP, to provide insights into the operational principles of IT privacy metrics as privacy measures. +We begin by reviewing the definitions of the IT privacy metrics studied in this paper. First, we introduce +f-divergences [45], [46], which are a general class of statistical distances measuring the divergence +between two probability distributions over the same probability space. +Definition 3 (f-divergence). Let P and Q be two probability measures over a sample space Ω such +that P is absolutely continuous w.r.t. Q. For a convex function f : [0, ∞) → R such that f(1) = 0, the +f-divergence from the reference measure Q to P is +Df(P ∥ Q) = +� +Ω +f +� dP +dQ +� +dQ. +(15) +Many of the common statistical divergences are special cases generated by different choices of function +f. For example, total variation (TV) distance, Kullback-Leibler (KL) divergence and χ2-divergence are +associated with generating functions f(x) = |x − 1|, f(x) = x log x and f(x) = x2 − 1, respectively. +Given two density functions p and q over Z, the total variation distance between p and q is +TV(p, q) = +� +Z +|p(z) − q(z)| dz, +the KL divergence between p and q is +DKL (p ∥ q) = +� +Z +p(z) log p(z) +q(z) dz, +and the χ2-divergence between p and q is +χ2(p ∥ q) = +� +Z +p(z) +q(z)p(z) dz − 1. +We restrict our discussion to the above three types of f-divergences. IT privacy metrics formed by the +f-divergences between the joint distribution and the product of the marginal distributions of the private +variable and the sanitized variable are widely used to quantify inference privacy [3], [33], [37], [38], +[47]. +Definition 4 (f-divergence privacy metrics). Denote +qS,Y (s, y) = pS(s)pY (y), ∀ (s, y) ∈ S × Y. +January 23, 2023 +DRAFT + +13 +For η ≥ 0, we say that +• S given Y satisfies η f-divergence privacy if +Df(pS,Y ∥ qS,Y ) = E +� +Df(pY |S(· | S) ∥ pY (·)) +� +≤ η. +(16) +• S given Y satisfies strong η f-divergence privacy if for almost surely all s ∈ S, +Df(pY (·) ∥ pY |S(· | s)) ≤ η. +(17) +Strong f-divergence privacy is tailored for privacy problems with |S| < ∞ because only in this case +is (17) numerically tractable for every s ∈ S. +Note that (16) with the KL divergence is the mutual information between S and Y , which is a quantity +of statistical dependence between S and Y [26]. The reference distribution qS,Y for the f-divergences in +(16) is chosen according to this analogy. Conversely, the choice of the reference distribution in (17) does +not follow this rule. As shown in Corollary 1, this choice leads to the conclusion that strong f-divergence +privacy implies strong (ϵ, δ)-IP. In what follows, we present the main result of this paper: f-divergence +privacy implies (ϵ, δ)-IP. +Theorem 1. The following η f-divergence privacies based on TV distance, KL divergence and χ2- +divergence, imply (ϵ, δ)-IP, for any ϵ > 0 and δ specified by η and ϵ as follows. +(a) If TV(pS,Y , qS,Y ) ≤ η, then S given Y achieves (ϵ, δ)-IP with δ = +η +1 − e−ϵ . +(b) If DKL (pS,Y ∥ qS,Y ) ≤ η, then S given Y achieves (ϵ, δ)-IP with δ = ζ(ϵ) + ζ(−ϵ), where +ζ(ϵ) = sup +� +p ∈ [0, 1] : (1 − p) log 1 − p +eϵ − p ≤ η − ϵ +� +. +(c) If χ2(pS,Y ∥ qS,Y ) ≤ η, then S given Y achieves (ϵ, δ)-IP with +δ = +e−ϵη +(e−ϵ − 1)2 + η + +eϵη +(eϵ − 1)2 + η. +Proof: The proofs of claims (a) to (c) are presented in Appendices A to C, respectively. +Theorem 1 gives a characterization of f-divergence privacy from the perspective of probabilistic IP, thus +allowing us to assign the operational interpretations of probabilistic IP to these f-divergence privacies. +For a given level of f-divergence privacy, Theorem 1 casts light on which level ϵ-IP or ϵ-DP is protected +with high probability. Note the δ in (ϵ, δ)-IP resulting from f-divergence privacy is coupled with ϵ. For a +fixed η, increasing ϵ decreases δ, and for a fixed ϵ, increasing η increases δ. Although ϵ can be evaluated +at any positive value, the resulting δ may become trivial if δ ≥ 1. +Taking the results in Theorem 1 further, we show that strong f-divergence privacy implies strong +(ϵ, δ)-IP. +January 23, 2023 +DRAFT + +14 +Corollary 1. Suppose |S| < ∞ and Df(pY ∥ pY |S(· | s)) ≤ η for all s ∈ S. For ϵ > 0, S given +Y achieves strong (ϵ, δ|S|)-IP, with the same δ given in Theorem 1 for total variation distance, KL +divergence and χ2-divergence, respectively. +Proof: For each s ∈ S, let +Γs = {ω ∈ Ω : d(s, Y (ω)) ≥ eϵ} ∪ +� +ω ∈ Ω : d(s, Y (ω)) ≤ e−ϵ� +. +Retracing the proof steps of Theorem 1, it can be deduced that if Df(pY ∥ pY |S=s) ≤ η for each of the +f-divergences in Theorem 1, we have P(Γs) ≤ δ with δ given in Theorem 1. Note this is true only if +pY |S=s acts as the reference distribution. Recall that S given Y achieves strong (ϵ, δ)-IP if +P +� +Y −1 ◦ Y (Lϵ ∪ Rϵ) +� +≤ δ. +For any ω ∈ Y −1 ◦ Y (Lϵ ∪ Rϵ), there exists s ∈ S such that d(s, Y (ω)) ≥ eϵ or d(s, Y (ω)) ≤ e−ϵ. +Therefore, we must have +Y −1 ◦ Y (Lϵ ∪ Rϵ) ⊂ +� +s∈S +Γs. +(18) +As a result, we have +P +� +Y −1 ◦ Y (Lϵ ∪ Rϵ) +� +≤ +� +s∈S +P(Γs) ≤ δ|S|. +The proof is now complete. +Remark 2. Following Theorem 1, one may be interested in whether it is possible to translate probabilistic +IP into f-divergence privacy. The answer is positive for the total variation distance as shown in Lemma 4 +below. However, the question remains to be explored for the other f-divergences. +Lemma 4. If S given Y achieves (ϵ, δ)-IP, we have +TV(pS,Y , qS,Y ) ≤ 2(eϵ − 1 + δ). +Proof: See Appendix D. +Apart from the IT privacy metrics based on f-divergences, maximal correlation [48], [49] defined +in Definition 5 below has also been extensively employed as a measure of privacy leakage from an +estimation-theoretic point of view [35], [50]–[52]. +January 23, 2023 +DRAFT + +15 +Definition 5 (The Hirschfeld-Gebeléin-Renyi Maximal Correlation). Let Z ∈ Z and W ∈ W be +jointly distributed random variables. Denote H(pZ) = +� +f : EZ∼pZ[f(Z)] = 0, EZ∼pZ +� +f(Z)2� += 1 +� +. +The maximal correlation between Z and W is +ρm(Z, W) = +sup +f∈H(pZ) +g∈H(pW ) +E[f(Z)g(W)]. +The following result shows the relationship between χ2-divergence and maximal correlation. +Lemma 5. The following inequalities hold: +χ2(pS,Y ∥ qS,Y ) +min{|S|, |Y|} − 1 ≤ ρm(S, Y )2 ≤ χ2(pS,Y ∥ qS,Y ). +Proof: If both S and Y are infinite alphabets, the lower bound holds vacuously. Therefore, we assume +at least one is finite. The rest of the proof is in Appendix E. +The IT privacy metrics and maximal correlation are formal measures of the statistical dependence +between S and Y . They possess desirable properties such as vanishing if and only if S and Y are +independent (perfect privacy). The usage of IT privacy metrics in a privacy configuration is typically to +form a loss function along with a utility measure for optimizing a privatization mechanism. +With the availability of several privacy metrics studied in this paper, a natural question arises: which +privacy metric should one choose? While there does not exist a unified answer as the choice often depends +on the problem domain, it is possible to compare these privacy metrics in a universal sense as follows +[5]. +Definition 6. We say type A privacy metric is stronger than type B privacy metric if for any valid privacy +budget η, there exists η′ such that any S given Y that achieves η′ type A privacy also satisfies η type B +privacy. If two privacy metrics are stronger than each other, we say they are equivalently strong. +From the Pinkster’s inequality [26], we have +TV(pS,Y , qS,Y )2 ≤ DKL (pS,Y ∥ qS,Y ) . +From Jensen’s inequality, we have +DKL (pS,Y ∥ qS,Y ) = E +� +log +�pS,Y (S, Y ) +qS,Y (S, Y ) +�� +≤ log E +�pS,Y (S, Y ) +qS,Y (S, Y ) +� += log(χ2(pS,Y ∥ qS,Y ) + 1). +Using Definition 6, χ2-divergence privacy metric is, therefore, stronger than the privacy metrics formed +by KL divergence and total variation distance. Furthermore, Lemma 5 indicates that χ2-divergence and +maximal correlation are equivalently strong if the private variable is discrete. In general, χ2-divergence +is the strongest privacy metric amongst the IT privacy metrics referenced in this section. +January 23, 2023 +DRAFT + +16 +A. Translating to Weak DP +In Corollary 1, it has been shown that strong IT privacy metrics imply strong probabilistic IP, and +Lemma 1 shows that strong probabilistic IP implies weak DP (when the private variable S has finite +support). By chaining these two results, we immediately obtain a lower bound of weak DP that is +guaranteed by the IT privacy metric. In what follows, we illustrate this lower bound using an example +of the Gaussian mechanism of DP [15]. +Consider a private variable S = {s0, s1} and a continuous sanitized variable Y whose distribution is +specified by +pY |S=s0 = N +� +µ0, σ2� +, +pY |S=s1 = N +� +µ1, σ2� +. +From the Gaussian mechanism, S given Y achieves (ϵ, δ)-DP if +σ2 = 2(µ0 − µ1)2 +ϵ2 +log(1.25/δ). +For an illustration, see Fig. 1. +Fig. 1. Gaussian mechanism. When µ0 and µ1 are close, it is almost equally likely for most realizations of Y (except for the +tail part) to be generated from S = s0 and S = s1. +Now we fix ϵ and δ for the Gaussian mechanism, and compute the χ2-divergence privacy for S and Y . +Note that DP disregards the prior distribution of S. The χ2-divergence between two normal distributions +can be computed analytically: +χ2(pY |S=s0 ∥ pY |S=s1) = exp +�(µ0 − µ1)2 +σ2 +� +. +January 23, 2023 +DRAFT + +0.4 +PYIS=$0 +0.35 +-PYIS=S1 +0.3 +0.25 +0.2 +0.15 +0.1 +0.05 +μo +;μ1 +0 +-4 +-3 +-2 +-1 +0 +1 +2 +3 +4 +517 +From pY = pS(s0)pY |S=s0 + pS(s1)pY |S=s1, it can be verified that +χ2(pY ∥ pY |S=s0) = pS(s1)2χ2(pY |S=s1 ∥ pY |S=s0), +χ2(pY ∥ pY |S=s1) = pS(s0)2χ2(pY |S=s0 ∥ pY |S=s1). +Based on the χ2-divergence privacy determined by (ϵ, δ)-DP, we firstly use Corollary 1 to quantify the +strong IP, and then apply Lemma 1 to compute the (ϵ′, δ′)-DP bound. We compare the derived (ϵ′, δ′)-DP +bounds with the baseline (ϵ, δ)-DP. In Figs. 2a and 2b, we set δ = 0.1 and δ = 0.05, respectively, and +vary ϵ from 0.1 to 1.2, while fixing pS(s0) = pS(s1) = 0.5. Note Theorem 1 indicates that δ′ is a +function of ϵ′ for the (ϵ′, δ′)-DP bound, and we can evaluate ϵ′ at any value. Letting ϵ′ be the sum of ϵ +and a small positive value, we obtain δ′. It can be seen that δ′ decreases as ϵ′ increases, implying that +weaker privacy protection always comes with a higher probability. The (ϵ′, δ′) bound becomes tighter +when (ϵ, δ) is closer to (0, 0). In Fig. 2c, we vary pS to verify its impact on DP. The results are consistent +with Lemma 1, which states the level of DP under probabilistic IP is related to mins∈S pS(s). The bound +tends to be looser when the prior of S is unbalanced. +IV. DATA-DRIVEN PRIVACY METRIC +In this section, we propose a practical implementation of χ2-divergence based on a variational form +and show that the proposed empirical estimate is asymptotically consistent. This lays a foundation of the +data-driven privacy-preserving framework in Section V. +One prominent advantage of an f-divergence privacy metric is that it can be estimated from data without +the need to estimate the data distribution, which is particularly useful for high-dimensional and continuous +data. This stands in striking contrast to DP, which is unmanageable in such cases. The variational form +views f-divergence from an optimization perspective, for which approximation is feasible by restricting +the search function space to be from a parametric family represented by neural networks. +In what follows, we review the dual representation of χ2-divergence and propose a tighter and reg- +ularized representation. Let p and q be two probability distributions over Z. A common variational +formulation of (15) is obtained via the Legendre-Fenchel duality [53]. The conjugate of the convex +function f : [0, ∞) → R in (15) is defined as +f∗(w) = sup +z∈R+ +{zw − f(z)}. +Note f∗∗ = f when f is convex and closed. This yields a dual representation of f-divergence [53]: +Df(p ∥ q) = sup +g∈H +{EZ∼p[g(Z)] − EZ∼q[f∗(g(Z))]}, +January 23, 2023 +DRAFT + +18 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +(a) (ϵ′, δ′)-DP bounds with varying ϵ and δ = 0.05. +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0.18 +0.2 +(b) (ϵ′, δ′)-DP bounds with varying ϵ and δ = 0.1. +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +(c) (ϵ′, δ′)-DP bounds with varying prior pS. +Fig. 2. (ϵ′, δ′)-DP bounds derived from the χ2-divergence privacy for (ϵ, δ)-DP Gaussian mechanism. +where H includes all measurable functions from Z to R such that the last expectation term is finite. In +particular, the χ2-divergence admits the following the dual representation [53]: +χ2(p ∥ q) = sup +g∈H +� +EZ∼p[g(Z)] − EZ∼q +� +g(Z) + g(Z)2/4 +�� +, +(19) +where the supremum is achieved at g(z) = 2 +�p(z) +q(z) − 1 +� +. +In Proposition 1, we present an improved variational form of χ2-divergence [54]. We note that the +optimal g in (19) must satisfy the regularization EZ∼p[g(Z)] = 1, whereas this is not required in (20). +Proposition 1. Let H = +� +g : Z → R +�� 0 < EZ∼q +� +g(Z)2� +< ∞ +� +. Assume q > 0 almost surely. χ2- +divergence admits the following variational form: +χ2(p ∥ q) = sup +g∈H +(EZ∼p[g(Z)] − EZ∼q[g(Z)])2 +EZ∼q[g(Z)2] +. +(20) +January 23, 2023 +DRAFT + +19 +Proof: From the Cauchy-Schwarz inequality, we have +EZ∼q +�� +1 − p(Z) +q(Z) +� +g(Z) +�2 +≤ EZ∼q +�� +1 − p(Z) +q(Z) +�2� +EZ∼q +� +g(Z)2� += χ2(p ∥ q)EZ∼q +� +g(Z)2� +, +where the inequality becomes equality when g(z) ∝ 1 − p(z) +q(z) for z ∈ Z almost everywhere. +Now suppose we are given two sets of samples {wi}m +i=1 and {zi}m +i=1 drawn independently from p and q, +respectively, and we want to estimate the χ2-divergence (20) using these samples. To ensure computational +tractability, we let H = {gφ : φ ∈ Φ} in which gφ is a neural network function parameterized by trainable +weights vector φ ∈ Φ. Replacing the expectations in (20) with their respective sample averages, χ2(p ∥ q) +can be estimated as +ˆχ2 +m(p ∥ q) = sup +φ∈Φ +� 1 +m +�m +i=1 gφ(wi) − 1 +m +�m +i=1 gφ(zi) +�2 +1 +m +�m +i=1 gφ(zi)2 + λm +, +(21) +where λm → 0 is a regularization term for countering a vanishing denominator. +The convergence of the empirical estimates to their corresponding population statistics with increasing +sample size is important to justify the method. We show that the estimate (21) converges to (20) in +probability (denoted as “ +p +−→”) if some mild assumptions are satisfied. +Theorem 2. The estimate ˆχ2 +m(p ∥ q) +p +−→ χ2(p ∥ q) as m → ∞ if the following conditions hold: +(a) There exists φ ∈ Φ such that gφ(z) ∝ 1 − p(z) +q(z). +(b) gφ(z) is smooth w.r.t. φ ∈ Φ and continuous w.r.t. z ∈ Z. +(c) gφ(z) ̸= 0 for almost everywhere z ∈ Z. +(d) Φ and Y are compact. +Proof: From condition (a), in (20), we can restrict to g = gφ for some φ ∈ Φ. Let its objective +function be denoted as γ(φ) and let γm(φ) be the objective function of (21). It suffices to prove +sup +φ∈Φ +γm(φ) +p +−→ sup +φ∈Φ +γ(φ). +(22) +From the generic uniform convergence theorem [55, Theorem 1], (22) is ensured by the following +conditions: +i. Φ is compact. +ii. γm(φ) a.s. +−→ γ(φ) for all φ ∈ Φ. +iii. γm(φ) is stochastically equicontinuous for all m ≥ 1, i.e., for any ϵ > 0, there exists σ > 0 such +that +lim +m→∞ P +� +sup +∥φ−φ′∥<δ +��γm(φ) − γm(φ′) +�� > η +� +(23) +January 23, 2023 +DRAFT + +20 +Note condition i is given by condition (d) and condition ii follows from the strong law of large numbers. +We only need to prove condition iii, which needs an auxiliary Lemma 6. +Lemma 6. If conditions (b), (c) and (d) are satisfied, there exists a sequence of random variables +(Bm)m≥1 and a constant b < ∞ such that +lim +m→∞ P(Bm − b > ϵ) = 0, +for any ϵ > 0 and +|λm(φ) − λm(φ′)| ≤ Bm∥φ − φ′∥, +for all φ, φ′ ∈ Φ, in which ∥·∥ is the Euclidean norm. +Proof: See Appendix F. +Applying Lemma 6 to (23) and letting δ = +η +b + ϵ with ϵ > 0, we have +(23) ≤ lim +m→∞ P +� +sup +∥φ−φ′∥<δ +Bm +��φ − φ′�� > η +� +≤ lim +m→∞ P(Bmδ > η) += lim +m→∞ P(Bm > b + ϵ) = 0. +The theorem is now proved. +In Theorem 2, condition (a) is implied by the universal approximation property of neural networks +[56] for φ in a sufficiently high dimensional convex set Φ. The conditions (b)-(d) can be satisfied by +choosing proper activation functions for the neural networks. +A. Data-Driven Privacy-Preserving Framework +The empirical estimate of the χ2-divergence empowers us to compute the privacy quantity from data +without the need to estimate the distribution of data. In what follows, we employ the χ2-divergence as +a privacy metric and present a data-driven framework for trading off privacy and utility. +A privacy-preserving framework comprises three components: sanitizer, privacy function and utility +function. A sanitizer takes the raw data X as input and produces the sanitized data Y , in an attempt to +remove the statistical information about the private variable S from X. In practice, a sanitizer can be +realized by a noisy transformation: +Y = hθ(X, N), +(24) +January 23, 2023 +DRAFT + +21 +where hθ is a neural network function parameterized by θ, and N is the noise perturbation. A naive +sanitizer is a constant function, which, however, deprives Y of any utility. It is necessary to reach a +compromise between privacy and utility, e.g., requiring that Y is maximally informative about a utility +task while not containing an excessive amount of information about S. +To learn the optimal sanitizer parameter θ, we need a privacy function to quantify the information +between S and Y . In this paper, the square root version of χ2-divergence (21) is adopted as the privacy +function (taking the square root to counter the vanishing gradient problem). Given a set of samples +{si, xi}m +i=1 drawn from (S, X), we generate yi = hθ(xi, ni) (with ni being a random perturbation) to +obtain DS,Y = {si, yi}m +i=1. Then the privacy function P(θ; φ) is formulated as: +max +φ +P(θ; φ) := +� +ˆχ2m(pS,Y ∥ qS,Y ). +Note that each yi is parameterized by the trainable parameter θ. For a fixed θ, maximizing P(θ; φ) +over φ yields an estimate of the dependence between S and Y . +On the other hand, a utility function measures the usefulness of the sanitized variable Y w.r.t. a utility +variable U of interest. We denote the utility function as L(θ; τ), in which τ is the trainable parameter of +the utility model. For example, L(θ; τ) can be the reconstruction loss of X from Y by letting U = X, +and τ is the vector of model weights. Minimizing L(θ; τ) over τ yields the minimum reconstruction +error. +With the privacy and utility functions at hand, optimizing the sanitizer parameter θ can be formulated +as an unconstrained optimization (Fig. 3): +min +θ,τ +� +L(θ; τ) + λ max +� +max +φ +P(θ; φ), √η +�� +, +(25) +where η is the privacy budget for χ2-divergence privacy and λ is a constant to reflect the significance of +privacy protection. The work [57] proposed an alternating algorithm to optimize (25), which is reproduced +Sanitizer hθ(X) +Utility model L (θ; τ) +Privacy model P(θ; ϕ) +X +Y +Loss +U +S +λ +Fig. 3. The privacy-preserving framework. +in Algorithm 1. Firstly, we freeze θ and optimize P(θ; φ) and L(θ; τ), respectively. Then, we fix τ +and φ and update θ. These two steps are repeated until an equilibrium is reached. +January 23, 2023 +DRAFT + +22 +Algorithm 1 Minibatch stochastic gradient algorithm +1: Initialize θ, φ, τ. +2: repeat +3: +Sample a mini-batch set from a training set. +4: +Optimize L(θ; τ) over τ and optimize P(θ; φ) over φ. +5: +Optimize (25) to update θ. +6: until θ converges +It is worth noting that the optimization strategy in Algorithm 1 is analogous to the empirical risk +approach [58], [59], where finding the optimal sanitization scheme is formulated as a competing game +between a sanitizer and an adversary. We demonstrate in Section V that such approaches are prone to +failure as the sanitizer can be fooled by an adversary. Our framework based on χ2-divergence privacy does +not assume that the adversary uses a particular attack model and is thus agnostic to the adversarial attack +model. From a theoretical perspective, if the data distribution is known, the χ2-divergence privacy should +be satisfied regardless of the attack that the adversary can muster. Since our framework is data-driven +with unknown data distribution, we use the estimate of χ2-divergence. +V. NUMERICAL EXPERIMENTS +In this section, we conduct experiments on the proposed privacy-preserving framework in Section IV-A +to demonstrate the efficacy of the χ2-divergence privacy metric. After training the privacy-preserving +framework, we simulate the worst-case privacy attacks (in which the sanitization scheme is known to the +attacker). We train an attack model and evaluate the level of privacy protection by the attacker’s inference +loss of the private variable from the sanitized data. +A. Privacy-Preserving Hypothesis Testing +In this experiment, we let S = {−1, 1} and U = {−1, 1} be two binary hypotheses, which are +statistically dependent on a noisy measurement X. The task is to learn the sanitized data Y from X such +that the detection error of U is minimized while making it difficult for an unknown attacker to detect S +from Y . +The noisy measurement is generated as X = A +� +S′2, U′2, S′U ′, S′, U′�⊺ +, where A ∈ R5×5 is a +randomly generated matrix and S′ ∼ N (S, 1) and U ′ ∼ N (U, 1) are noisy observations. +January 23, 2023 +DRAFT + +23 +1) Network architecture: The sanitizer function is Y = hθ(X, N) = X + h′ +θ(N), where h′ +θ is a +multilayer perceptron of 5 layers with LeakyRelu activation and N is a Gaussian white noise as a +perturbation. The utility function is exactly the loss of a neural classifier w.r.t. U: +L(θ; τ) = +m +� +j=1 +2 +� +i=1 +f(ui | yj, τ) log pU|X(ui | xj), +where f(ui | y, τ) is the output of the neural classifier, with τ denoting the trainable parameter and +p(· | y) denotes the one-hot encoding of the class of input xi, i.e., p(ui | xj) = 1 if xj is labeled with +class ui. The neural classifier is a multilayer perceptron of 5 layers with tanh activation. The generating +function gφ for the χ2-divergence privacy metric (21) is a multilayer perceptron of 5 layers with ELU +activation. +We draw 4000 samples and apply the Adam optimizer with learning rate 10−4 and batch size 500 to +train the sanitizer according to Algorithm 1. +2) Experimental results: To simulate the privacy attack, we train a neural classifier to detect S from +Y after obtaining the sanitizer. We gradually increase the privacy budget η and plot the utility loss on U +and the attack loss on S (measured in terms of classification accuracy) in Figs. 4a and 4b. In Fig. 4a, S +and U are independent with pS,U(s, u) = 0.25 for each u and s. In Fig. 4b, S and U are correlated with +pS,U(1, 1) = pS,U(−1, −1) = 0.4 and pS,U(−1, 1) = pS,U(1, −1) = 0.1. It can be seen that a higher +level of privacy protection is at the cost of less utility when S and U are correlated, while the utility is +not affected by increasing privacy when U is independent of S. A diminishing χ2-divergence leads to +an increasing classification loss on S. This suggests that IT privacy metrics can defend against unknown +adversarial attacks as alluded to in Section III. +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +(a) S and U are independent. +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +1.1 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +(b) S and U are correlated. +Fig. 4. The privacy-utility trade-offs for hypothesis testing. The percentages shown are classification accuracies. +January 23, 2023 +DRAFT + +24 +B. Privacy-Preserving Auto-Encoders +In this experiment, we impose the χ2-divergence privacy metric on variational auto-encoders (VAE) +[60] and our task is to learn latent representations of images that are insensitive to a chosen private +attribute associated with the images. We compare our method against the generative adversarial privacy +(GAP) [59], the variational fair autoencoder (VFAE) [61] and the invariant representation learning (IRL) +[62] on the UTKface [63] and CelebA dataset [64] dataset. +UTKface is a face attribute dataset with annotations of age, gender and ethnicity. CelebA is a large-scale +face attributes dataset with more than 200,000 celebrity images, each with 40 binary attribute annotations. +We choose the gender attribute as the private variable for UTKface and the smiling attribute as the private +variable for CelebA. +1) Preliminaries: Given a high-dimensional input variable X, a VAE learns a continuous latent variable +Y of the input X = x through a reparameterization of the variational lower-bound of log pX(x): +L(x; θ, τ) = E +� +log pX|Y (x | Y ) +� +− DKL +� +qY |X(· | x) ∥ pY +� +, +(26) +where qY |X is the variational encoder (parameterized by θ) that approximates the intractable posterior +distribution and pX|Y is the decoder (parameterized by τ). In this case, the encoder is equivalent to the +notion of sanitizer, the utility is the reconstruction loss (U = X), and the latent variable Y is the sanitized +data. For tractability, it is assumed that Y ∼ N (0, I) and +qY |X = N (µ(X), diag(σ(X))) , +pX|Y = N (ν(Y ), I) , +in which µ(·) and σ(·) are neural network functions with their collective trainable weights denoted by +θ. The function ν(·) is a neural network function with trainable weights denoted by τ. Given a training +set {xi}m +i=1, the utility function can be written as +L(θ; τ) = − +m +� +i=1 +L(xi; θ, τ), +which is to be minimized over θ and τ. Following the framework (25), the χ2-divergence privacy metric +is used for encouraging the disentanglement of S and Y . +The original VAE serves as the baseline. The GAP framework differs from our χ2-divergence method +(25) in that GAP quantifies privacy using the empirical risk of an adversary model [59] instead of +an agnostic privacy function. The VFAE and IRL, which are variants of VAEs, aim to factor out a +sensitive variation from the latent variable and are thus on a comparable basis with our method. In +contrast to our method and GAP, the encoders of the VFAE and IRL (i.e., pY |S,X(Y | S, X)) take +January 23, 2023 +DRAFT + +25 +an additional input of the private attribute. Therefore, the sanitizer (i.e., the encoder) needs to know +the label of S for X. To penalize privacy leakage, the VFAE uses the maximum mean discrepancy +between pY |S(· | si) and pY |S(· | sj) for si ̸= sj, while the IRL uses the pairwise KL divergences +DKL +� +pY |S,X(· | si, xi) ∥ pY |S,X(· | sj, xj) +� +for i ̸= j. For detailed VFAE and IRL frameworks, we refer +readers to [61] and [62], respectively. The privacy function is multiplied by a constant λ (similar to λ in +(25)). +2) Experimental Setup: The VAE encoder networks µ(·) and log σ(·) share 6 down-sampling ResNet +blocks [65] followed by two separate dense layers. The VAE decoder network ν(·) is made of a dense +layer and 6 up-sampling convolutional layers that recover the input image size. The dimension of the latent +variable Y is 4608. This network architecture also applies to VFAE and IRL except that an additional +channel for feeding S is required at the input of the encoder and decoder. The generating function gφ for +the χ2-divergence is made of 4 MLPs with hidden units (2304, 1152, 576, 1) with Instance Normalization. +The adversary (for GAP) and attack models (for evaluating privacy leakage) are MLPs of 4 layers with +hidden units (2304, 1152, 576, 1). +For training, we use the Adam optimizer with 10−4 learning rate and 0.5 (reps. 0.99) momentum for +running average mean and (resp. square). +3) Experimental Results: The mean square error (MSE) for reconstruction and the attack loss and +accuracy for UTKface are shown in Table I.1 A ↓ symbol means a smaller value is better and vice versa +for the ↑ symbol. Samples of the reconstructed images are displayed in Fig. 5. We set λ = 20 for GAP +and choose √η = 0.1 and λ = 5 for our method so that it has an attack performance similar to that of +GAP. From the reconstruction MSE, it can be seen that GAP and our method generate a similar utility +loss. However, the adversary model in GAP is identical to the attack model. If we replace the batch +normalization with instance normalization for the adversary model in GAP (whose results are shown in +GAP-A), the level of privacy protection dropped significantly as indicated by the attack performance. +Therefore, privacy cannot be ensured by the empirical risk if the adversary model in GAP does not match +the attack model. +The results for CelebA are shown in Table II with samples of reconstructed images displayed in +Fig. 6. In this case, we include an additional utility task of classifying gender in the learning architecture. +Retaining the λ and η used for UTKface, our method outperforms the GAP (where the adversary model +and attack model are the same) in terms of privacy protection. Adversarial training is known to be +unstable and the quality of privacy sanitization is determined by the capability of the chosen adversarial +1Abbreviations. Prv.: Private, Attr.: Attribute, Acc.: Accuracy, Util.: Utility. +January 23, 2023 +DRAFT + +26 +neural network, which in practice cannot incorporate all possible adversarial strategies. In contrast, the +χ2-divergence privacy metric captures statistical information from data without assuming an adversary +model. +In both cases, VFAE failed to remove the private attributes while severely distorting the data (leading +to a large reconstruction error). We made attempts to improve the VFAE performance by changing λ. +However, the privacy protection offered by the VFAE is not controllable by λ. IRL with λ = 50 achieves +its best privacy protection across different values of λ but is still weaker than our method and GAP. +The accuracy of classifying the utility variable is better preserved for our method when compared to the +VAE baseline. Results in Section V-A suggest that a utility variable can be preserved almost intact if it +is independent of the private variable. +TABLE I +UTKFACE DATASET. +VAE +VFAE +IRL +GAP +GAP-A +χ2 +Prv. Attr. Acc. ↓ +88% +98% +84% +70% +83% +69% +Prv. Attr. Loss ↑ +0.29 +0.07 +0.37 +0.56 +0.37 +0.58 +Util. MSE ↓ +0.026 +0.08 +0.029 +0.057 +0.041 +0.07 +TABLE II +CELEBFACES DATASET. +VAE +VFAE +IRL +GAP +GAP-A +χ2 +Prv. Attr. Acc. ↓ +85% +99.5% +75% +79% +82% +66% +Prv. Attr. Loss ↑ +0.36 +0.015 +0.48 +0.45 +0.41 +0.61 +Util. MSE ↓ +0.04 +0.12 +0.036 +0.06 +0.05 +0.075 +Util. Attr. Acc. ↑ +99.7% +93% +98% +98.8% +99% +98% +Util. Attr. Loss ↓ +0.006 +0.17 +0.057 +0.035 +0.03 +0.06 +VI. CONCLUSION +In this paper, we have made connections between probabilistic IP and weak DP and shown that imposing +this privacy notion leads to error lower bounds for detecting and estimating the private variable from the +sanitized variable. Based on probabilistic IP, we characterized several well-known IT privacy metrics given +by f-divergences. We argued that χ2-divergence privacy is stronger than TV and KL divergence privacy +January 23, 2023 +DRAFT + +27 +Fig. 5. Reconstructed UTKface images from the latent space where gender is the private attribute. +Fig. 6. Reconstructed Celebfaces images from the latent space where smiling is the private attribute and gender classification +is the utility task. +January 23, 2023 +DRAFT + +Raw +VAE +IRL +VFAE +GAP +.2 +XRaw +VAE +IRL +VFAE +GAP +X +228 +metrics. Therefore, we used χ2-divergence to develop a data-driven privacy-preserving framework. In this +paper, we have not investigated the analytical bounds for privacy-utility trade-offs under χ2-divergence +privacy. An interesting future work is to consider different utility measures and derive fundamental trade- +off bounds if they exist. +APPENDIX A +PROOF OF THEOREM 1(a) +Since Lϵ ∩ Rϵ = ∅, we have +TV(pS,Y , qS,Y ) = +� +S×Y +|pS,Y (s, y) − pS(s)pY (y)| ds dy +≥ +� +Lϵ∪Rϵ +|pS,Y (s, y) − pS(s)pY (y)| ds dy +≥ (eϵ − 1) +� +Lϵ +pS,Y (s, y) ds dy + (1 − e−ϵ) +� +Rϵ +pS,Y (s, y) ds dy += (eϵ − 1)P(Lϵ) + (1 − e−ϵ)P(Rϵ) +≥ (1 − e−ϵ)P(Lϵ) + (1 − e−ϵ)P(Rϵ) += (1 − e−ϵ)P(Lϵ ∪ Rϵ), +where the last inequality is due to eϵ − 1 ≥ 1 − e−ϵ. Finally, we have +TV(pS,Y , qS,Y ) ≤ η =⇒ P(Lϵ ∪ Rϵ) ≤ +η +1 − e−ϵ , +and the proof is complete. +APPENDIX B +PROOF OF THEOREM 1(b) +For an arbitrary event A ∈ F, consider a channel that produces a Bernoulli random variable W based +on the following law: pW|S,Y (1 | s, y) = 1 if S−1(s) ∩ Y −1(y) ∩ A ̸= ∅ and 0 otherwise. Then the +distribution of W, when (S, Y ) is generated by pS,Y , is pW (1) = p, where +p = +� +A +pS,Y (s, y) ds dy. +And the distribution of W, when (S, Y ) is generated by qS,Y , is qW (1) = q, where +q = +� +A +qS,Y (s, y) ds dy. +January 23, 2023 +DRAFT + +29 +From the data processing inequality, we have +DKL (pS,Y ∥ qS,Y ) ≥ DKL (pW ∥ qW ) += p log p +q + (1 − p) log 1 − p +1 − q . +(27) +Let γ = p +q and the right-hand side of (27) can be written as +f(p, γ) = log γ + (1 − p) log 1 − p +γ − p. +The partial derivatives of f(p, γ) are +∂f(p, γ) +∂p += 1 − γ +γ − p − log 1 − p +γ − p ≥ 0, +∂f(p, γ) +∂γ += p(γ − 1) +γ(γ − p) +� +� +� +≤ 0 +if γ < 1, +≥ 0 +otherwise, +where the inequalities are due to γ = p +q > p. Therefore, it can be concluded that +• For any fixed γ > 0, f(p, γ) is non-decreasing w.r.t. p ∈ [0, 1]. +• For any fixed p ∈ [0, 1], f(p, γ) is non-increasing w.r.t. γ < 1, and non-decreasing w.r.t. γ ≥ 1. +Now letting A = Lϵ, we have γ ≤ e−ϵ and p = P(Lϵ). From the claim assumption and (27), we have +f(p, γ) ≤ η. Consequently, we obtain +P(Lϵ) = p ≤ sup +� +p′ ∈ [0, 1] : f(p′, e−ϵ) ≤ η +� +. +On the other hand, letting A = Rϵ, we have γ ≥ eϵ and p = P(Rϵ). Similarly, we must have +P(Rϵ) = p ≤ sup +� +p′ ∈ [0, 1] : f(p′, eϵ) ≤ η +� +. +The proof is completed by noting that P(Lϵ ∪ Rϵ) = P(Lϵ) + P(Rϵ). +APPENDIX C +PROOF OF THEOREM 1(c) +The proof exploits the geometric property of χ2-divergence. Let A ∈ F be an arbitrary event. From +Sedrakyan’s inequality (which is a direct consequence of the Cauchy-Schwarz inequality), we have +χ2(pS,Y ∥ qS,Y ) = +� +A∪Ac +pS,Y (s, y)2 +pS(s)pY (y) ds dy − 1 +≥ p2 +q + (1 − p)2 +1 − q +− 1, +(28) +January 23, 2023 +DRAFT + +30 +where +p = +� +A +pS,Y (s, y) ds dy = P(A), +q = +� +A +pS(s)pY (y) ds dy. +Let γ = p +q . Substituting q = p +γ into (28) and from the assumption χ2(pS,Y ∥ qS,Y ) ≤ η, we obtain +pγ + (1 − p)2 +1 − p/γ − 1 ≤ η. +Rearranging the above inequality, we have +P(A) = p ≤ g(γ, η) +(29) +where +g(γ, η) = +γη +(γ − 1)2 + η. +The following properties about g(γ, η) can be verified by checking its derivatives. (For the reader’s +convenience, we visualize g(γ, η) by plotting its numerator and denominator as functions of γ in Fig. 7.) +• For a fixed η > 0, g(γ, η) is monotonically increasing w.r.t. γ2 ∈ [0, 1 + η] and monotonically +decreasing w.r.t. γ2 ∈ (1 + η, ∞). +• g(γ, η) ≥ 1 for γ ∈ [1, 1 + η]. +Now we substitute Lϵ and Rϵ for A in (29). It can be verified that γ ≤ e−ϵ when A = Lϵ, and γ ≥ eϵ +when A = Rϵ. From the monotonicity property of g(γ, η), we have +P(Lϵ) ≤ +e−ϵη +(e−ϵ − 1)2 + η, ∀ ϵ > 0, +P(Rϵ) ≤ +eϵη +(eϵ − 1)2 + η, ∀ ϵ > log(1 + η). +Note that the second inequality above also holds true for ϵ ∈ [0, log(1 + η)] because P(Rϵ) ≤ 1 while +its right-hand side is greater than 1. The proof for claim (c) is now complete. +APPENDIX D +PROOF OF LEMMA 4 +Let +γ(A) = +� +A +(pS,Y (s, y) − pS(s)pY (y)) ds dy, +January 23, 2023 +DRAFT + +31 +0 +1 +2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +Fig. 7. The denominator and numerator of g(γ, η). +and denote +Ψ = +� +ω : e−ϵ ≤ d(S(ω), Y (ω)) ≤ 1 +� +, +Γ = {ω : 1 ≤ d(S(ω), Y (ω)) ≤ eϵ}. +Firstly, we have +γ(Γ) − γ(Ψ) ≤ (1 − e−ϵ)P(Γ) + (eϵ − 1)P(Ψ) +≤ (eϵ − 1) (P(Γ) + P(Ψ)) +≤ eϵ − 1. +(30) +Moreover, we have +γ(Rϵ) ≤ +� +Rϵ +pS,Y ds dy ≤ δ. +From γ(Lϵ) + γ(Rϵ) + γ(Γ) + γ(Ψ) = γ(Ω) = 0 and (30), we obtain +−γ(Lϵ) = γ(Γ) + γ(Ψ) + γ(Rϵ) +≤ γ(Γ) − γ(Ψ) + δ +≤ eϵ − 1 + δ. +Finally, we obtain +TV(pS,Y , qS,Y ) = γ(Γ) − γ(Ψ) + γ(Rϵ) − γ(Lϵ) +≤ 2(eϵ − 1 + δ), +and the proof is complete. +January 23, 2023 +DRAFT + +32 +APPENDIX E +PROOF OF LEMMA 5 +Let L2(pS) (resp. L2(pY )) be the space of all real-valued functions of S (resp. Y ) with finite variance. +Define a linear operator T : L2(pY ) → L2(pS) such that for f ∈ L2(pY ), +[Tf](s) = E[f(Y ) | S = s]. +It is associated with an adjoint operator [T ∗g](y) = E[g(S) | Y = y] for g ∈ L2(pS). Let (σi)i≥1 be +a sequence of singular values of the operator T in descending order. From the definition of maximal +correlation, it is well-known that σ1 = 1 and σ2 = ρm(S, Y ) [66]. Moreover, we have +∥T∥2 +HS = +� +i≥1 +σ2 +i , +(31) +where ∥·∥2 +HS is the Hilbert-Schmidt norm. +On the other hand, we can rewrite T as +[Tf](s) = +� +Y +f(y)k(s, y)pY (y) dy, +in which k(s, y) : S × Y → R is a kernel: +k(s, y) = pS,Y (s, y) +pS(s)pY (y). +From [67, Lemma 4.8], we have +∥T∥2 +HS = +� +Y +� +S +k(s, y)2pS(s)pY (y) ds dy += χ2(pS,Y ∥ qS,Y ) + 1. +(32) +The proof is completed by combining (31) and (32). +APPENDIX F +PROOF OF LEMMA 6 +Let vm(φ) be the numerator of γm(φ) and dm(φ) = +1 +m +�m +i=1 gφ(zi)2. The gradient of γm(φ) can +then be written as ∇γm(φ) = +km(φ) +(dm(φ) + λm)2 with +km(φ) = dm(φ)∇vm(φ) − vm(φ)∇dm(φ). +By assumption, gφ(x) is a smooth function w.r.t. φ and continuous w.r.t. x. Therefore, ∂gφ(x) +∂φ +is also +a continuous function, which is thus uniformly bounded by some constant due to the compactness of Φ +January 23, 2023 +DRAFT + +33 +and Y. 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Jennrich, “Asymptotic properties of non-linear least squares estimators,” Ann. Math. Statist., vol. 40, no. 2, pp. +633–643, Apr. 1969. +January 23, 2023 +DRAFT + diff --git a/ytFAT4oBgHgl3EQfBBwz/content/tmp_files/load_file.txt b/ytFAT4oBgHgl3EQfBBwz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5fe8e66939b8aacc16ca3121357188415620d677 --- /dev/null +++ b/ytFAT4oBgHgl3EQfBBwz/content/tmp_files/load_file.txt @@ -0,0 +1,1442 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf,len=1441 +page_content='1 On the Relationship Between Information-Theoretic Privacy Metrics And Probabilistic Information Privacy Chong Xiao Wang and Wee Peng Tay, Senior Member, IEEE Abstract Information-theoretic (IT) measures based on f-divergences have recently gained interest as a mea- sure of privacy leakage as they allow for trading off privacy against utility using only a single-value characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' However, their operational interpretations in the privacy context are unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In this paper, we relate the notion of probabilistic information privacy (IP) to several IT privacy metrics based on f-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We interpret probabilistic IP under both the detection and estimation frameworks and link it to differential privacy, thus allowing a precise operational interpretation of these IT privacy metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We show that the χ2-divergence privacy metric is stronger than those based on total variation distance and Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, we further develop a data-driven empirical risk framework based on the χ2-divergence privacy metric and realized using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This framework is agnostic to the adversarial attack model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Empirical experiments demonstrate the efficacy of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Index Terms Inference privacy, privacy measure, f-divergence, differential privacy, χ2-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' INTRODUCTION The past decades have witnessed the proliferation of digital services such as cloud computing, which necessitates the collection of prodigious amounts of data from a myriad of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The concomitant risk of exposing sensitive information arouses the antipathy of data owners towards external access to their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For example, studies have shown that users’ personal information such as sexual orientation and The authors are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' E-mails: {wangcx, wptay}@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='sg January 23, 2023 DRAFT arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='08401v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='IT] 20 Jan 2023 2 political affiliation can be accurately inferred from their activities on social networking platforms [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Data providers must privatize or sanitize the data to mitigate the tension between the need to share data and the need to protect sensitive information [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Data privacy involves the proper collection and dissemination of data in ways that conceal the identity or attribute of any individual datum while inference privacy [3]–[8] seeks to prevent the disclosure of sensitive information that is statistically dependent on the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The key distinction is that inference privacy is completely built upon a statistical inference framework, while ingredients of data privacy can be partially or totally non-stochastic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For both cases, a major challenge in developing privacy- preserving methodologies is to formally quantify the amount of privacy leakage, given all possible auxiliary information the adversary may have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This quantification plays a crucial role in designing privatization schemes as an indicator of the necessary amount of perturbation needed for a desirable level of privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Privacy Metrics Privacy notions that have gained wide visibility trace back to the concept of group-based anonymization, which hides individual records by reducing the granularity of data in a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A popular technique is k-anonymity [9], which guarantees that the identity of an individual whose data is contained in a database is indistinguishable from at least k−1 other individual participants when projected on the quasi- identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' However, attackers can still make inferences about sensitive values that exhibit homogeneity within an anonymized group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Subsequently, ℓ-diversity [10] is proposed to overcome the weakness of the anonymized database by additionally requiring the sensitive fields in an equivalence class to have at least ℓ well-represented values to maintain diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' One problem with ℓ-diversity is that it does not consider semantic meanings of sensitive values and hence is not immune to attacks with global knowledge about the sensitive attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The definition of t-closeness [11] refines ℓ-diversity by taking into account the distributions of the sensitive attributes in an equivalence class and the whole database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Over the past decade, differential privacy (DP) [12]–[15] has emerged out of attempts to withhold individual information when releasing aggregate information about a database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Owing to its rigorous approach and formal privacy guarantees, DP has become the mainstream data privacy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It formalizes the idea that the presence or absence of an individual in a database does not appreciably affect the distribution of a randomized inquiry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Compared to k-anonymity and ℓ-diversity, which are semantic, DP is algorithmic and provides semantic privacy guarantees [16], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' One of the extraordinary characteristics of DP is that it abstracts away the attacker’s auxiliary information about the data, and DP is thus proof against an attacker with arbitrary side information [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' However, enforcing this strict guarantee comes January 23, 2023 DRAFT 3 with a price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A differentially private algorithm in practice can significantly distort data, thus diminishing the overall utility of the privatized results [5], [19], [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It should be noted that DP is independent of the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Going beyond this, many privacy works leverage the distribution of the data to obtain interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For instance, references [21], [22] relate t-closeness to DP by making assumptions about the prior and posterior views of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The work [23] demonstrates that under proper choices of the prior, responding to queries using samples from the posterior is sufficient to guarantee DP, and the work [24] generalizes DP by choosing prior distribution families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Because the data distribution is often available to the attacker as side information, privacy mechanisms can take advantage of the uncertainty of the data in a probabilistic manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For example, Bayesian DP proposed by [25] calibrates noise perturbation to the data distribution to provide practical DP guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Quantifiers from information theory [26] that measure the uncertainty of a random variable from observing another random variable become a natural choice to formalize the measure of privacy leakage as well as utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The reader is referred to the survey [27] for a detailed history of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Works like [3], [28]–[30] cast the privacy-utility trade-off as a modified rate-distortion problem [31] or the opposite of the information bottleneck problem [32], in which finding the privatization scheme is formulated as an optimization over a privacy-assuring probabilistic mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The most well-known information-theoretic (IT) privacy metrics include mutual information, total variation distance [33], chi-square information and maximal correlation [34]–[38], which are the subjects of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' There is a growing interest in IT privacy metrics as each typically uses a single-value characterization of privacy leakage (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', mutual information), whereas the number of constraints to formulate DP is contingent on the size of data, thus making it unwieldy in optimization frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Due to their concise formulations, IT privacy metrics can be combined with a utility measure as a loss function for finding an optimal sanitizer while maintaining computational tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, IT privacy metrics are more accessible to many application domains that emphasize optimal privacy-utility trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' On the other hand, DP suffers from several practical problems and limitations [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For example, employing DP as a privacy measure for learning an arbitrary sanitizer [40] requires the data distribution to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The differentially private mechanism of adding Laplacian noise can significantly decrease the utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In contrast, in practical cases where the data is continuous and high-dimensional and its distribution is unavailable, it is possible to derive an estimate of an IT privacy metric from a finite number of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' On the downside [41], IT privacy metrics do not come with a cogent operational interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Although operational interpretations of some IT privacy metrics like mutual information do arise in transmission and compression settings and are related to statistical dependency between variables, they are not explicit January 23, 2023 DRAFT 4 operational interpretations like those provided by privacy notions like DP and information privacy (IP) [3]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This paper aims to bridge this gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Contributions The goal of this paper is to provide an interpretation of IT privacy metrics formed by f-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This is achieved by relating to the notion of probabilistic IP [6], which confines an adversary’s posterior belief about the private variable with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' While it has been shown that DP can bound IT privacy metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', ϵ-DP ensures ϵ-mutual information privacy) [42], how IT privacy metrics can imply (weak) DP has not been identified yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The authors in [43], [44] investigated the relationship between mutual information and DP based on their impact on data distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To the best of our knowledge, our work is the first paper that examines the connections between f-divergence IT privacy metrics and probabilistic IP (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Corollary 1) and thus weak DP (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Lemma 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Our contributions are summarized as follows: We review the probabilistic IP concept, which is consistent with an axiomatic view of a leakage measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We show that probabilistic IP implies weak DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Probabilistic IP is premised on a Bayesian model, which allows us to exploit the adversary’s uncertainty about data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The key to probabilistic IP is restricting the coverage of privacy protection to typical scenarios (which contain the events that are likely to happen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We show how probabilistic IP is related to the decision error under the detection framework and the mean square estimation error under the estimation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We derive the relationship of several IT privacy metrics formed by f-divergences to probabilistic IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The f-divergences we study are the total variation (TV) distance, Kullback-Leibler (KL) divergence and χ2-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We show that the IT privacy metric that is strongest amongst them is the χ2- divergence privacy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We consider practical cases where data distribution is not available and propose a statistically consistent estimator of the χ2-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Based on that, we develop a data-driven framework for learning a neural network sanitizer, which can be instantiated appropriately depending on the problem domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The focus of this paper is on the interpretation of IT privacy metrics via their relationships to probabilistic IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It is expected that some of our results are useful in studying privacy-utility trade-offs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The latter study is interesting future work and beyond the scope of the current paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The rest of the paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Section II, we bring in the notion of probabilistic IP and derive its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Section III, we characterize IT privacy metrics using probabilistic IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Section IV, we present an estimate of the χ2-divergence which converges in the large sample size January 23, 2023 DRAFT 5 regime and propose a data-driven privacy-preserving framework using the χ2-divergence privacy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Section V, we conduct experiments for privacy-utility trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Finally, we make conclusions in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Notations: We use capital letters like X to denote random variables or vectors, and lowercase letters like x for deterministic scalars or vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Throughout this paper, all random variables are defined on the same probability space with probability measure P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We use E[X] := � X dP to denote the expectation of X and E[X | Y ] is the conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We assume that every random variable has a (generalized) probability density function (pdf) (for discrete random variables, this specializes to a probability mass function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We use pX(·) to denote the pdf of X, and pX|Y (· | ·) to denote the conditional pdf of X given Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We use X ∼ p to say that the random variable X follows a pdf p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We use EX∼p[X] := � xp(x) dx to emphasize that the expectation is with respect to (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=') X with pdf p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We use ◦ to denote function composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The Cartesian product of two sets A and B are denoted as A × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The indicator function 1A(x) takes value 1 if and only if x belongs to set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Γc denotes the complement of the set Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We denote |a| as the absolute value of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The inverse function of a function f is f−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The logarithm log is the natural logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' PROBABILISTIC INFORMATION PRIVACY In this section, we review the probabilistic IP definition and concept [3], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We characterize the properties of probabilistic IP under a statistical framework and show that probabilistic IP implies DP with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Our goal is to relate probabilistic IP with IT privacy metrics that are based on f-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' These are introduced in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Consider a probability space (Ω, F, P), where Ω is the sample space, F is a σ-algebra of events and P is a probability measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A random element X is a measurable function from (Ω, F) to (X, B(X)), where X is a topological space X takes values in and B(X) denotes the Borel σ-algebra generated by the open sets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Recall that for any subset A ⊂ Ω, X(A) = {X(ω) ∈ X : ω ∈ A} is the image of A under X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For any subset B ⊂ X, the inverse set map X−1(B) = {ω ∈ Ω : X(ω) ∈ B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We use a random element S taking values in some set S to typify the private variable to be protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A random element X ∈ X denotes the raw data, which is supposed to be released but is correlated with S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Releasing X will inevitably disclose information about S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To preserve the privacy of S, we let X pass through a noisy channel pY |X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This generates a sanitized variable Y ∈ Y to replace X as the released data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The process of generating Y from X is called the privatization mechanism or data sanitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note that S, X and Y form a Markov chain S − X − Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 6 When sanitizing X to produce Y , the utility of Y should also be taken into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' However, measuring utility is not the focus of this paper, and we simply quantify it by the empirical risk in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The discussion of privacy definitions involves the random elements S and Y only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In this paper, for simplicity, we assume that all random elements have probability density functions or probability mass functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', there exists a dominating probability measure w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' we can take Radon-Nikodym derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Accordingly, pS(·) and pY (·) denote the marginal distributions of S and Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We assume that pS(s) > 0 for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Definition of probabilistic IP A privacy metric provides a formal measure of the amount of privacy “leakage” when publishing the sanitized variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In a general statistical framework, the prior distribution (before the release of any information) of the private variable S is known to an adversary, which constitutes the adversary’s side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For each (s, y) ∈ (S, Y), the relative disparity between the posterior belief (after observing Y = y) and the prior about S = s is defined as d(s, y) = pS|Y (s | y) pS(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For ϵ > 0, S given Y achieves ϵ-information privacy (ϵ-IP) [3], [6] if for almost surely all (s, y), we have e−ϵ ≤ d(s, y) ≤ eϵ, (1) where ϵ is called the privacy budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The privacy budget limits the adversary’s posterior belief about S when observing Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We note that ϵ-IP provides the worst-case privacy guarantee in at least two senses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' First, inequality (1) must hold for every s ∈ S, meaning that the privacy for almost surely every s ∈ S is protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Second, inequality (1) requires that the bounds hold for almost surely every possible sanitization outcome y ∈ Y, even if y occurs only with very low probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This can be unwieldy in many practical learning settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For example, the privatization mechanism designer may not have global knowledge about the population of S or Y but has access to only data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Moreover, an excessive utility trade-off may be needed to account for the rare cases of (s, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Probabilistic IP is a relaxation of ϵ-IP by imposing the privacy constraint (1) on the most probable occurrences (which are referred to as typical scenarios).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' As a consequence, it is possible but unlikely for an adversary to gain information about the private variable S from observing the sanitized variable Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We give the formal definition of probabilistic IP, or, equivalently, (ϵ, δ)-IP as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 7 Definition 1 ((ϵ, δ)-IP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For ϵ > 0 and 0 ≤ δ ≤ 1, we say S given Y achieves (ϵ, δ)-IP if P �� ω ∈ Ω : e−ϵ ≤ d(S(ω), Y (ω)) ≤ eϵ�� ≥ 1 − δ, (2) and achieves strong (ϵ, δ)-IP if P �� s∈S � ω ∈ Ω : e−ϵ ≤ d(s, Y (ω)) ≤ eϵ� � ≥ 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (3) There is a subtle but non-trivial difference between (2) and (3) in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The event in (2) includes the randomness of both S and Y , whereas, in (3), the event of interest is w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' the randomness of Y only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', the former is a union of events while the latter is an intersection of events).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The motivation behind (3) is the observation that in a majority of practical problems we desire that a sanitized variable Y does not disclose information about S, regardless of the realization of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In this case, we only require that this happens with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' By taking δ → 0, (ϵ, δ)-IP degenerates to ϵ-IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Either decreasing ϵ or δ yields a stronger privacy guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To facilitate our analysis, we define two useful “tail” events in which the sanitized variable Y leaks information about S: Lϵ = � ω ∈ Ω : d (S(ω), Y (ω)) < e−ϵ� , (4) Rϵ = {ω ∈ Ω : d (S(ω), Y (ω)) > eϵ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (5) Note that (ϵ, δ)-IP is equivalent to P(Lϵ ∪ Rϵ) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Since Y (Lϵ ∪ Rϵ) is the set of values Y (ω) in Y for some ω ∈ Lϵ ∪ Rϵ, we have Y −1 ◦ Y (Lϵ ∪ Rϵ) = � s∈S � ω ∈ Ω : e−ϵ ≤ d(s, Y (ω)) ≤ eϵ�c, (6) and strong (ϵ, δ)-IP in (3) is equivalent to P � Y −1 ◦ Y (Lϵ ∪ Rϵ) � ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It is obvious that strong (ϵ, δ)-IP implies (ϵ, δ)-IP because P � Y −1 ◦ Y (Lϵ ∪ Rϵ) � ≥ P(Lϵ ∪ Rϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If S given Y achieves (ϵ, δ)-IP, it also achieves (ϵ′, δ′)-IP for any ϵ′ > ϵ and δ′ > δ because Lϵ′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Rϵ′) is a subset of Lϵ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Rϵ) for ϵ′ ≥ ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We wish to make connections between probabilistic IP and weak DP or (ϵ, δ)-DP since DP is deemed a gold standard within the privacy research community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We recall the concept of (ϵ, δ)-DP, whose goal is to simultaneously withhold information about an individual record in a database when releasing aggregate January 23, 2023 DRAFT 8 information about the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A randomized query is differentially private if it is almost equally likely to be from any two databases that differ in a single individual data record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In the following, we adopt a stronger notion of neighbors in our inference framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Definition 2 ((ϵ, δ)-Differential privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Any s ∈ S and s′ ∈ S are said to be neighbors if they take distinct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We say S given Y achieves (ϵ, δ)-DP if for every pair of neighbors s, s′ ∈ S and all A ∈ B(Y), we have P(Y ∈ A | S = s) ≤ eϵP � Y ∈ A �� S = s′� + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If δ = 0, we say that S given Y achieves ϵ-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If S ⊂ Rn (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', in a database), the typical definition of neighbors s = (s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' , sn) and s′ = (s′ 1, s′ 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' , s′ n) in the DP framework require that s and s′ differ only in one component, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', si ̸= s′ i for some i and sj = s′ j for all j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In our inference framework, S is not necessarily embedded in an n-dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Hence, we consider any distinct s and s′ to be neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Nevertheless, our framework can also accommodate database privacy using the usual definition of neighbors in DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For every run of the privatization algorithm pY |X, ϵ-DP ensures that Y is almost equally likely to be observed on every pair of neighboring private data, simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In practice, ϵ-DP can be too strong to satisfy in some scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A commonly used relaxation is to allow a small error probability such that it is possible but unlikely that ex post facto an observation of Y will be much more or much less likely to be generated when S = s than when S = s′ (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' [15, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' As opposed to DP, which is independent of the prior distribution of S, probabilistic IP makes use of pS to model the side information of an adversary [23], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In addition, the interpretation of δ in DP is somewhat problematic due to taking the probability space over the privatization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' As pointed out by [16], the probability that a privacy breach occurs is not bounded by δ in DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In contrast, δ in probabilistic IP explicitly amounts to the probability over the “tail” scenarios out of the coverage of privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It is easy to see that ϵ-IP immediately leads to 2ϵ-DP [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In what follows, we show that strong (ϵ, δ)-IP can guarantee a certain level of (ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Suppose α = infs∈S pS(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If S given Y achieves strong (ϵ, δ)-IP, it is also (2ϵ, δ/α)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: Let Ψ = Y −1 ◦ Y (Lϵ ∪ Rϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For y ∈ Y (Ψc) and any neighbors s, s′ ∈ S with s ̸= s′, we January 23, 2023 DRAFT 9 have pY |S(y | s) pY |S(y | s′) = pS|Y (s | y) pS(s) pS(s′) pS|Y (s′ | y) ≤ e2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, for any B ⊂ Ψc, we have P(B | S = s) ≤ e2ϵP � B �� S = s′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (7) On the other hand, we have P(Ψ | S = s) ≤ P � Ψ ∩ S−1(s) � P(S = s) ≤ P(Ψ) P(S = s) ≤ δ/α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (8) Finally, for any A ∈ B(Y), we have P � Y −1(A) �� S = s � = P � Y −1(A) ∩ Ψc �� S = s � + P � Y −1(A) ∩ Ψ �� S = s � ≤ e2ϵP � Y −1(A) ∩ Ψc �� S = s′� + P(Ψ | S = s) ≤ e2ϵP � Y −1(A) �� S = s′� + P(Ψ | S = s), where the last equality follows from (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From (8), the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the proof of Lemma 1, we also have that (ϵ, δ)-IP ensures 2ϵ-DP with probability 1 − δ (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' the randomness over S and Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Error Bounds The goal of invoking a privacy definition is to limit an adversary’s capability of inferring S based on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, a quantitative characterization of this capability is important to justify the appropriateness of the privacy definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We show that (ϵ, δ)-IP indeed lower-bounds the detection error and estimation error of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The following Lemma 2 provides a non-trivial bound to the probability of error under the detection framework when enforcing (ϵ, δ)-IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Suppose S and Y are finite alphabets, and S given Y achieves (ϵ, δ)-IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Then, for any decision rule γ : Y → S, we have P(γ(Y ) ̸= S) ≥ 1 − δ − eϵ max s∈S pS(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: It is known that the maximum a posteriori rule minimizes P(γ(Y ) ̸= S), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', the optimal decision rule γ is given by γ(y) = arg max s∈S pS|Y (s | y), ∀ y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 10 Let Γy = S(Y −1(y) ∩ Rϵ) for y ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Firstly, we have � y∈Y pY (y) max s∈Γy pS|Y (s | y) (9) ≤ � y∈Y � s∈Γy pS,Y (s, y) = � y∈Y � s∈Γy P � S−1(s) ∩ Y −1(y) � = P(Rϵ) ≤ δ, where the last inequality is due to P(Rϵ) ≤ P(Lϵ ∪ Rϵ) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Secondly, we have � y∈Y pY (y) max s∈Γc y pS|Y (s | y) (10) ≤ � y∈Y pY (y) max s∈Γc y {eϵpS(s)} ≤ eϵ max s∈S pS(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Finally, the proof is completed by noting that sup γ P(γ(Y ) = S) = � y∈Y pY (y) max s∈S pS|Y (s | y) ≤ (9) + (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Either decreasing ϵ or δ elevates the lower bound of the error probability, which suggests a lower accuracy for the Bayes classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This observation is consistent with the claim that a smaller ϵ or δ provides stronger privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In the extreme case where ϵ = δ = 0, it is no surprise that the bound reaches the largest Bayes error of 1 − maxs∈S pS(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Next, we provide a bound for the estimation error when enforcing (ϵ, δ)-IP for continuous S and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note that estimation error is defined w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' the variable range while (ϵ, δ)-IP is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To relate them, we need to assume a regularity condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Suppose S ⊂ R≥0 and Y ⊂ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let M = (Lϵ ∪ Rϵ)c and Γy = S(Y −1(y) ∩ M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Suppose for y ∈ Y (M) and α ∈ {1, 2}, the following regularity condition holds: E[Sα] = E � Sα �� S−1(Γy) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (11) If S given Y achieves (ϵ, δ)-IP, then for any estimator γ : Y → S, we have E � (S − γ(Y ))2� ≥ (1 − δ)e−2ϵE � S2� − e2ϵE[S]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 11 Proof: Firstly, we have E � E[S1M | Y ]2� = � Y (M) �� Γy spS|Y (s | y) ds �2 pY (y) dy ≤ � Y (M) � eϵ � Γy spS(s) ds �2 pY (y) dy ≤ e2ϵ � Y (M) E[S | Γy]2pY (y) dy = e2ϵE[S]2 � Y (M) pY (y) dy ≤ e2ϵE[S]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (12) Secondly, we have E � (S1M)2� = � M s2pS,Y (s, y) ds dy = � Y (M) pY (y) � Γy s2pS|Y (s | y) ds dy ≥ e−ϵ � Y (M) pY (y) � Γy s2pS(s) ds dy = e−ϵE � S2� � Y (M) � Γy pY (y)pS(s) ds dy ≥ e−2ϵE � S2� � M pS,Y (s, y) ds dy ≥ (1 − δ)e−2ϵE � S2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (13) Finally, we have E � (S − γ(Y ))2� ≥ E � (S1M − γ(Y )1M)2� ≥ E � (S1M − E[S1M | Y ])2� = E � (S1M)2� − E � E[S1M | Y ]2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (14) The proof is completed by substituting (12) and (13) into (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To interpret the regularity condition in Lemma 3, note for y ∈ Y (M), Γy = � s : e−ϵ ≤ d(s, y) ≤ eϵ� ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Thus, Γy contains all points in S that are protected by (ϵ, δ)-IP when conditioned on Y = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The regularity condition (11) ensures that the first and second moments of S on Γy are consistent with that over Γc y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The regularity condition is always satisfied for strong (ϵ, δ)-IP because Γy = S for y ∈ Y (M) by Definition 1 and hence S−1(Γy) = Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' When S is independent of Y , the estimation error bound reaches January 23, 2023 DRAFT 12 its maximum value (which equals the variance of S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' One can enlarge this error bound by decreasing δ or ϵ to provide stronger privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' FROM IT PRIVACY METRICS TO PROBABILISTIC IP In this section, we present the relationship of several well-known IT privacy metrics with probabilistic IP, to provide insights into the operational principles of IT privacy metrics as privacy measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We begin by reviewing the definitions of the IT privacy metrics studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' First, we introduce f-divergences [45], [46], which are a general class of statistical distances measuring the divergence between two probability distributions over the same probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Definition 3 (f-divergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let P and Q be two probability measures over a sample space Ω such that P is absolutely continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For a convex function f : [0, ∞) → R such that f(1) = 0, the f-divergence from the reference measure Q to P is Df(P ∥ Q) = � Ω f � dP dQ � dQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (15) Many of the common statistical divergences are special cases generated by different choices of function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For example, total variation (TV) distance, Kullback-Leibler (KL) divergence and χ2-divergence are associated with generating functions f(x) = |x − 1|, f(x) = x log x and f(x) = x2 − 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Given two density functions p and q over Z, the total variation distance between p and q is TV(p, q) = � Z |p(z) − q(z)| dz, the KL divergence between p and q is DKL (p ∥ q) = � Z p(z) log p(z) q(z) dz, and the χ2-divergence between p and q is χ2(p ∥ q) = � Z p(z) q(z)p(z) dz − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We restrict our discussion to the above three types of f-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' IT privacy metrics formed by the f-divergences between the joint distribution and the product of the marginal distributions of the private variable and the sanitized variable are widely used to quantify inference privacy [3], [33], [37], [38], [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Definition 4 (f-divergence privacy metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Denote qS,Y (s, y) = pS(s)pY (y), ∀ (s, y) ∈ S × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 13 For η ≥ 0, we say that S given Y satisfies η f-divergence privacy if Df(pS,Y ∥ qS,Y ) = E � Df(pY |S(· | S) ∥ pY (·)) � ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (16) S given Y satisfies strong η f-divergence privacy if for almost surely all s ∈ S, Df(pY (·) ∥ pY |S(· | s)) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (17) Strong f-divergence privacy is tailored for privacy problems with |S| < ∞ because only in this case is (17) numerically tractable for every s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note that (16) with the KL divergence is the mutual information between S and Y , which is a quantity of statistical dependence between S and Y [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The reference distribution qS,Y for the f-divergences in (16) is chosen according to this analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Conversely, the choice of the reference distribution in (17) does not follow this rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' As shown in Corollary 1, this choice leads to the conclusion that strong f-divergence privacy implies strong (ϵ, δ)-IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In what follows, we present the main result of this paper: f-divergence privacy implies (ϵ, δ)-IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The following η f-divergence privacies based on TV distance, KL divergence and χ2- divergence, imply (ϵ, δ)-IP, for any ϵ > 0 and δ specified by η and ϵ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (a) If TV(pS,Y , qS,Y ) ≤ η, then S given Y achieves (ϵ, δ)-IP with δ = η 1 − e−ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (b) If DKL (pS,Y ∥ qS,Y ) ≤ η, then S given Y achieves (ϵ, δ)-IP with δ = ζ(ϵ) + ζ(−ϵ), where ζ(ϵ) = sup � p ∈ [0, 1] : (1 − p) log 1 − p eϵ − p ≤ η − ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (c) If χ2(pS,Y ∥ qS,Y ) ≤ η, then S given Y achieves (ϵ, δ)-IP with δ = e−ϵη (e−ϵ − 1)2 + η + eϵη (eϵ − 1)2 + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: The proofs of claims (a) to (c) are presented in Appendices A to C, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Theorem 1 gives a characterization of f-divergence privacy from the perspective of probabilistic IP, thus allowing us to assign the operational interpretations of probabilistic IP to these f-divergence privacies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For a given level of f-divergence privacy, Theorem 1 casts light on which level ϵ-IP or ϵ-DP is protected with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note the δ in (ϵ, δ)-IP resulting from f-divergence privacy is coupled with ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For a fixed η, increasing ϵ decreases δ, and for a fixed ϵ, increasing η increases δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Although ϵ can be evaluated at any positive value, the resulting δ may become trivial if δ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Taking the results in Theorem 1 further, we show that strong f-divergence privacy implies strong (ϵ, δ)-IP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 14 Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Suppose |S| < ∞ and Df(pY ∥ pY |S(· | s)) ≤ η for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For ϵ > 0, S given Y achieves strong (ϵ, δ|S|)-IP, with the same δ given in Theorem 1 for total variation distance, KL divergence and χ2-divergence, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: For each s ∈ S, let Γs = {ω ∈ Ω : d(s, Y (ω)) ≥ eϵ} ∪ � ω ∈ Ω : d(s, Y (ω)) ≤ e−ϵ� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Retracing the proof steps of Theorem 1, it can be deduced that if Df(pY ∥ pY |S=s) ≤ η for each of the f-divergences in Theorem 1, we have P(Γs) ≤ δ with δ given in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note this is true only if pY |S=s acts as the reference distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Recall that S given Y achieves strong (ϵ, δ)-IP if P � Y −1 ◦ Y (Lϵ ∪ Rϵ) � ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For any ω ∈ Y −1 ◦ Y (Lϵ ∪ Rϵ), there exists s ∈ S such that d(s, Y (ω)) ≥ eϵ or d(s, Y (ω)) ≤ e−ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, we must have Y −1 ◦ Y (Lϵ ∪ Rϵ) ⊂ � s∈S Γs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (18) As a result, we have P � Y −1 ◦ Y (Lϵ ∪ Rϵ) � ≤ � s∈S P(Γs) ≤ δ|S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Following Theorem 1, one may be interested in whether it is possible to translate probabilistic IP into f-divergence privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The answer is positive for the total variation distance as shown in Lemma 4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' However, the question remains to be explored for the other f-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If S given Y achieves (ϵ, δ)-IP, we have TV(pS,Y , qS,Y ) ≤ 2(eϵ − 1 + δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: See Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Apart from the IT privacy metrics based on f-divergences, maximal correlation [48], [49] defined in Definition 5 below has also been extensively employed as a measure of privacy leakage from an estimation-theoretic point of view [35], [50]–[52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 15 Definition 5 (The Hirschfeld-Gebeléin-Renyi Maximal Correlation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let Z ∈ Z and W ∈ W be jointly distributed random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Denote H(pZ) = � f : EZ∼pZ[f(Z)] = 0, EZ∼pZ � f(Z)2� = 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The maximal correlation between Z and W is ρm(Z, W) = sup f∈H(pZ) g∈H(pW ) E[f(Z)g(W)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The following result shows the relationship between χ2-divergence and maximal correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The following inequalities hold: χ2(pS,Y ∥ qS,Y ) min{|S|, |Y|} − 1 ≤ ρm(S, Y )2 ≤ χ2(pS,Y ∥ qS,Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: If both S and Y are infinite alphabets, the lower bound holds vacuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, we assume at least one is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The rest of the proof is in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The IT privacy metrics and maximal correlation are formal measures of the statistical dependence between S and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' They possess desirable properties such as vanishing if and only if S and Y are independent (perfect privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The usage of IT privacy metrics in a privacy configuration is typically to form a loss function along with a utility measure for optimizing a privatization mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' With the availability of several privacy metrics studied in this paper, a natural question arises: which privacy metric should one choose?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' While there does not exist a unified answer as the choice often depends on the problem domain, it is possible to compare these privacy metrics in a universal sense as follows [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We say type A privacy metric is stronger than type B privacy metric if for any valid privacy budget η, there exists η′ such that any S given Y that achieves η′ type A privacy also satisfies η type B privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If two privacy metrics are stronger than each other, we say they are equivalently strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the Pinkster’s inequality [26], we have TV(pS,Y , qS,Y )2 ≤ DKL (pS,Y ∥ qS,Y ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From Jensen’s inequality, we have DKL (pS,Y ∥ qS,Y ) = E � log �pS,Y (S, Y ) qS,Y (S, Y ) �� ≤ log E �pS,Y (S, Y ) qS,Y (S, Y ) � = log(χ2(pS,Y ∥ qS,Y ) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Using Definition 6, χ2-divergence privacy metric is, therefore, stronger than the privacy metrics formed by KL divergence and total variation distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Furthermore, Lemma 5 indicates that χ2-divergence and maximal correlation are equivalently strong if the private variable is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In general, χ2-divergence is the strongest privacy metric amongst the IT privacy metrics referenced in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 16 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Translating to Weak DP In Corollary 1, it has been shown that strong IT privacy metrics imply strong probabilistic IP, and Lemma 1 shows that strong probabilistic IP implies weak DP (when the private variable S has finite support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' By chaining these two results, we immediately obtain a lower bound of weak DP that is guaranteed by the IT privacy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In what follows, we illustrate this lower bound using an example of the Gaussian mechanism of DP [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Consider a private variable S = {s0, s1} and a continuous sanitized variable Y whose distribution is specified by pY |S=s0 = N � µ0, σ2� , pY |S=s1 = N � µ1, σ2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the Gaussian mechanism, S given Y achieves (ϵ, δ)-DP if σ2 = 2(µ0 − µ1)2 ϵ2 log(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='25/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For an illustration, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Gaussian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' When µ0 and µ1 are close, it is almost equally likely for most realizations of Y (except for the tail part) to be generated from S = s0 and S = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Now we fix ϵ and δ for the Gaussian mechanism, and compute the χ2-divergence privacy for S and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note that DP disregards the prior distribution of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The χ2-divergence between two normal distributions can be computed analytically: χ2(pY |S=s0 ∥ pY |S=s1) = exp �(µ0 − µ1)2 σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 PYIS=$0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='35 PYIS=S1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='05 μo ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='μ1 0 4 3 2 1 0 1 2 3 4 517 From pY = pS(s0)pY |S=s0 + pS(s1)pY |S=s1, it can be verified that χ2(pY ∥ pY |S=s0) = pS(s1)2χ2(pY |S=s1 ∥ pY |S=s0), χ2(pY ∥ pY |S=s1) = pS(s0)2χ2(pY |S=s0 ∥ pY |S=s1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Based on the χ2-divergence privacy determined by (ϵ, δ)-DP, we firstly use Corollary 1 to quantify the strong IP, and then apply Lemma 1 to compute the (ϵ′, δ′)-DP bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We compare the derived (ϵ′, δ′)-DP bounds with the baseline (ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 2a and 2b, we set δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='05, respectively, and vary ϵ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2, while fixing pS(s0) = pS(s1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note Theorem 1 indicates that δ′ is a function of ϵ′ for the (ϵ′, δ′)-DP bound, and we can evaluate ϵ′ at any value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Letting ϵ′ be the sum of ϵ and a small positive value, we obtain δ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It can be seen that δ′ decreases as ϵ′ increases, implying that weaker privacy protection always comes with a higher probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The (ϵ′, δ′) bound becomes tighter when (ϵ, δ) is closer to (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 2c, we vary pS to verify its impact on DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The results are consistent with Lemma 1, which states the level of DP under probabilistic IP is related to mins∈S pS(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The bound tends to be looser when the prior of S is unbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' DATA-DRIVEN PRIVACY METRIC In this section, we propose a practical implementation of χ2-divergence based on a variational form and show that the proposed empirical estimate is asymptotically consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This lays a foundation of the data-driven privacy-preserving framework in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' One prominent advantage of an f-divergence privacy metric is that it can be estimated from data without the need to estimate the data distribution, which is particularly useful for high-dimensional and continuous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This stands in striking contrast to DP, which is unmanageable in such cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The variational form views f-divergence from an optimization perspective, for which approximation is feasible by restricting the search function space to be from a parametric family represented by neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In what follows, we review the dual representation of χ2-divergence and propose a tighter and reg- ularized representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let p and q be two probability distributions over Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A common variational formulation of (15) is obtained via the Legendre-Fenchel duality [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The conjugate of the convex function f : [0, ∞) → R in (15) is defined as f∗(w) = sup z∈R+ {zw − f(z)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note f∗∗ = f when f is convex and closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This yields a dual representation of f-divergence [53]: Df(p ∥ q) = sup g∈H {EZ∼p[g(Z)] − EZ∼q[f∗(g(Z))]}, January 23, 2023 DRAFT 18 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 (a) (ϵ′, δ′)-DP bounds with varying ϵ and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 (b) (ϵ′, δ′)-DP bounds with varying ϵ and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='7 (c) (ϵ′, δ′)-DP bounds with varying prior pS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (ϵ′, δ′)-DP bounds derived from the χ2-divergence privacy for (ϵ, δ)-DP Gaussian mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' where H includes all measurable functions from Z to R such that the last expectation term is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In particular, the χ2-divergence admits the following the dual representation [53]: χ2(p ∥ q) = sup g∈H � EZ∼p[g(Z)] − EZ∼q � g(Z) + g(Z)2/4 �� , (19) where the supremum is achieved at g(z) = 2 �p(z) q(z) − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Proposition 1, we present an improved variational form of χ2-divergence [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We note that the optimal g in (19) must satisfy the regularization EZ∼p[g(Z)] = 1, whereas this is not required in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let H = � g : Z → R �� 0 < EZ∼q � g(Z)2� < ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Assume q > 0 almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' χ2- divergence admits the following variational form: χ2(p ∥ q) = sup g∈H (EZ∼p[g(Z)] − EZ∼q[g(Z)])2 EZ∼q[g(Z)2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (20) January 23, 2023 DRAFT 19 Proof: From the Cauchy-Schwarz inequality, we have EZ∼q �� 1 − p(Z) q(Z) � g(Z) �2 ≤ EZ∼q �� 1 − p(Z) q(Z) �2� EZ∼q � g(Z)2� = χ2(p ∥ q)EZ∼q � g(Z)2� , where the inequality becomes equality when g(z) ∝ 1 − p(z) q(z) for z ∈ Z almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Now suppose we are given two sets of samples {wi}m i=1 and {zi}m i=1 drawn independently from p and q, respectively, and we want to estimate the χ2-divergence (20) using these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To ensure computational tractability, we let H = {gφ : φ ∈ Φ} in which gφ is a neural network function parameterized by trainable weights vector φ ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Replacing the expectations in (20) with their respective sample averages, χ2(p ∥ q) can be estimated as ˆχ2 m(p ∥ q) = sup φ∈Φ � 1 m �m i=1 gφ(wi) − 1 m �m i=1 gφ(zi) �2 1 m �m i=1 gφ(zi)2 + λm , (21) where λm → 0 is a regularization term for countering a vanishing denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The convergence of the empirical estimates to their corresponding population statistics with increasing sample size is important to justify the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We show that the estimate (21) converges to (20) in probability (denoted as “ p −→”) if some mild assumptions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The estimate ˆχ2 m(p ∥ q) p −→ χ2(p ∥ q) as m → ∞ if the following conditions hold: (a) There exists φ ∈ Φ such that gφ(z) ∝ 1 − p(z) q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (b) gφ(z) is smooth w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ ∈ Φ and continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (c) gφ(z) ̸= 0 for almost everywhere z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (d) Φ and Y are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: From condition (a), in (20), we can restrict to g = gφ for some φ ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let its objective function be denoted as γ(φ) and let γm(φ) be the objective function of (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It suffices to prove sup φ∈Φ γm(φ) p −→ sup φ∈Φ γ(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (22) From the generic uniform convergence theorem [55, Theorem 1], (22) is ensured by the following conditions: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Φ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' γm(φ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' −→ γ(φ) for all φ ∈ Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' γm(φ) is stochastically equicontinuous for all m ≥ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', for any ϵ > 0, there exists σ > 0 such that lim m→∞ P � sup ∥φ−φ′∥<δ ��γm(φ) − γm(φ′) �� > η � (23) January 23, 2023 DRAFT 20 Note condition i is given by condition (d) and condition ii follows from the strong law of large numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We only need to prove condition iii, which needs an auxiliary Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If conditions (b), (c) and (d) are satisfied, there exists a sequence of random variables (Bm)m≥1 and a constant b < ∞ such that lim m→∞ P(Bm − b > ϵ) = 0, for any ϵ > 0 and |λm(φ) − λm(φ′)| ≤ Bm∥φ − φ′∥, for all φ, φ′ ∈ Φ, in which ∥·∥ is the Euclidean norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Proof: See Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Applying Lemma 6 to (23) and letting δ = η b + ϵ with ϵ > 0, we have (23) ≤ lim m→∞ P � sup ∥φ−φ′∥<δ Bm ��φ − φ′�� > η � ≤ lim m→∞ P(Bmδ > η) = lim m→∞ P(Bm > b + ϵ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The theorem is now proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Theorem 2, condition (a) is implied by the universal approximation property of neural networks [56] for φ in a sufficiently high dimensional convex set Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The conditions (b)-(d) can be satisfied by choosing proper activation functions for the neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Data-Driven Privacy-Preserving Framework The empirical estimate of the χ2-divergence empowers us to compute the privacy quantity from data without the need to estimate the distribution of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In what follows, we employ the χ2-divergence as a privacy metric and present a data-driven framework for trading off privacy and utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A privacy-preserving framework comprises three components: sanitizer, privacy function and utility function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A sanitizer takes the raw data X as input and produces the sanitized data Y , in an attempt to remove the statistical information about the private variable S from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In practice, a sanitizer can be realized by a noisy transformation: Y = hθ(X, N), (24) January 23, 2023 DRAFT 21 where hθ is a neural network function parameterized by θ, and N is the noise perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A naive sanitizer is a constant function, which, however, deprives Y of any utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It is necessary to reach a compromise between privacy and utility, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', requiring that Y is maximally informative about a utility task while not containing an excessive amount of information about S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To learn the optimal sanitizer parameter θ, we need a privacy function to quantify the information between S and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In this paper, the square root version of χ2-divergence (21) is adopted as the privacy function (taking the square root to counter the vanishing gradient problem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Given a set of samples {si, xi}m i=1 drawn from (S, X), we generate yi = hθ(xi, ni) (with ni being a random perturbation) to obtain DS,Y = {si, yi}m i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Then the privacy function P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ) is formulated as: max φ P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ) := � ˆχ2m(pS,Y ∥ qS,Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note that each yi is parameterized by the trainable parameter θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For a fixed θ, maximizing P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ) over φ yields an estimate of the dependence between S and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' On the other hand, a utility function measures the usefulness of the sanitized variable Y w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' a utility variable U of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We denote the utility function as L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ), in which τ is the trainable parameter of the utility model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For example, L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ) can be the reconstruction loss of X from Y by letting U = X, and τ is the vector of model weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Minimizing L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ) over τ yields the minimum reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' With the privacy and utility functions at hand, optimizing the sanitizer parameter θ can be formulated as an unconstrained optimization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 3): min θ,τ � L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ) + λ max � max φ P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ), √η �� , (25) where η is the privacy budget for χ2-divergence privacy and λ is a constant to reflect the significance of privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The work [57] proposed an alternating algorithm to optimize (25), which is reproduced Sanitizer hθ(X) Utility model L (θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ) Privacy model P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' ϕ) X Y Loss U S λ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The privacy-preserving framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Firstly, we freeze θ and optimize P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ) and L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Then, we fix τ and φ and update θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' These two steps are repeated until an equilibrium is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 22 Algorithm 1 Minibatch stochastic gradient algorithm 1: Initialize θ, φ, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 2: repeat 3: Sample a mini-batch set from a training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 4: Optimize L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ) over τ and optimize P(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ) over φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 5: Optimize (25) to update θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 6: until θ converges It is worth noting that the optimization strategy in Algorithm 1 is analogous to the empirical risk approach [58], [59], where finding the optimal sanitization scheme is formulated as a competing game between a sanitizer and an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We demonstrate in Section V that such approaches are prone to failure as the sanitizer can be fooled by an adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Our framework based on χ2-divergence privacy does not assume that the adversary uses a particular attack model and is thus agnostic to the adversarial attack model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From a theoretical perspective, if the data distribution is known, the χ2-divergence privacy should be satisfied regardless of the attack that the adversary can muster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Since our framework is data-driven with unknown data distribution, we use the estimate of χ2-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' NUMERICAL EXPERIMENTS In this section, we conduct experiments on the proposed privacy-preserving framework in Section IV-A to demonstrate the efficacy of the χ2-divergence privacy metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' After training the privacy-preserving framework, we simulate the worst-case privacy attacks (in which the sanitization scheme is known to the attacker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We train an attack model and evaluate the level of privacy protection by the attacker’s inference loss of the private variable from the sanitized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Privacy-Preserving Hypothesis Testing In this experiment, we let S = {−1, 1} and U = {−1, 1} be two binary hypotheses, which are statistically dependent on a noisy measurement X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The task is to learn the sanitized data Y from X such that the detection error of U is minimized while making it difficult for an unknown attacker to detect S from Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The noisy measurement is generated as X = A � S′2, U′2, S′U ′, S′, U′�⊺ , where A ∈ R5×5 is a randomly generated matrix and S′ ∼ N (S, 1) and U ′ ∼ N (U, 1) are noisy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 23 1) Network architecture: The sanitizer function is Y = hθ(X, N) = X + h′ θ(N), where h′ θ is a multilayer perceptron of 5 layers with LeakyRelu activation and N is a Gaussian white noise as a perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The utility function is exactly the loss of a neural classifier w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' U: L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ) = m � j=1 2 � i=1 f(ui | yj, τ) log pU|X(ui | xj), where f(ui | y, τ) is the output of the neural classifier, with τ denoting the trainable parameter and p(· | y) denotes the one-hot encoding of the class of input xi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', p(ui | xj) = 1 if xj is labeled with class ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The neural classifier is a multilayer perceptron of 5 layers with tanh activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The generating function gφ for the χ2-divergence privacy metric (21) is a multilayer perceptron of 5 layers with ELU activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We draw 4000 samples and apply the Adam optimizer with learning rate 10−4 and batch size 500 to train the sanitizer according to Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 2) Experimental results: To simulate the privacy attack, we train a neural classifier to detect S from Y after obtaining the sanitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We gradually increase the privacy budget η and plot the utility loss on U and the attack loss on S (measured in terms of classification accuracy) in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 4a and 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 4a, S and U are independent with pS,U(s, u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='25 for each u and s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 4b, S and U are correlated with pS,U(1, 1) = pS,U(−1, −1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 and pS,U(−1, 1) = pS,U(1, −1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It can be seen that a higher level of privacy protection is at the cost of less utility when S and U are correlated, while the utility is not affected by increasing privacy when U is independent of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' A diminishing χ2-divergence leads to an increasing classification loss on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This suggests that IT privacy metrics can defend against unknown adversarial attacks as alluded to in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='9 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='9 1 (a) S and U are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='9 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='9 (b) S and U are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The privacy-utility trade-offs for hypothesis testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The percentages shown are classification accuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 24 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Privacy-Preserving Auto-Encoders In this experiment, we impose the χ2-divergence privacy metric on variational auto-encoders (VAE) [60] and our task is to learn latent representations of images that are insensitive to a chosen private attribute associated with the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We compare our method against the generative adversarial privacy (GAP) [59], the variational fair autoencoder (VFAE) [61] and the invariant representation learning (IRL) [62] on the UTKface [63] and CelebA dataset [64] dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' UTKface is a face attribute dataset with annotations of age, gender and ethnicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' CelebA is a large-scale face attributes dataset with more than 200,000 celebrity images, each with 40 binary attribute annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We choose the gender attribute as the private variable for UTKface and the smiling attribute as the private variable for CelebA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 1) Preliminaries: Given a high-dimensional input variable X, a VAE learns a continuous latent variable Y of the input X = x through a reparameterization of the variational lower-bound of log pX(x): L(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' θ, τ) = E � log pX|Y (x | Y ) � − DKL � qY |X(· | x) ∥ pY � , (26) where qY |X is the variational encoder (parameterized by θ) that approximates the intractable posterior distribution and pX|Y is the decoder (parameterized by τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In this case, the encoder is equivalent to the notion of sanitizer, the utility is the reconstruction loss (U = X), and the latent variable Y is the sanitized data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For tractability, it is assumed that Y ∼ N (0, I) and qY |X = N (µ(X), diag(σ(X))) , pX|Y = N (ν(Y ), I) , in which µ(·) and σ(·) are neural network functions with their collective trainable weights denoted by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The function ν(·) is a neural network function with trainable weights denoted by τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Given a training set {xi}m i=1, the utility function can be written as L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' τ) = − m � i=1 L(xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' θ, τ), which is to be minimized over θ and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Following the framework (25), the χ2-divergence privacy metric is used for encouraging the disentanglement of S and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The original VAE serves as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The GAP framework differs from our χ2-divergence method (25) in that GAP quantifies privacy using the empirical risk of an adversary model [59] instead of an agnostic privacy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The VFAE and IRL, which are variants of VAEs, aim to factor out a sensitive variation from the latent variable and are thus on a comparable basis with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In contrast to our method and GAP, the encoders of the VFAE and IRL (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', pY |S,X(Y | S, X)) take January 23, 2023 DRAFT 25 an additional input of the private attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, the sanitizer (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=', the encoder) needs to know the label of S for X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' To penalize privacy leakage, the VFAE uses the maximum mean discrepancy between pY |S(· | si) and pY |S(· | sj) for si ̸= sj, while the IRL uses the pairwise KL divergences DKL � pY |S,X(· | si, xi) ∥ pY |S,X(· | sj, xj) � for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For detailed VFAE and IRL frameworks, we refer readers to [61] and [62], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The privacy function is multiplied by a constant λ (similar to λ in (25)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 2) Experimental Setup: The VAE encoder networks µ(·) and log σ(·) share 6 down-sampling ResNet blocks [65] followed by two separate dense layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The VAE decoder network ν(·) is made of a dense layer and 6 up-sampling convolutional layers that recover the input image size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The dimension of the latent variable Y is 4608.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' This network architecture also applies to VFAE and IRL except that an additional channel for feeding S is required at the input of the encoder and decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The generating function gφ for the χ2-divergence is made of 4 MLPs with hidden units (2304, 1152, 576, 1) with Instance Normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The adversary (for GAP) and attack models (for evaluating privacy leakage) are MLPs of 4 layers with hidden units (2304, 1152, 576, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For training, we use the Adam optimizer with 10−4 learning rate and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5 (reps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='99) momentum for running average mean and (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' square).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 3) Experimental Results: The mean square error (MSE) for reconstruction and the attack loss and accuracy for UTKface are shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 A ↓ symbol means a smaller value is better and vice versa for the ↑ symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Samples of the reconstructed images are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We set λ = 20 for GAP and choose √η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='1 and λ = 5 for our method so that it has an attack performance similar to that of GAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the reconstruction MSE, it can be seen that GAP and our method generate a similar utility loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' However, the adversary model in GAP is identical to the attack model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' If we replace the batch normalization with instance normalization for the adversary model in GAP (whose results are shown in GAP-A), the level of privacy protection dropped significantly as indicated by the attack performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, privacy cannot be ensured by the empirical risk if the adversary model in GAP does not match the attack model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The results for CelebA are shown in Table II with samples of reconstructed images displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In this case, we include an additional utility task of classifying gender in the learning architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Retaining the λ and η used for UTKface, our method outperforms the GAP (where the adversary model and attack model are the same) in terms of privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Adversarial training is known to be unstable and the quality of privacy sanitization is determined by the capability of the chosen adversarial 1Abbreviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Prv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' : Private, Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' : Attribute, Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' : Accuracy, Util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' : Utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 26 neural network, which in practice cannot incorporate all possible adversarial strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In contrast, the χ2-divergence privacy metric captures statistical information from data without assuming an adversary model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In both cases, VFAE failed to remove the private attributes while severely distorting the data (leading to a large reconstruction error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We made attempts to improve the VFAE performance by changing λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' However, the privacy protection offered by the VFAE is not controllable by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' IRL with λ = 50 achieves its best privacy protection across different values of λ but is still weaker than our method and GAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The accuracy of classifying the utility variable is better preserved for our method when compared to the VAE baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Results in Section V-A suggest that a utility variable can be preserved almost intact if it is independent of the private variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' TABLE I UTKFACE DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' VAE VFAE IRL GAP GAP-A χ2 Prv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' ↓ 88% 98% 84% 70% 83% 69% Prv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Loss ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='58 Util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' MSE ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='07 TABLE II CELEBFACES DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' VAE VFAE IRL GAP GAP-A χ2 Prv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' ↓ 85% 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='5% 75% 79% 82% 66% Prv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Loss ↑ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='61 Util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' MSE ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='075 Util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' ↑ 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='7% 93% 98% 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8% 99% 98% Util.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Attr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Loss ↓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='06 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' CONCLUSION In this paper, we have made connections between probabilistic IP and weak DP and shown that imposing this privacy notion leads to error lower bounds for detecting and estimating the private variable from the sanitized variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Based on probabilistic IP, we characterized several well-known IT privacy metrics given by f-divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' We argued that χ2-divergence privacy is stronger than TV and KL divergence privacy January 23, 2023 DRAFT 27 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Reconstructed UTKface images from the latent space where gender is the private attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Reconstructed Celebfaces images from the latent space where smiling is the private attribute and gender classification is the utility task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT Raw VAE IRL VFAE GAP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 XRaw VAE IRL VFAE GAP X 228 metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, we used χ2-divergence to develop a data-driven privacy-preserving framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' In this paper, we have not investigated the analytical bounds for privacy-utility trade-offs under χ2-divergence privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' An interesting future work is to consider different utility measures and derive fundamental trade- off bounds if they exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' APPENDIX A PROOF OF THEOREM 1(a) Since Lϵ ∩ Rϵ = ∅, we have TV(pS,Y , qS,Y ) = � S×Y |pS,Y (s, y) − pS(s)pY (y)| ds dy ≥ � Lϵ∪Rϵ |pS,Y (s, y) − pS(s)pY (y)| ds dy ≥ (eϵ − 1) � Lϵ pS,Y (s, y) ds dy + (1 − e−ϵ) � Rϵ pS,Y (s, y) ds dy = (eϵ − 1)P(Lϵ) + (1 − e−ϵ)P(Rϵ) ≥ (1 − e−ϵ)P(Lϵ) + (1 − e−ϵ)P(Rϵ) = (1 − e−ϵ)P(Lϵ ∪ Rϵ), where the last inequality is due to eϵ − 1 ≥ 1 − e−ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Finally, we have TV(pS,Y , qS,Y ) ≤ η =⇒ P(Lϵ ∪ Rϵ) ≤ η 1 − e−ϵ , and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' APPENDIX B PROOF OF THEOREM 1(b) For an arbitrary event A ∈ F, consider a channel that produces a Bernoulli random variable W based on the following law: pW|S,Y (1 | s, y) = 1 if S−1(s) ∩ Y −1(y) ∩ A ̸= ∅ and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Then the distribution of W, when (S, Y ) is generated by pS,Y , is pW (1) = p, where p = � A pS,Y (s, y) ds dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' And the distribution of W, when (S, Y ) is generated by qS,Y , is qW (1) = q, where q = � A qS,Y (s, y) ds dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 29 From the data processing inequality, we have DKL (pS,Y ∥ qS,Y ) ≥ DKL (pW ∥ qW ) = p log p q + (1 − p) log 1 − p 1 − q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (27) Let γ = p q and the right-hand side of (27) can be written as f(p, γ) = log γ + (1 − p) log 1 − p γ − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The partial derivatives of f(p, γ) are ∂f(p, γ) ∂p = 1 − γ γ − p − log 1 − p γ − p ≥ 0, ∂f(p, γ) ∂γ = p(γ − 1) γ(γ − p) � � � ≤ 0 if γ < 1, ≥ 0 otherwise, where the inequalities are due to γ = p q > p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, it can be concluded that For any fixed γ > 0, f(p, γ) is non-decreasing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' p ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' For any fixed p ∈ [0, 1], f(p, γ) is non-increasing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' γ < 1, and non-decreasing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' γ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Now letting A = Lϵ, we have γ ≤ e−ϵ and p = P(Lϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the claim assumption and (27), we have f(p, γ) ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Consequently, we obtain P(Lϵ) = p ≤ sup � p′ ∈ [0, 1] : f(p′, e−ϵ) ≤ η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' On the other hand, letting A = Rϵ, we have γ ≥ eϵ and p = P(Rϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Similarly, we must have P(Rϵ) = p ≤ sup � p′ ∈ [0, 1] : f(p′, eϵ) ≤ η � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The proof is completed by noting that P(Lϵ ∪ Rϵ) = P(Lϵ) + P(Rϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' APPENDIX C PROOF OF THEOREM 1(c) The proof exploits the geometric property of χ2-divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let A ∈ F be an arbitrary event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From Sedrakyan’s inequality (which is a direct consequence of the Cauchy-Schwarz inequality), we have χ2(pS,Y ∥ qS,Y ) = � A∪Ac pS,Y (s, y)2 pS(s)pY (y) ds dy − 1 ≥ p2 q + (1 − p)2 1 − q − 1, (28) January 23, 2023 DRAFT 30 where p = � A pS,Y (s, y) ds dy = P(A), q = � A pS(s)pY (y) ds dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let γ = p q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Substituting q = p γ into (28) and from the assumption χ2(pS,Y ∥ qS,Y ) ≤ η, we obtain pγ + (1 − p)2 1 − p/γ − 1 ≤ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Rearranging the above inequality, we have P(A) = p ≤ g(γ, η) (29) where g(γ, η) = γη (γ − 1)2 + η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The following properties about g(γ, η) can be verified by checking its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (For the reader’s convenience, we visualize g(γ, η) by plotting its numerator and denominator as functions of γ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=') For a fixed η > 0, g(γ, η) is monotonically increasing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' γ2 ∈ [0, 1 + η] and monotonically decreasing w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' γ2 ∈ (1 + η, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' g(γ, η) ≥ 1 for γ ∈ [1, 1 + η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Now we substitute Lϵ and Rϵ for A in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It can be verified that γ ≤ e−ϵ when A = Lϵ, and γ ≥ eϵ when A = Rϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the monotonicity property of g(γ, η), we have P(Lϵ) ≤ e−ϵη (e−ϵ − 1)2 + η, ∀ ϵ > 0, P(Rϵ) ≤ eϵη (eϵ − 1)2 + η, ∀ ϵ > log(1 + η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Note that the second inequality above also holds true for ϵ ∈ [0, log(1 + η)] because P(Rϵ) ≤ 1 while its right-hand side is greater than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The proof for claim (c) is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' APPENDIX D PROOF OF LEMMA 4 Let γ(A) = � A (pS,Y (s, y) − pS(s)pY (y)) ds dy, January 23, 2023 DRAFT 31 0 1 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The denominator and numerator of g(γ, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' and denote Ψ = � ω : e−ϵ ≤ d(S(ω), Y (ω)) ≤ 1 � , Γ = {ω : 1 ≤ d(S(ω), Y (ω)) ≤ eϵ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Firstly, we have γ(Γ) − γ(Ψ) ≤ (1 − e−ϵ)P(Γ) + (eϵ − 1)P(Ψ) ≤ (eϵ − 1) (P(Γ) + P(Ψ)) ≤ eϵ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (30) Moreover, we have γ(Rϵ) ≤ � Rϵ pS,Y ds dy ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From γ(Lϵ) + γ(Rϵ) + γ(Γ) + γ(Ψ) = γ(Ω) = 0 and (30), we obtain −γ(Lϵ) = γ(Γ) + γ(Ψ) + γ(Rϵ) ≤ γ(Γ) − γ(Ψ) + δ ≤ eϵ − 1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Finally, we obtain TV(pS,Y , qS,Y ) = γ(Γ) − γ(Ψ) + γ(Rϵ) − γ(Lϵ) ≤ 2(eϵ − 1 + δ), and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' January 23, 2023 DRAFT 32 APPENDIX E PROOF OF LEMMA 5 Let L2(pS) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' L2(pY )) be the space of all real-valued functions of S (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Y ) with finite variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Define a linear operator T : L2(pY ) → L2(pS) such that for f ∈ L2(pY ), [Tf](s) = E[f(Y ) | S = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' It is associated with an adjoint operator [T ∗g](y) = E[g(S) | Y = y] for g ∈ L2(pS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let (σi)i≥1 be a sequence of singular values of the operator T in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the definition of maximal correlation, it is well-known that σ1 = 1 and σ2 = ρm(S, Y ) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Moreover, we have ∥T∥2 HS = � i≥1 σ2 i , (31) where ∥·∥2 HS is the Hilbert-Schmidt norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' On the other hand, we can rewrite T as [Tf](s) = � Y f(y)k(s, y)pY (y) dy, in which k(s, y) : S × Y → R is a kernel: k(s, y) = pS,Y (s, y) pS(s)pY (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From [67, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='8], we have ∥T∥2 HS = � Y � S k(s, y)2pS(s)pY (y) ds dy = χ2(pS,Y ∥ qS,Y ) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' (32) The proof is completed by combining (31) and (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' APPENDIX F PROOF OF LEMMA 6 Let vm(φ) be the numerator of γm(φ) and dm(φ) = 1 m �m i=1 gφ(zi)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The gradient of γm(φ) can then be written as ∇γm(φ) = km(φ) (dm(φ) + λm)2 with km(φ) = dm(φ)∇vm(φ) − vm(φ)∇dm(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' By assumption, gφ(x) is a smooth function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' φ and continuous w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, ∂gφ(x) ∂φ is also a continuous function, which is thus uniformly bounded by some constant due to the compactness of Φ January 23, 2023 DRAFT 33 and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Therefore, for any m > 0, km(φ) (consisting of the mean of bounded functions) is bounded by a constant vector c1 with c < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Let Bm = c (minφ∈Φ dm(φ) + λm)2 , which yields ∇γm(φ) ≤ Bm1 followed by |γm(φ) − γm(φ′)| ≤ Bm1⊺|φ − φ′| ≤ Bm∥φ − φ′∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' From the uniform law of large numbers [68], we have min φ∈Φ dm(φ) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' −→ a := min φ∈Φ E[dm(φ)], where a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' As a result, we have lim m→∞ P � Bm > c a2 � = lim m→∞ P � min φ∈Φ dm(φ) < a − λm � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' The proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' REFERENCES [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytFAT4oBgHgl3EQfBBwz/content/2301.08401v1.pdf'} +page_content=' Fire, R.' metadata={'source': 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